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CMC-trinoids with properly embedded annular ends [Elektronische Ressource] / Philipp Lang

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TECHNISCHE UNIVERSITAT MUNCHENZentrum MathematikCMC-Trinoids with Properly Embedded Annular EndsPhilipp LangVollst andiger Abdruck der von der Fakult at fur Mathematik der Technischen Universit at Munc hen zurErlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. J. ScheurlePrufer der Dissertation:1. Hon.-Prof. Dr. J. Dorfmeister2. Univ.-Prof. Dr. T. N. Ho mann3. Dr. F. Pedit, Eberhard Karls Universit at TubingenDie Dissertation wurde am 28.01.2010 bei der Technischen Universit at Munc hen eingereicht und durchdie Fakult at fur Mathematik am 08.06.2010 angenommen.AbstractWe consider CMC-trinoids in Euclidian three-space with properly embedded annular ends. Startingwith a holomorphic potential ~ and a special solution to the di erential equation d = ~ , we char-acterize all solutions to this di erential equation which produce CMC-trinoids with properly embeddedannular ends via the loop group method. Moreover, we give a classi cation of CMC-trinoids with properlyembedded annular ends with respect to their symmetry properties in terms of the monodromy matricesof the solution associated with the trinoid ends.3Contents1 Introduction 62 Outline of the loop group method 112.1 Loop Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Published 01 January 2010
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TECHNISCHEUNIVERSIT¨ATM¨UNCHEN
MathematiktrumZen

CMC-TrinoidswithProperlyEmbeddedAnnularEnds

angLPhilipp

Vollst¨andigerAbdruckdervonderFakult¨atf¨urMathematikderTechnischenUniversit¨atM¨unchenzur
ErlangungdesakademischenGradeseines

DoktorsderNaturwissenschaften(Dr.rer.nat.)

Dissertation.genehmigten

Vorsitzender:Univ.-Prof.Dr.J.Scheurle
Pr¨uferderDissertation:
DorfmeisterJ.Dr.Hon.-Prof.1.2.Univ.-Prof.Dr.T.N.Hoffmann
3.Univ.-Prof.Dr.F.Pedit,EberhardKarlsUniversit¨atT¨ubingen

DieDissertationwurdeam28.01.2010beiderTechnischenUniversit¨atM¨uncheneingereichtunddurch
dieFakult¨atf¨urMathematikam08.06.2010angenommen.

Abstract

WeconsiderCMC-trinoidsinEuclidianthree-spacewithproperlyembeddedannularends.Starting

withaholomorphicpotentialη˜andaspecialsolutionΨtothedifferentialequationdΨ=Ψη˜,wechar-

acterizeallsolutionstothisdifferentialequationwhichproduceCMC-trinoidswithproperlyembedded

annularendsviatheloopgroupmethod.Moreover,wegiveaclassificationofCMC-trinoidswithproperly

embeddedannularendswithrespecttotheirsymmetrypropertiesintermsofthemonodromymatrices

ofthesolutionΨassociatedwiththetrinoidends.

3

tstenCon6ductiontroIn12Outlineoftheloopgroupmethod11
2.1LoopGroups...........................................11
2.2Iwasawadecomposition.....................................11
2.3Holomorphicpotentials.....................................11
2.4Theloopgroupmethod.....................................12
2.5Monodromy............................................14
2.6Delaunaysurfaces.........................................15
19rinoidsT33.1TrinoidsonthedomainM=C\{0,1}.............................19
˜3.2TheuniversalcoverMofM...................................20
˜3.3ThefundamentalgroupΓofManditsmonodromyactiononM..............23
33.4Thesu(2)modelofR......................................26
3.5Thetrinoidpotential.......................................27
3.6Thestandardizedtrinoidpotential...............................32
3.7TheFuchsianODE........................................34
3.8SolvingdΦ=Φη.........................................36
3.9Simultaneousunitarizationofthemonodromymatrices....................43
52symmetriesrinoidT44.1Definitions.............................................52
4.2Theextendedframe.......................................52
4.3Trinoidswithproperlyembeddedannularends........................55
4.4Theextendedframesymmetrytransformations........................56
4.5Theextendedframemonodromyrelations...........................68
4.6Trinoidsymmetries........................................69
5Rotationalsymmetrywithrespecttothetrinoidnormal80
5.1Definition.............................................80
5.2Implicationsofrotationalsymmetrywithrespecttothetrinoidnormal...........80
5.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerota-
tionallysymmetricwithrespecttothetrinoidnormal....................83
5.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerotationallysym-
metricwithrespecttothetrinoidnormal...........................87
2005.5Solvingζζ=4sin(πµ)−1..................................93
6Rotationalsymmetrywithrespecttoatrinoidaxis104
6.1Definition.............................................104
6.2Implicationsofrotationalsymmetrywithrespecttoatrinoidaxis.............104
6.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerota-
tionallysymmetricwithrespecttoatrinoidaxis.......................106
6.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerotationallysym-
metricwithrespecttoatrinoidaxis..............................111
7Reflectionalsymmetrywithrespecttothetrinoidplane121
7.1Definition.............................................121
7.2Implicationsofreflectionalsymmetrywithrespecttothetrinoidplane...........121
7.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharereflec-
tionallysymmetricwithrespecttothetrinoidplane.....................122
7.4Normalizedtrinoidswithproperlyembeddedannularends,whicharereflectionallysym-
metricwithrespecttothetrinoidplane............................125
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8Reflectionalsymmetrywithrespecttoatrinoidnormalplane131
8.1Definition.............................................131
8.2Implicationsofreflectionalsymmetrywithrespecttoatrinoidnormalplane........131
8.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharereflec-
tionallysymmetricwithrespecttoatrinoidnormalplane..................134
8.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerefletionallysym-
metricwithrespecttoatrinoidnormalplane.........................138
9Rotoreflectionalsymmetrywithrespecttothetrinoidnormal150
9.1Definition.............................................150
9.2Implicationsofrotoreflectionalsymmetrywithrespecttothetrinoidnormal........150
9.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerotore-
flectionallysymmetricwithrespecttothetrinoidnormal...................153
AAppendix:BasicTopology157
A.1Topologicalspaces,continuousmappingsandpaths......................157
A.2Thefundamentalgroup.....................................157
A.3Theautomorphismgroup....................................158
A.4Themonodromyactionofthefundamentalgroup.......................159
BAppendix:Thefunctionµ=XX164
jjjCAppendix:Proofoflemma3.37168
DAppendix:OnthenecessityoftheunitarizingmatrixT172
EAppendix:Amendmentstotheproofoftheorem3.53173
FAppendix:Proofofremark3.55177
GAppendix:Proofofremark3.56178
HAppendix:Proofoftheorem5.16180
IAppendix:Proofsoflemma5.21andlemma5.24183
186References

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ductiontroIn1AmongthesurfacesofconstantmeancurvatureH=0,CMC-surfacesforshort,onlyafewsubclasses
havDelaunaebeenysurfaces.classified.TheyThewerefirstfoundoneshavalmostebee200nytheearsagosurfaces[7],ofandrevareolutionstillofinamongterest,thesinceevCMC-surfaces,eryproperlythe
embMoreeddedanngenerallyular,endofaCMC-immersionsCMC-surfaceofisroundasympcylinderstoticallyintoaR3Delaunaareyfairlysurfacewellu[25].nderstood.Theclassof
CMC-torihasbeeninvestigatedextensivelyusingdifferentmathematicaltechniques[31],[3],[23]andis
clearlysofarthebestinvestigatedoneamongallCMC-immersions.
groupsAllthearepsurfaceerhapsthclassesosewhimenchtionedaresofreefarandhavhaeveanabonlyeliantwofundamengenerators.talThgroup.usitTheseemstosimplestbenon-abparticularlyelian
impsphereortanTtintotoR3.understandtheCMC-trinoids,i.e.CMC-immersionsofthethrice-puncturedRiemannian
3AmongtheCMC-trinoidsT3→R3clearlytheembeddingsareofparticularinterest.Itseemsto
beBraucdifficultkmann,toKusnerclassifyandthisSuclassllivanofhaveCMC-immersions.classifiedtheHowAlexandroever,vinemabbeddedeautifulpieceCMC-surfacesofwork,T3→Große-R3
[21].In[27]itwasshown,however,thatthereareCMC-trinoidsT3→R3,whichhaveproperlyembedded
annularendsbutarenotAlexandrovembedded.Furtherexamplesofsuchsurfaceshavebeengivenin
[17]CMC-trinoidsusingthelowithoppropgrouperlymethoembdedded[15]forannaularcertainendsclassofencompassesstartingthepotenclasstials.oftheNaturally(globally),thepropclasserlyof
embeddedtrinoids.Inthissense,theinvestigationofCMC-trinoidswithproperlyembeddedannular
endsseemstobeanaturalnextstepfortheunderstandingofallCMC-trinoids.
In[8]itisshownthatallCMC-trinoidsT3→R3withproperlyembeddedannularendscanbe
aobtainedclassificationviatheofloallopgroupCMC-trinoidsmethodwhicfromhcanthebpeotenobtainedtialsofvia[17].theloBasedopongroupthismethoresult,dthisfromthethesispprootenvidestials
of[17],andthusinparticularaclassificationofallCMC-trinoidswithproperlyembeddedannularends,
intermsofthemonodromymatricesassociatedwiththetrinoidends.
WegiveallpossibletriplesofmonodromymatricesassociatedwiththeendsofaCMC-trinoidT3→R3
awhicgivhencanCMbeC-triobtainednoidwithfromproptheperlyotenembtialsedofded[17].annularMoreovendser,weunderinvestigateEuclideanthepmotionsossibleinR3symmetriesandchar-of
acterizethesesymmetriesintermsofthecorrespondingmonodromymatrices.I.e.,westatenecessary
andsufficientconditionsonthemonodromymatricesofagivenCMC-trinoidT3→R3withproperly
embeddedannularends,3suchthatthe(imageofthe)givenCMC-trinoidisinvariantunderaspecific
.RinmotionEuclideanInholomorphicsectionp2otenwetials.reviewAtheloholomorphicopgrouppotenmethotialdη˜isfromasl(2[15],Cfor)-valuedconstructingdifferentialCMC-immersionsone-form,whichfromis
definedontheuniversalcoverM˜ofaRiemann˜surfaceM∗.Furthermore,η˜involvesaloopparameter
λanddependsholomorphicallyonbothz∈Mandλ∈C.Givenaholomorphicpotentialη˜,thefirst
stepmappingoftheΨloonopM˜,groupsatisfyimethongdsomeconsistsinitialinsolvingconditiontheΨ(z∗)differen=Ψ0tial.ΨalsoequationdepdΨends=onΨη˜λforandatheSL(2,formC)-vofaluedthis
dependenceisdeterminedbytheinitialconditionΨ0.AssumingthatΨ0(andthusΨ)isdefinedforallλ
fromsomer-circleC(r),0<r≤1,onecanproceedwiththesecondstepoftheloopgroupmethod.This
involves(foreachz0∈M˜)anr-Iwasawadecomposition(rof)theλ-dependentloopΨ(z0):C(r)→SL(2,C),
opi.e.ena(pannointuluswise)r<|λ|factorization<1andofisΨintounitaryaloonopFtheonunitC,circlewhicS1h,canandbaeloopextendedB+onC(r),holomorphicallywhichcantobthee
extendedholomorphicallyrtothedisc|λ|<r,Ψ=FB+.ThefactorFproducesinthethirdandfinalstep
ofthelo˜opgroupmethod,byevaluatingthesocalledSym-Bobenkoformulaforλ=1,aCMC-immersion
ψonM.ψ“descends”toaCMC-immersionφonM˜ifandonlyifthemonodromymatricesM(γ˜,λ)of
Ψ2.11).assoInciatedparticular,withtheallcovmonoeringdromymtransformationsatricesM(γ˜γ˜,ofλ)MofΨsatisfyneedtocertainbeunitary“closingforλconditions”fromS1.(cf.1theorem
AchangeintheinitialconditionΨ0correspondstomodifyingthesolutionΨbyadressingmatrix
T=T(λ)toobtainanewsolutionΨˆ=TΨ.ThisnewsolutionΨˆproducesanewCMC-immersionψˆonM˜
viatheloopgroupmethod.Sincethereisˆ(ingeneral)noobviousrelationbetweencorrespondingfactors
inofthedressingIwasawtheadecompsolutionΨositionsbyTofinΨtoaandnewΨ,soluresptionectivΨˆelyon,ittheis(inlevelofgeneral)thenotcorresppossibleondingtocontrolCMC-immersionstheeffect
how1ever,Actuallyif,notthestatedSym-Bobotherwise,enkoweformulaconsiderprotheducescahoiceλ=1.CMC-immersionψλonM˜foreachλ∈S1.Throughoutthisthesis

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ψandψˆ.However,thechangeinthemonodromymatricesiswellunderstood:Ifthemonodromymatrices
ofΨaregivenbyM(γ˜,λ),themonodromymatricesofΨˆ=TΨaregivenbyMˆ(γ˜,λ)=TM(γ˜,λ)T−1.
Ho(Notewever,thatforthistrinoids,onlyholdsthisifisthealwaysoriginalthecase.)CMC-immersionBasedonψthisassofact,ciatedonewithderivesΨhastheanfolloumwingbilicp“recipointe”[14].for
theconstructionofCMC-immersionsonaRiemannsurfaceM:
˜1.tialGivenaequationholomorphicdΨ=Ψη˜pforotenatialη˜solution,whicΨhiswithdefinedsomeoninitialthevunivalueΨ(ersalz∗)cov=erΨM0.ofMDenote,solvtheethemonodifferen-dromy
matricesofΨunderthecoveringtransformationsγ˜ofM˜byM(γ˜,λ).
ˆ2.TM(Determineγ˜,λ)T−a1satisfydressingthematrixcondiTtions=Tof(λ),theoremsuchthat2.11.theInp“dressed”articular,onemononeedsdromtoyensumatricesrethatM(Mˆγ˜,(λ˜γ),λ=)
isunitaryforλfromS1andforallcoveringtransformationsγ˜.Then,the“dressed”solutionΨˆ=TΨ
producesviatheloopgroupmethodforλ=1aCMC-immersionψonM˜,whichdescendstoa
.MonφCMC-immersionSection2isarrangedasfollows:In2.1,wedefinethedifferentloopgroupswhichserveasdomains
ofdefinitionforthefactorsoccuringintheIwasawadecomposition,whichispresentedin2.2.Section
2.3introducesholomorphicpotentials,whichformtheinitialdatafortheloopgroupmethodexplicated
in2.4.constructionIn2.5,ofweintroCMC-immersionsducetheonmonoa(notdromynecessarilymatricessiandmplycitefromconnected)[11]ourRiemannbasicsurtheoremfaceM.2.11Infor2.6,wthee
applysurfaces,theloopCMC-surfacesgroupmeofthorevdandolutionintheoremR32.11toparametrizedexplicitlybyctheonstrucpuncturedtthealreadycomplexmenplanetionedC∗=CDelaun\{0a}y.
Insection3,weapplytheframeworkbuiltupinsection2toCMC-trinoids,i.e.toCMC-immersions
inR3parametrizedbyT3,thetwo-sphereS2={x∈R3;|x|=1}withthreepointsremoved.Asfitto
ourpurposes,weidentifyT3viastereographicprojectionwiththetwice-puncturedcomplexplane(or,
equivalently,thethrice-puncturedextendedcomplexplaneCˆ=C∪{∞}),M=C\{0,1}=Cˆ\{0,1,∞},
andactuallyinterpretaCMC-trinoidasaCMC-immersionφ:M→R3.TheuniversalcoverofMis
givenbyM˜=H,theupperhalf-planeinC.ˆ
(annular)Thethreeendspofointhetszj=surfacej,jφ(=M0).,1,While∞,wremoevalloedwfromarbitraryCareself-insingularitiestersectionsofofφtheandthsurfaceusainducewayfromthree
itssmallends,wepuncturedrequireneightheborhoendsodtobearoundpropeacerlyhembesingularitdded.ytheMorepreciseimmersionly,φweisarequireproperthatonCMC-emabsufficienedding.tly
Therefore,accordingto[25],theendsasymptoticallyshowthebehaviourof(unduloidal)Delaunaysur-
whicfaces.hnearBasedeaconhthissingularitfactyandzjtakfolloethewingform[17],ofweain“ptroderturbuceaed”familyDelaunaofypotenholomorphictialηˆj.potentialsηonM,
By[17],thecorrespondingholomorphicpotentialη˜=π∗ηonM˜,obtainedfromηviapullbackbythe
universalcoveringmapπ,yieldsviatheloopgroupmethodaCMC-trinoidwiththreeproperlyembedded
annularendsatzj,j=0,1,∞,showingtheasymptoticbehaviouroftherespectiveDelaunaysurface
prosolutionducedΨbytothe(thepullbdifferenacktialof)ηˆjequationviathedΨlo=opΨη˜,grouptheremethoexistsd.aMoredressingpreciselymatrix(cf.T=Ttheorem(λ)3.14),generatingforaagivnewen
solutionTΨ,whichproducesviatheloopgroupmethodaCMC-trinoidwiththreeproperlyembedded
annularends.Notethatitisclaimedin[8]thatalltrinoidswithproperlyembeddedannularendscan
beconstructedfromourpotentials.
ΦtoThethemaindifferenfeaturestialofequationsectiondΦ3=areΦηthe.Tofolloacwing:hieveFirst,this,wweeuseexplicitlythewellcomputeknownafact“startinthatg”thesolutiongiven
differentialequationcanberetracedtoa(scalar)hypergeometricdifferentialequation,whosesolutions
arewellknownandcanbeexpressedintermsofhypergeometricfunctions.Moreover,weknowby[17],
thatneareachsingularityzjthedifferentialequationdΦˆj=ΦˆjηˆjpossessesasolutionΦˆjofaspecial
ˆtheseform,twocalledfacts,anΦcanEDP-solutionbeexp.Φlicitlyislocallycomputedaround(cf.zjlemmarelatedto3.37).ΦjbyMoreoavgaugeer,thematrixmonoV+,jdrom.yCommatricesbining
M(γ˜,λ)ofthecorrespondingpullbackΨ=π∗ΦsolvingdΨ=Ψη˜aredetermined.
Thesecondfeatureofsection3consistsinthecharacterizationofallpossibledressingmatricesT
renderingΨintoanewsolutionTΨtodΨ=Ψη˜,whichproducesaCMC-trinoidviatheloopgroup
ˆTMmetho(γ˜,d.λ)AsT−1indicatedsatisfybtheefore,thconditionsisisacofhievedtheoremby2.11.ensuringInthatthisthecontext,dressedwemonorestrictdromyourmatricesconsiderationM(γ˜,λs)to=
thethreemonodromymatricesMj(λ):=M(γ˜j,λ),j=0,1,∞,correspondingtothethreecovering
transformationsγ˜j,j=0,1,∞,ofM˜,whichrepresentthreesimpleloopsinM,surroundingexactlyonce
thesingularityzj(counter-clockwise)withoutenclosingtheothertwosingularities,respectively.Since

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γ˜j,j=0,1,∞,generatethegroupofcoveringtransformationsγ˜ofM˜(actually,eventwoofthemdo),
amatrixTdressesthestartingsolutionΨintoanewsolutionTΨtodΨ=Ψη˜,whichproducesaCMC-
trinoidviatheloopgroupmethod,ifandonlyifthethreedressedmonodromymatricesMˆj=TMjT−1
satisfytheconditionsoftheorem2.11.Explicitly,TΨproducesaCMC-trinoid,ifandonlyifthethree
monodromymatricesMˆjareoftheform
Mˆj=−cos(2πµj)0110+isin(2πµj)qpjj−qpjj
withλ-dependentfunctionspj,pj,qjandqjsatisfyinganumberofconditions,whicharesummarizedin
3.59.theoremSection3isorganizedasfollows:Insection3.1,weintroduceCMC-trinoidsasCMC-immersionsof
thetwice-puncturedcomplexplaneM=C\{0,1}.Section3.2presentstheuniversalcoverM˜=HofM
anddefinesthecorrespondingcovering˜mapπ:M˜→M.Insection3.3,westudythemonodromyaction
ofthefundamentalgroupofMonM,whichconstitutesthebasisforthedefinitionofthemonodromy
matrices3later.Sincetheloopgroupmetho3dactuallyproducesCMC-immersionsintosu(2)ratherthan
tointotheR,wedefinitionneedtoofidenthetifytrinoidsu(2)pandotentRials.ηThis(onisMdone)andinη˜section(onM˜3.4.).OurSections3.5startingand3.6solutionareΦtodedicatedthe
differentialequationdΦ=Φη,alongwiththecorrespondingstartingsolutionΨtodΨ=Ψη˜,isexplicitly
computedinsections3.7and3.8.Finally,section3.9dealswiththepossibledressingmatricesT=T(λ)
transformingSection4ΨopinenstoathenewsecondsolutionpartTofΨ,thiswhichthesis:givesaHavingCMC-trinoidsofarviathedeterminedloopallgroupsolutionsmethod.Ψtothe
todifferenasktotialwhatequationextentdΨ=Ψgeometricalη˜,whichpropgenerateertiesofaCMC-tCMC-trinoidrinoidsviaφtheproloopducedgrcanoupbemethoreadd,offitseethemsrespnaturalective
generatingsolutionΨ.Inparticular,onecanaskhowsymmetrypropertiesof(theimageφ(M)of)a
giventrinoidφ:M→R3showinthecorrespondinggeneratingsolutionΨ.Inthisthesis,wegivea
comprehensiveanswertothisquestion3inthecaseofCMC-trinoidswithproperlyembeddedannularends.
φasGivanenaEuclideanCMC-trinmotionoidφT:,Mi.e.→anRwithorthogonalproperlyemtransformationbeddedfolloannwularedbyends,awetranslationdefineaonR3symmetry,whicofh
preservestheimageofφinR3:T(φ(M))=φ(M).Denoting,asbefore,byπtheuniversalcovering
M˜→M,TalsodefinesasymmetryoftheCMC-immersionψ:=φ◦π:M˜→R3,i.e.T(ψ(M˜))=ψ(M˜).
Apriori,duetotheIwasawadecompositionofΨinthesecondstepoftheloopgroupmethod,itis
difficulttoretraceanysymmetrypropertiesofφ(resp.ψ)backtothecorrespondinggeneratingsolution
Ψfromtoψthe,whicdifferenhtialreturnsψequationbydΨinsertion=Ψη˜in.totheThoughitSym-Bobisposenkosibleformtoulaforreconstructλ=1the[10],extendethedfrsubsequenameFt
stepofreconstructingΨfromF(i.e.,ofreversingtheIwasawadecompositionΨ=FB+)ishighlynon-
trivial.Nevertheless,weobservethatthemonodromymatricesMjofΨwithrespecttothecovering
transformationsγ˜jalsooccuras“monodromymatrices”ofF:
F(γ˜j(z),λ)=±MˆjF(z,λ)kj(z),
wherekdependsdirectlyonγ˜.
Thisjobservationformsthejbasisforourapproachofretracinganysymmetrypropertiesofφ(resp.
ψ)tothelevelofthegeneratingsolutionΨ:Wetranslatethegivensymmetrypropertiesofφ(resp.ψ)
to“monothelevdromelyofthematrices”extendedofFframe(sectionsFof5ψto9).(sectionInthis4)wanday,wededuceactuallycertainobtain“symmetry“symmetryrestrictions”restrictions”ontheon
themonodromymatricesofΨ,whichinturndeterminesthesolutionstothedifferentialequationdΨ=Ψη˜
generatingtrinoidswithproperlyembeddedannularendswiththerespectivesymmetryproperties.
Indetail,weproceedasfollows:Letφ:M→R3beaCMC-trinoidwithproperlyembeddedannular
ends,whichissymmetric3withrespecttotheEuclideanmotionT.Denotebyψthecorresponding
orientationCMC-immersionorrevM˜erses→Rorien,whitationchisonalsoR3,itcansymmetricbeshowithwnre(cf.specttotheoremT.4.9,DepbasedendingononresultswhetherfromT[12])preservthates
thereexistapairofbiholomorphic(resp.bi-antiholomorphic)mappingsγ:M→Mandγ˜:M˜→M˜
translatingthesymmetryTtoMandM˜,respectively:
T◦φ=φ◦γ,(1.0.1)
T◦ψ=ψ◦γ˜.(1.0.2)
Moreoassumptionver,γthatandγ˜φpareossesseslinkedpbyroptheerlyemrelationbeddedπ◦γ˜ann=γular◦π.ends(Notetakesthat,ineffect.)orderBytotheobtainrelationsthisaboresult,ve,thethe

8

symmetrypropertiesofφ(resp.ψ)translatetotheleveloftheassociatedextendedframeasfollows(cf.
theorem4.17):IfTpreservesorientation,then
F(γ˜(z),λ)=Mγ˜(λ)F(z,λ)kT,γ˜(z),(1.0.3)
where1kT,γ˜dependsonγ˜andMγ˜(λ)denotesaλ-dependentmatrix,whichisunitaryontheunitcircle,
λ∈S.IfTreversesorientation,then
F(γ˜(z),λ−1)=Mγ˜(λ)F(z,λ)kT,γ˜(z),(1.0.4)
unitwherekcircle,T,γ˜,λas∈abSo1v.e,Independsparticular,onγ˜inandtheMγ˜(caseλ)Tdenotes=I,atheλ-depidentitendenytmappingmatrix,onwhicRh3isandunitarynaturallyonthea
symmetryofφ(resp.ψ),weobtainthe“monodromyrelations”forFgivenabove,involvingthecovering
transformationsγ˜j,j=0,1,∞,associatedwithT=IinthesensethatI◦ψ=ψ◦γ˜j.
Thebiholomorphic(resp.bi-antiholomorphic)mappingγ:M→Massociatedwiththetrinoid
symmetryTcanbeextendedtoabiholomorphic(resp.bi-antiholomorphic)mappingγextd:Cˆ→Cˆ,
whichnecessarilypermutesthethreepointszj=j∈{0,1,∞}accordingtoapermutationσoftheset
{0,1,∞},i.e.γextd(zj)=zσ(j).Inthisway,however,γextd(andthusγ)iscompletelydeterminedbyσand
canthusbeexplicitlycomputed.Astherearesixpossibilitiesforσ,weobtaintwelvepossibilitiesforγ,six
trinoidbiholomorphicsymmetriesonesT,andsixsixorienbi-antationtiholomorphicpreservingones.onesOneandsixeasilyorieninferstationthatrevetherersingareones.onlytwMoreoelvevper,ossiblethese
canbeexplicitlydeterminedaswell(cf.theorem4.31)andcharacterizedbytheirrespectivepermutation
behaviourconcerningthetrinoidends:Thesixp2ossibleπorientationpreservingtrinoidsymmetriesarethe
Riden−1tityandthemappingthreeI,therotationrsotationRjbRybtheytheangleangleπ,±eac3hpresrotatingervingthethetrinoidtrinoidendsendinattotheeachsinguother,larityitszjinvwhileerse
Sinintercsomehangingplane,thepreseotherrvtwingo.eacThehofsixthepossiblethreeorientrinoidtationends,revtheersingthreetrinoidreflectionssymmetriesSj,eacarhethepreservingreflectionthe
trinoidendatthesingularit−1yzjwhileinterchangingtheothertwo,therotoreflection2SˆcomposedofR
andS,anditsinverseSˆ.
fromOncethegivrelationentheπt◦wγ˜elv=eγp◦π.ossibilities(Noteforthatγ,γ˜isoneonlycancomputedeterminedtheuniquelyassociateduptoleftmappingscompγ˜:ositionM˜→withM˜
acoveringtransformationM˜→M˜.)Thus,weexplicitlyobtainforeachpossibletrinoidsymmetryT
abiholomorphic(resp.bi-antiholomorphic)mappingγ˜T:M˜→M˜,whichweputintorelationwiththe
coforvereacinghsymmetrytransformationstypeinγ˜jsectionscorresp5toonding9.toGenerallythesp“monoeaking,dromythematrices”relationsofbeFtw.eenThisγ˜Tisanddonethecovseparatelyering
transformationsγ˜jtranslatebyuseofthe“monodromyrelations”andthe“symmetryrelations”ofF
givenaboveintorelationsonthe“monodromymatrices”MˆjofF,involvingtherespective“symmetry
monodromymatrix”Mγ˜T.Thelatterrelationstranslatedirectlyintofurtherconstraintsonthemappings
pj,differenqjotialccurringequationinMˆjdΨand=Ψthη˜,us,whicbyhprotheoremduces3.59theintorespectivconstrainetstrinoidonthewithpropgeneratingerlyembsolutioneddedΨanntoularthe
ends,whichissymmetricwithrespecttoT.Theexplicitresultsaregivenintheorems5.9,6.6,7.5,8.6
and9.7,respectively.
Ingeneral,thefurtherconstraintsonthemappingspj,qjoccurringinMˆj,whichareobtainedby
evaluatingthededucedrelationsbetweenthemonodromymatricesMˆjandtherespective“symmetry
monodromymatrix”Mγ˜T,alsointroducemoreparameters,namelythe(λ-dependent)entriesofMγ˜T.
However,thisadditionalfreedomcanbefixedbyanappropriate“normalization”oftheextendedframe
F:First,observethatduringthereconstructionoftheextendedframeFfromψ,Fis“normalized”,
suchthatF(z∗,λ)=Iforallλ∈S1atanarbitrarilychosenbasepointz∗∈M˜.Thisnormalizationof
Fthelocorrespopgroupondstomethoaλd.-depNoendenw,cthoosingrotation(ifpandossible)shiftofforthez∗∈assoM˜aciatedfixedfamilypointψλofoftheF,obtainedbiholomorphicfromF(resp.via
bi-antiholomorphic)mappingγ˜T:M˜→M˜associatedwithaconsideredtrinoidsymmetryT,weinfer
fromthe“symmetryrelation”ofFgivenabovethat
Mγ˜(λ)=(kT,γ˜(z∗))−1,(1.0.5)
i.e.knowwneobtai(sincenkaT,γ˜(z∗“symmetry)is).Thmonous,dromtheyadditionalmatrix”Mγ˜Tparameterswhichismenactuallytionedinabdepoveendenaretoffixed,λandandtheexplicitlycor-
respondingadditionalconstraintsonthemappingspj,qjoccurringinMˆjbecomebyfarmoreexplicit.
32rotationArotorRinefleR3ctionfollowedefinesdbyaarEuclideeflectionanSmotion,inR3,i.e.suanchthatisometrytherofefleRction(cf.seplanectionofS4.1),iswhichorthoisgonalcomptoosethedrofanotationarbitraxisaryof
.R

9

(Notethat,however,suchan“appropriatenormalization”ofFisonlypossible,iftheconsideredmapping
γ˜TpossessesanyfixedpointinM˜.ˆItturnsoutˆ−1thatthisisthecaseforallpossibletrinoidsymmetries
exceptfortherotoreflectionsT=SandT=S.)
Ifthe“appropriatenormalization”ofFasdescribedaboveispossible,wecanexplicitlytranslatea
givensymmetrypropertyofatrinoidφ:M→R3withproperlyembeddedannularends(resp.ofthe
correspondingCMC-immersionψ:M˜→R3),intoconstraintsonthemappingspj,qjoccurringinthe
monodromymatricesMˆjofthegeneratingsolutionΨtothedifferentialequationdΨ=Ψ˜η.Theexplicit
resultsaregivenintheorems5.13,6.9,7.8and8.9,respectively.Moreover,itturnsoutthattheobtained
constraintsonthemappingspj,qjarenotonlynecessarybutalsosufficientforΨgeneratingatrinoid
withproperlyembeddedannularendsandtherespectivesymmetry.Thisisprovedintheorems5.14,
6.10,7.9and8.10,respectively.
Section4isorganizedasfollows:First,wegivethedefinitionofatrinoidsymmetryin3section4.1.
TheprocedureofrecoveringtheextendedframeFfromtheCMC-immersionψ:M˜→Rassociated
withatrinoidφ:M→R3isstudiedinsection4.2.Actually,thisisdoneinthegeneralizedsetting
ofanarbitraryRiemannsurfaceMwithuniversalcoverM˜andapairofconformalCMC-immersions
φ:M→R3andψ:M˜→R3linkedviatheuniversalcoveringπ:M˜→M,ψ=φ◦π.Sections4.3and
4.4explicatethetranslationofagivensymmetrypropertyofatrinoidφwithproperlyembeddedannular
ends(resp.ofthecorrespondingCMC-immersionψ)totheleveloftheextendedframeFofψ.Section
4.5appliesthistranslationtothesymmetryT=Iandthecoveringtransformationsγ˜j,j=0,1,∞,
associatedwiththemonodromymatricesofΨtoobtainthe“monodromyrelations”forFmentioned
earlier.Finally,section4.6providestheexplicitformsofthebiholomorphic(resp.bi-antiholomorphic)
mappingsγ:M→Masssociatedwiththepossibletrinoidsymmetries,aswellasthetwelvepossible
es.themselvsymmetriestrinoidSections5to9translatethe“symmetryrelation”oftheextendedframeFassociatedwithapar-
ticulartrinoidsymmetryTofatrinoidφwithproperlyembeddedannularendsintoconstraintsonthe
monodromymatricesMˆjofthesolutionΨtothedifferentialequationdΨ=Ψη˜,whichproducesφ(or,
moreprecisely,thecorrespondingCMC-immersionψwhich“descends”toφ).Section5dealswiththe
symmetriesRandR−1,section6treatsthesymmetriesRj,j=0,1,∞,section7discussesS,section8
studiesthesymmetriesSj,j=0,1,∞,andsection9isconcernedwiththesymmetriesSˆandSˆ−1.
Throughoutthisthesis,weactontheassumptionthatthereaderisfamiliarwiththebasicnotionsof
differentialgeometry.Inparticular,thisinvolves:(parametrized)surfacesinR3,themeancurvature
ofasurface,(differentiable)manifolds,differentiablemappingsbetweenmanifolds,thedifferentialofa
differentialmapping,Riemannianmanifolds.Acomprehensiveintroductiontodifferentialgeometrycan
befoundin[6].Thefollowingnotionsareofparticularinterestforourconcerns:
Adifferentiablemappingf:M→R3onaRiemannianmanifoldMiscalledanimmersion,ifthe
correspondingdifferentialateachpointp∈M,df:TpM→R3,isinjective.Animmersionf:M→R3
onaRiemannianmanifoldMiscalledanembedding,ifitisahomeomorphismontoitsimage,i.e.ifthe
mappingf:M→f(M)iscontinuousandinjectivewithacontinuousinversemappingf−1:f(M)→M.
LetMbeaRiemannianmanifoldwithdifferentiablestructure{Uα,xα}.Animmersionf:M→R3
onMiscalledaCMC-immersion(oraCMC-H-immersion),if,foreachα,themappingf◦xα:Uα→R3
isa(parametrized)surfaceofconstantmeancurvatureH.
Furthermore,thisthesisinvolvesbasictopologicaldefinitionsandresults,whichhavebeenassembled
inappendixA.Foradetailedintroductiontoalgebraictopology,thereaderisreferredto[20].

tswledgmenknoAcIwouldliketothankProf.J.Dorfmeister,whointroducedmeintothefieldofCMC-surfacesandhas
alwaysbeenafriendlyandhelpfuladvisor.Inparticular,IthankProf.Dorfmeisterforhissupport,his
guidanceandallthetimehedevotedtomyresearch.
IalsoowemythankstoProf.F.PeditandN.Schmittforhelpfuldiscussionsonmywork.
Finally,IthankmycolleaguesattheTechnischeUniversit¨atM¨unchenforanamicableatmosphere
ort.supptheirand

10

2Outlineoftheloopgroupmethod
Webeginbygivingabriefreviewofthe“loopgroupmethod”fortheconstructionofconstantmean
tocurvasaturethesurfac“DPW-methoesfromd”.)Thisholomorphicreviewpotenwilltialsinvolvaseinpresentrotedduciningthe[15].basic(Thisconceptsmethod(loisopoftengroupsalso,Iwasareferredwa
decomposition,holomorphicpotentials)aswellasgivinganoutlineoftheloopgroupmethoditself,
illustratedbythesimpleexampleofthealreadymentionedDelaunaysurfaces.
GroupsopLo2.101LetSL(2,C)denotethespeciallineargroupofcomplex2×2matricesandσ:SL(2,C)→SL(2,C)denote
theconjugationbythePaulimatrixσ3=0−1.Foreachr∈(0,1],wedefineby
ΛrSL(2,C)σ={γ:C(r)→SL(2,C)smooth;γ(−λ)=σ(γ(λ))}(2.1.1)
the(twisted)loopgroupofsmoothmapsfromther-cirleC(r)={λ∈C;|λ|=r}intoSL(2,C).TheLie
algebraΛrsl(2,C)σofΛrSL(2,C)σisgivenby
Λrsl(2,C)σ={x:C(r)→sl(2,C)smooth;x(−λ)=σ3x(λ)σ3},(2.1.2)
wheresl(2,C)denotestheLiealgebraofSL(2,C).Theconditionsγ(−λ)=σ(γ(λ))andx(−λ)=
σtiv3x(elyλ.)σ3Notewillbthatethereferredtwistingtoastheconditiontwistingforcaondsmoitionsothformatrixγ∈ΛrfunctionSL(2,γC):σC(andr)x→∈ΛSL(2r,sl(2C,)C)σ(resp.,respforec-a
smoothmatrixfunctionx:C(r)→sl(2,C))ismetifandonlyiftheoff-diagonalentriesof(r)γ(resp.ofx)
areoddfunctions,whilethediagonal+entriesofγ(resp.ofx)areevenfunctionsofλ∈C.
phicallyFtourthermore,theopwenediscdenoteI(rb)y=Λ{rλ∈SL(2C,;|Cλ)|σ<ther},andsubgroup-byofabusemapsofγ∈ΛnotationrSL(2-,bCy)ΛσthatSU(2)extendtheholomsubgroupor-
σrofmapsγ∈ΛrSL(2,C)σthatextendholomorphicallytotheopenannulusA(r)={λ∈C;r<|λ|<r1}
andtakevaluesinthespecialunitarygroupSU(2)ontheunitcircleS1=C(1).
Inthecaser=1,weomitanysubscripts“r”,simplydenotingthegroupsΛ1SL(2,C)σ,Λ+SL(2,C)σ,
Λ1SU(2)σbyΛSL(2,C)σ,Λ+SL(2,C)σ,ΛSU(2)σ,respectively.Wedealanalogouslywiththe1correspond-
algebras.LieingRemark2.1.ThetopologyintroducedabovefortheloopgroupsandLiealgebrasisaFrechettopology.
SometimesitispreferabletoworkwithBanachstructuresinsteadofwithFrechetstructures.Inthiscase
(cf.,onee.g.,could[9]).require,Forthee.g.,purpthatosesallofmatrixthiswcoorkefficthienetstopareologyconoftainethedingrouthepswillWienerplayaalgebraminoronrole.theunitcircle
2.2Iwasawadecomposition
Itisknownfrom[32]thatthemultiplicationmapΛrSU(2)σ×Λr+SL(2,C)σ→ΛrSL(2,C)σissurjective,
thatis,anyγ∈ΛrSL(2,C)σmaybewrittenas
γ=γuγ+,(2.2.1)
whereγ∈ΛSU(2)andγ∈Λ+SL(2,C).Thesplitting(2.2.1)iscalledanr-Iwasawadecomposition
ofγ∈ΛurSL(2r,C)σ,σor,ifr+=1,jrustIwasawaσdecompositionofγ.Byadditionallyrequiringthatγ+(0)
isdiagonalwithpositiverealentries,thefactorsofthesplitting(2.2.1)areuniquelydetermined.Inthis
casethemultiplicationmapisareal-analyticdiffeomorphism,andwewillthereforespeakoftheunique
r-Iwasawadecomposition(resp.uniqueIwasawadecomposition)ofγ.Foraproofofthis,thereaderis
[30].and[32]toreferredtialsotenpHolomorphic2.33HNext=0weonawillsimplyoutlinehoconnectedwonedomainobtainsM˜⊆fromCanansl(2imme,C)-vrsionaluedψ:M˜→holomorphicRofdifferenconstantialtmeanone-formcurvonatureM˜
involvingaloopparameterλ∈C∗=C\{0},thesocalledholomorphicpotentialη˜.

11

Letψ:M˜→R3beaCMC-immersion.ConsidertheextendedframeF:M˜→ΛSU(2)σcorresponding
toψasdefinedin[15].3Accordingto[15],thereexistsB+:M˜→Λ+SL(2,C)σsuchthat
Ψ=FB+isholomorphicinbothz∈M˜andλ∈C∗(2.3.1)
Ψiscalledanholomorphicframeassociatedwithψ.ThecorrespondingMaurer-Cartanform
η˜=Ψ−1dΨ(2.3.2)
isholomorphicinbothz∈M˜andλ∈C∗aswellandiscalledtheholomorphicpotentialassociatedwith
.ψimmersiontheRemark2.2.TheextendedframeFassociatedwithψisnotdetermineduniquely,butonlyuptothe
choiceofsomeinitialvalueF(z∗,λ)∈ΛSU(2)σforsomez∗∈M˜.Itisinparticularalwayspossibleto
achieveF(z∗,λ)=Iforachosenbasepointz∗∈M˜byreplacingagivenframeF0(z,λ)byF(z,λ):=
F0(z∗,λ)−1F0(z,λ).
InthefollowingsectionswerecapitulatetheprocedureofconstructingCMC-immersionsφ:M→R3
ofaRiemannsurfaceMintoR3fromagivenholomorphicpotentialη˜,whichisdefinedontheuniversal
coverM˜ofthe(notnecessarilysimplyconnected)RiemannsurfaceM.Inordertoconstructφ,we
proceedasfollows:First,we3applytheloopgroupmethodtotheholomorphicpotentialη˜toobtaina
CMC-immersionψ:M˜→R(cf.section2.4).Moreprecisely,theloopgroupmethodwillproducea
wholefamilyψλofCMC-immersionsM˜→R3,parametrizedbyaloopparameterλ∈S1.Second,we
turntothequestionunderwhichcircumstancesψλdescendstoaCMC-immersionφλ:M→R3,atleast
foraspecialchoiceoftheloopparameterλ(seesection2.5).

2.4Theloopgroupmethod
Asindicatedabo˜ve,wecanconstructimmersionsofconstantmeancurvatureH=0definedonthe
universalcoverMofaRiemannsurfaceMfromholomorphicpotentialsintroducedinsection2.3by
applyingthe“loopgroupmethod”presentedin[15].Carryingoutthisprocedureinvolvesthefollowing
steps:three1.Givenaholomorphicpotentialη˜,solvethedifferentialequation
dΨ=Ψ˜η.(2.4.1)
2.Perform(foreachz∈M˜)anr-Iwasawadecomposition
Ψ=FB+.(2.4.2)
Notethat,byconstruction,Fin(vr)olvestheloopparameterλ1∈C(r)(1)andcanbeholomorphically
extended(inλ)totheannulusAcontainingthe1-sphereS:=C.
3.InterpretingFasanelementofΛSU(2)σandwritingλ=eiθfortheloopparameterλ∈S1,
evaluatetheSym-Bobenkoformula
SymBob(F)=−21H(∂∂θF∙F−1+2iFσ3F−1)(2.4.3)
foranyλ0∈S1toobtainaCMC-immersionψλ0definedonM˜.
endsRemarkfrom2.3.Fholomorphicorourppurpotenoses,tialsthat(cf.isforsectionthe3),weconstructioncanthinofkoftrinoidsthewithstartingproppotenerlyetialmbη˜eddonedM˜annasularthe
˜[9]).pullbacThkusofwesomeensurepotenthattialη˜ηisandefinedinvariantonM=M/holomorphicΓ,wherepotenΓtial,denotesi.e.inthevariantfundamenundertalthgreoupactionofMofthe(cf.
fundamentalgroupΓonM˜.(Cf.sectionA.4˜ofappendixAforadetaileddiscussionofthementioned
actionofthefundamentalgroupΓofMonM.)
Moreprecisely,we3notethatthechoiceofaholomorphicpotentialassociatedwithagivenCMC-
aninimmersionvariantψ:M˜holomorphic→Rispnototenutialniqueη˜.(cf.The[15]).factBythatwetheoremcanth3.2usof[9],assumeitispw.l.o.g.ossiblethattoη˜assoisanciateinvwitharianψt
whetherholomorphicapotenCMC-immersiontialwillbMe˜→usefulR3inprotheducedfollofromwingη˜section,descendswhentoaweaddressCMC-immersionourselvesMto→theR3.question,
3WereviewtheprocedureofconstructingtheextendedframeFfromaCMC-immersionψ:M˜→R3insection4.4.

12

Bythetheoryofordinarydifferentialequations,thesolutionto(2.4.1)isuniquelydeterminedassoon
asweprescribeaninitialvalueconditionΨ(z)=Ψ(2.4.4)
0∗foranarbitrarybasepointz∗∈M˜andsome˜Ψ0∈ΛrSL(2,C)σforsomer∈(0,1].∗Whereasanysolution
Ψto(2.4.1)willbe∗holomorphicinz∈M,Ψwillonlybeholomorphicinλ∈CifandonlyifΨ0is
holomorphicinλ∈C.
Remark2.4.Inourdiscussionsthroughoutthisworkwewilldealwithsolutionsto(2.4.1),whoseinitial
valuesatz∗arenotexplicitlyknownandthereforemightbenot∗holomorphicinλ∈C∗.Inthecasethata
stepsolutionoftheΨtoloop(2.4.1)groupismethoactuallyd,ifΨsingularisatforleastcertainvholomorpaluesofhicλ(in∈Cλ),wonecansomeonlycircleproC(ceer0d),rwith0∈the(0,1].second(In
Ththisus,case,whenresumedealingthewithloopagroupsolutionmethoΨtodby(2.4.1)paterforminganytianmre,-IwweasawneedatodecompensureositionthatofΨΨiswithrholomorphic=r0.)
(inλ)onsomecircleC(r0),r0∈(0,1].
GivenasolutionΨ˜to(2.4.1)withinitialvalue˜Ψ(z∗),itiseasytoverifythatˆΨ(z):=Ψ0˜Ψ(z∗)−1˜Ψ(z)
alsosolves(2.4.1)and,moreover,meetstheinitialvaluecondition(2.4.4).Consequentlyanysolution
Ψto(2.4.1)canbemodified(byamultiplicationfromtheleftindependentofz)tomeetaprescribed
initialcondition.Inparticular,asolutionsingularforcertain∗valuesofλ∈C∗,caninsuchaw1aybe
holomorphictransformedinintoλaonneanwopensolution,neighbwhicorhohisodofSholomorph1).icSolutionsinλ∈toCor,(2.4.1),atwhicleast,hareholomorphicholomorphicinλin∈Sλ∈S(i.e.1
areofspecialinterestforthiswork(cf.theorem2.11).
Definition2.5.LetΨbeasolutionto(2.4.1).Then,theactionofreplacingΨby
Ψˆ=TΨ,(2.4.5)
whereTdenotessomez-independentloopinΛrSL(2,C)σ,isreferredtoasr-dressingorsimplydressing
.TybΨBydressingasolutionΨof(2.4.1),weobtainanewsolutionΨˆto(2.4.1),“only”changingtheinitial
condition.Suchachange,however,hasprofoundconsequencesinsteptwooftheloopgroupmethod,as
ˆthereisnotrivialrelationbetweentheframesFandFˆinvolvedintheIwasawadecompositionsofΨand
Ψ,respectively.Thismeans,thatdressingasolutionΨto(2.4.1)will(ingeneral)giverisetosignificant
changesmanipulationintheoftheCMC-immersioninitialvalueψΨ=0ψgivλ0enbygenerated(2.4.5)byturnsstepoutthreetoofbethecrucialloopforgroupourpurpmethooses,d.Inasitfact,platheys
thedecisiverolewhenitcomestodecidingwhetherψwilldescendtoaCMC-immersionφonMornot.
Thisissuewillbediscussedfurtherinthefollowingsection.
(2.4.1)Remarkis2.6.sometimesWewonlyouldlikdefinedetoforTremark,∈Λ+thatSL(2in,C)otherσ.ThisplacesisthemotivateddressingbytheactionfolloofTwingonaconsiderations:solutionto
rˆLetΨbe˜asolutionto(2.4.1)andT∈ΛrSL(2,C)σwithIwasawadecompositionT=TuT+.ˆDenoteby
˜Ψ(resp.Ψ)thenewˆsolutionˆtoˆ(2.4.1)obtainedfromdressingΨbˆyT(resp.byT+only):Ψ=TΨ,
ofΨΨ˜=T=+TΨ.u−1ΨˆMoreoisobver,viouslyletΨgiv=enFBby+bΨ˜e=theF˜IwB˜+asawwithaF˜decomp=Tu−1FˆositionandofB˜Ψ.+=ThBˆen+,.theThus,Iwasathewaextendeddecompositionframes
F˜andFˆinvolvedintheIwasawadecompositionsofΨ˜andΨˆdifferonlybyTu−1,whichisunitaryfor
λ∈S1.Itiseasytoverify,that,consequently,thefamiliesψ˜andψˆ,λ∈S1,ofCMC-immersions
M˜→R3obtainedfromF˜andFˆ,respectively,bythethirdstepλoftheloλopgroupmethoddifferonlyby
a(λ-dependent)rigidmotioninR3.Inthissense,dressingΨbyToronlybyT+yieldsnewsolutionsΨˆ
andΨ˜to(2.4.1),whichinduce“essentiallythesame”CMC-immersionsviatheloopgroupmethod.
Weendthissectionwiththefollowingobservation:GivenasolutionΨtothedifferentialequation
(2.4.1)andaloopg∈Λr+SL(2,C)σ,themappingΨˆ:=Ψgsolvestheequation
dΨˆ=ˆΨηˆ,(2.4.6)
whereηˆisgivenby
ηˆ=η˜#g:=g−1η˜g+g−1dg.(2.4.7)
Definition2.7.LetΨbeasolutionto(2.4.1).Then,theactionofreplacingΨby
Ψˆ=Ψg,(2.4.8)

13

(2.4.7)(2.4.8)

wheregdenotessomez-independentloopinΛr+SL(2,C)σ,isreferredtoasr-gaugingorsimplygauging
Ψbyg.Thepotentialηˆassociatedwiththeholomorphicpotentialη˜by(2.4.7)iscalledthegauged
.otentialpTheuseoftheaboveobservationconsistsintherelationbetweentheunitaryfactorsintheIwasawa
+decompositionsofΨandΨˆ=Ψg,respectivelyˆ,duringthecourseoftheloopgroupmethod:Asg∈
ΛrSL(2,C)σ,ther-IwasawadecompositionofΨisgivenby
Ψˆ=F(B+g),(2.4.9)
ˆThwhicus,hΨmeansandΨˆthatprotheducetheunitarysamefactorsinCMC-immersionstheIwasawaviathedecomploopositionsgroupofmethoΨandd.ΨareConsequenactuallytly,thistheallosamews.
forreplacingagivenholomorphicpotentialη˜byacorrespondinggaugedpotentialηˆwithoutchanging
theCMC-immersionproducedbytheloopgroupmethod.

ydromMono2.5NextweinvestigateunderwhichcircumstancesagivenimmersionψontheuniversalcoverM˜ofa
RiemannsurfaceMwilldescendtoanimmersionφdefinedonM.Theanswertothisquestionis
closelylinkedtothebehaviouroftheholomorphicframeΨassociatedwithψ(cf.section2.3)underthe
coveringtransformationsγ˜correspondingtotheelements[γ]ofthefundamentalgroupΓofM.4This
transformationbehaviourofΨisexpressedbyaz-independentmatrix,themonodromymatrixM(γ,λ).
Wewillbrieflystatetheresultspertinenttothisarticle,formoredetailsseesection2.4of[17].
Lemma2.8.Givenaholomorphicpotentialη˜onM˜whichisinvariantunderΓinthesenseofremark
2.3andaclassofloops[γ]∈Γ,anysolutionΨ:M˜→ΛrSL(2,C)σto(2.4.1)willtransformunderthe
coveringtransformationγ˜:z→[γ]∙z(cf.sectionA.4)accordingto
Ψ(γ˜(z),λ)=M(γ,λ)Ψ(z,λ),(2.5.1)
whereM(γ,λ)denotessomeΛrSL(2,C)σmatrixdependingon[γ],butindependentofz.M(γ,λ)iscalled
themonodromymatrixofΨwithrespectto[γ].
Remark2.9.Wewouldliketoaddsomecommentsconcerningthepremisesoftheabovelemma.
TheoremA.14ofappendixAstateshowtoconstructthecoveringtransformationz→[γ]∙zfroman
element[γ]ofthefundamentalgroupofMatabasepointx.Thisconstructioninvolvesthechoiceof
apointy∈M˜,whichismappedtoxbytheuniversalcoveringπ:M˜→M.Thus,whenspeakingof
“the”coveringtransformationonM˜correspondingtoanelement[γ]∈Γ(likeintheabovelemma),we
tacitlyassumethatthenecessarychoiceshavealreadybeenmade:Firstofall,weassumethatwehave
chosenabasepointx∈M,whichallowsforrepresenting“the”fundamentalgroupΓofMbyπ1(M,x)
andthusforworkingwithloopsγbasedatx.Moreover,weassumethatwehavechosenapointy∈M˜,
suchthatπ(y)=x.InthisframeworkwecanapplytheoremA.14toobtainthecoveringtransformation
γ˜:z→[γ]∙z.
Remark2.10.Carryingoutanr-IwasawadecompositionofasolutionΨto(2.4.1),weobtain
Ψ=FB+,(2.5.2)
whereF∈ΛrSU(2)σandB+∈Λr+SL(2,C)σ.Furthermore,theabovelemmayields
F(γ˜(z),λ)=Ψ(γ˜(z),λ)B+−1(γ˜(z),λ)=M(γ,λ)Ψ(z,λ)B+−1(z,λ)B+(z,λ)B+−1(γ˜(z),λ)
=M(γ,λ)F(z,λ)B+(z,λ)B+−1(γ˜(z),λ),(2.5.3)
whereM(γ,λ)denotesthemonodromymatrixofΨwithrespectto[γ].Thus,incasethatM(γ,λ)∈
ΛrSU(2)σ,weobtain
F(γ˜(z),λ)=M(γ,λ)F(z,λ)k(z,γ˜,λ),(2.5.4)
4anytwNoteopointhattswineMassumeareisomorphicthroughouttothiseachworkotherthat(cf.MislemmapathA.4).connectedWewillandthatthereforethusw.l.o.g.thespfundameneakoftal“the”groupsfundamenofMtalat
groupofM,actuallyconsidering˜thefundamentalgroupofMatanarbitrarilychosenpointinM.Cf.sectionA.4formore
detailsontheactionofΓonM.

14

wherek(z,γ˜,λ):=B+(z,λ)B+−1(γ˜(z),λ).Thereby,forfixedz∈M˜,wehavek∈Λr+SL(2,C)σandatthe
sametimek∈ΛrSU(2)σ(sinceF◦γ,MandFareelementsofΛrSU(2)σ).Thus,forfixedz∈M˜,k
actuallydenotesadiagonalmatrixinSU(2),whichisindependentofλ:
F(γ˜(z),λ)=M(γ,λ)F(z,λ)k(z,γ˜).(2.5.5)
Undertheaboveassumption,i.e.inthecaseM(γ,λ)∈ΛrSU(2)σ,wewillsometimesspeakofthe
monodromymatrixM(γ,λ)ofF(withrespectto[γ]),expressinginthiswaythatF(γ˜(z),λ)islinkedto
F(z,λ)byM(γ,λ)asin(2.5.5).
Thebasictheoremforallourconsiderationsisobtainedfromtheorem2.7of[11]:
Theorem2.11.LetMbeaR˜iemannsurfacewithuniversalcoverM˜andfundamentalgroupΓ.Letη˜
bleteaΨbeaholomorphicsolutionptootential(2.4.1)onandM,letwhichψ:isM˜→invariantR3betheunderΓinCMC-immerthesensesionofrobtaineemarkdfr2.3.omΨFurbytherthemorloope,
groupmethodforλ0=1.Then,ψdescendstoaCMC-immersionφonM=M˜/Γifandonlyif
1.M(γ,λ)isunitaryforall[γ]∈Γ,λ∈S1and
2.M(γ,λ=1)=±Iforall[γ]∈Γand
3.∂λM(γ,λ)|λ=1=0forall[γ]∈Γ.
Theorem2.11providesthekeyfor“tuning”theloopgroupmethod,suchthatitwillgenerateaCMC-
immersionψ=ψ1onM˜thatdescendstoanimmersionφˆonM:GivenasolutionΨto(2.4.1),dressingit
bΨybTyM=(Tγ(,λλ)),∈[γΛ]r∈SL(2Γ,,theC)σmonowillprodromyduceamatricesnewofΨˆsolutionareΨthe=nTgivΨ.enbyDenotingMˆ(γ,λthe)=Tmono(λ)Mdrom(γ,yλ)(T(matricesλ))−1of,
[γ]∈Γ.(Again,notethatthisrelationonlyholdsinthesettingofCMC-immersionswithanumbilic
pointandthusinparticularinthetrinoidsetting.[14])
Thus,toobtainaCMC-immersionφ:M→R3fromagivenpotentialη˜,thestrategywillbetofind
[γan]∈Γ,appropriatesuchthatdressingthemonomatrixdromTythatmatriceswillmoMˆ(γdify,λ)aofgivΨˆen=TΨsolutionwillΨmeetwiththemonoconditionsdromymgivatrenicesinM(γtheorem,λ),
2.11.Inparticular,ifagivensolutionΨto(2.4.1)issingularforcertainvaluesofλ∈S1,alsoitsmonodromy
tomatricesfindaM(γdressing,λ),[γ]matrix∈Γ,T,willwhicnothev“remoenvbees”definedtheseforsingularities,thesevaluessuchofλthat.ΨˆTh=us,TinΨisthiscase,holomorphicweneedin
1.Sλ∈surfacesyDelauna2.6Aspointedoutintheintroduction,forthestudyofCMC-immersionswithproperlyembeddedannular
endsDelaunaysurfacesareofparticularimportance.TheseareCMC-surfacesofrevolutionaroundan
axisinR3,theDelaunayaxis,andparametrizedbythepuncturedcomplexplaneC∗:=C\{0}.Fora
detaileddiscussionofDelaunaysurfaces,wereferto[16].Here,weonlysummarizesomebasicresults,
whichwewilluseinthiswork.
Bysection3.2.1of[16]allDelaunaysurfaces(uptorigidmotions)canbeconstructedfromholomor-
phicpotentialsoftheform
η˜=Ddz=X00Xdz,(2.6.1)
whereX(λ)=sλ−1+tλ,X(λ)=sλ+tλ−1ands,t∈Rwith(s+t)2=41.ThematrixDiscalleda
.matrixDelaunaytheRemarkusual2.12.complexAsforconjugateXandofXX,(λw)eforhavλe∈S1adopted,weintheterpretnotationXforofv[17].aluesofNoteλ∈that,C∗\Swhile1asXthe(λ)uniqueyields
holomorphicextensionofthecomplexconjugateofX|1toC∗.ThismotivatesthedefinitionofXabove
forallλ∈C∗.Moreover,forλ∈C∗,XandXarelinkSedviatheformula
X(λ)=X(1).(2.6.2)
λ

15

(seeRemark[16]for2.13.details).ThecWehoiceareofespsecandiallytinwillteresteddetermineinembtheespddedecialDelaunashapeyofthesurfacespro,ducedwhichareDelaunaalsoyreferredsurface
toasunduloids.Thesecorrespondtosandtsuchthatst>0,i.e.eithers,t>0ors,t<0.However,
sincethepotentialsDdzand−Ddzaregaugeequivalent(andthusproducethesamesurfaces,1cf.section
2.4),allunduloidscanbeobtainedfrompotentialsoftheform(2.6.1)withs,t>0,s+t=2.
Givenapotentialη˜oftheform(2.6.1),itiseasytoverifythat
Ψ=ezD(2.6.3)
solvesthedifferentialequation(2.4.1).Aroundthepointz=0,ΨpicksuptheDelaunaymonodromy
matrixM(γ,λ):
Ψ(γ˜(z),λ)=M(γ,λ)Ψ(z,λ),(2.6.4)
whereγ˜:z→z+2πidenotesthecoveringtransformationcorrespondingtothesimplyclosedcurveγ
inC∗,whichenclosesthepointz=0,w.l.o.g.definedbyγ:[0,1]→C∗,t→e2πit.Notethatγalready
generatesthefundamentalgroupΓofC∗.Asimplecomputationyields
M(γ,λ)=e2πiD.(2.6.5)
Viatheloopgroupmethod,ΨgivesrisetoaCMC-immersionψ=ψ1definedontheuniversalcover
ofC\{0}.Applyingtheorem2.11,w2eπiDprovethatψdescendstoaCMC-surfaceφonC∗byshowingthat
theWemonorestrictdromytothematrixMunduloid(γ,λ)=casees,t>of0,Ψs+witht=resp1ect(cf.toγremarkmeetst2.13).heMoreoconditionsver,ofwetheoremassumes2.11.≥t.
Despitethoserestrictions,itturnsoutthatthecases2=tneedstobetreatedseparately.
Letfirsts=t.Inparticular,since(s+t)2=41,thisimpliesst>0.Inviewofremark2.13,wecan
assumes,t>0.Together,thisimpliess=t=41andthusX(λ)=X(λ)=41(λ−1+λ),whichallowsfor
writingD=1(λ−1+λ)01=1(λ−1+λ)S˜σ3S˜−1,(2.6.6)
4014whereS˜:=√1211−11.Consequently,wederivethat
πi(λ−1+λ)
e2πiD=S˜eπ2i(λ−1+λ)σ3S˜−1=S˜e20e−π2i(λ0−1+λ)S˜−1.(2.6.7)
ThisshowsthatM(γ,λ)=e2πiDisunitaryforallλ∈S1(asaproductofunitarymatrices),which
meansthatthefirstconditionoftheorem2.11ismet.Moreover,
iπe2πiD|λ=1=S˜e0e−0πiS˜−1=−I,(2.6.8)
whichprovesthatthesecondconditionoftheorem2.11ismet.Finally,wehave
πi−1
(∂λe2πiD)λ=1=S˜π2ie2(λ+λ)(1−λ−2)πiπi(λ−1+0λ)−2S˜−1=0,(2.6.9)
0−2e2(1−λ)λ=1
whicinduceshinmeansthethatcasealss=ottheathirdCMC-immersionconditionofφontheoremC∗.2.11issatisfied.Altogether,bytheorem2.11,Ψ
Letnows=t,i.e.s>t.Weconsiderthefunction
µ(λ)=X(λ)X(λ)(2.6.10)
fromappendixB,1which∗(inthepresentcases=t)isholomorphicandnon-zeroonasufficientlysmall
neighborhoodofSinC(cf.lemmaB.1andremarkB.2).Inviewoftherelation
(µ(λ))2=X(λ)X(λ)=s2(1+tλ2)(1+tλ−2),(2.6.11)
ssthatinferew

µ(λ)=sµ+(λ)µ−(λ),

16

(2.6.12)

where∈{±1}andthemappingsµ+andµ−,respectively,arewelldefined(recalls>t)by
µ+(λ)=1+tλ2:=1+1tλ2−1t22λ4±...,(2.6.13)
s2s8s
µ−(λ)=1+tλ−2:=1+1tλ−2−1t22λ−4±....(2.6.14)
s8s2sNotethatµ+is(atleast)holomorphicandnon-zeroforλ∈{λ∈C;|λ|<ts},whileµ−is(atleast)
holomorphicandnon-zeroforλ∈{λ∈C;|λ|>st}.Inparticular,bothµ+andµ−areholomorphic
andnon-zeroonasufficientlysmallneighborhoodofS1inC∗.Moreover,µ+andµ−,respectively,allow
forexplicitydefiningthecomplexsquareroots
µ+(λ):=1+1tλ2−3t2λ4±...,(2.6.15)
2s32s4µ−(λ):=1+1tλ−2−3t22λ−4±...,(2.6.16)
s32s4ofwhicS1hin(byC∗.analogousarguments)arealsoholomorphicandnon-zeroonasufficientlysmallneighborhood
Inviewoftheconsiderationsabove,wesetforµ=sµ+µ−with∈{±1}fromnowon
µ(λ):=˜√sµ+(λ)µ−(λ),(2.6.17)
2ts(µwhere+(λ˜))=21andifλ−=11X(λand)=˜s=(1i+iftλ=−2−)1.=s(µMoreo−(λv))er,2wtheedefinesquareforrootsthemappingsλX(λ)=s(1+sλ)=
sλX(λ):=√sµ+(λ),(2.6.18)
√λ−1X(λ):=sµ−(λ).(2.6.19)
Altogether,themappingsµ),√λXand√λ−1Xdefinedasaboveareholomorphicandnon-zeroon
asufficientlysmallneighborhoodofS1inC∗.Consequently,weinferthattheexpressions
√√1−√µλX,λ√µX(2.6.20)
arewelldefinedandholomorphiconasufficientlysmallneighborhoodUofS1inC∗.Thereforewecan
proceedbywritingforallλ∈U:
D=0X=µRSσ3S−1R−1,(2.6.21)
0Xwhere√√λX0−1
R=µ√λ√−1X,S=√12λ1−λ1.(2.6.22)
0µ√√(Notethat,sinceλXλ−1X=√sµ+√sµ−=µ,wehavedet(R)=1.Moreover,weobviouslyhave
det(S)=1.)
By(2.6.21)weinferthatforallλ∈U
e2πiD=RSe2πiµσ3S−1R−1=RSe2πiµ−20πiµS−1R−1.(2.6.23)
e011MSince(γ,λ)D=ise2πiDHermitianisunitaryforλfor∈λS∈,2Sπ1,iDwhicishskew-meansHermitianthattheforfirstλ∈Scondition.Thisofimpliestheorem2.11directlyisthatmet.
Moreover,usingµ(λ=1)=21fromlemmaB.3,weinferfrom(2.6.23)that
iπe2πiD|λ=1=R|λ=1S|λ=1e0e−0πi(S−1)|λ=1(R−1)|λ=1=−I,(2.6.24)
17

Rwhic,Rh−1p,roSvesandthatR−1thearesecondholomorphicconditionatofλ=1theoremtogether2.11iswithmet.(∂λµ(λFinally))λ,=1=using0thefromfactlemmathatB.3,thewmeatrhaicvees

(∂λe2πiD)λ=1=(∂λR)Se2πiµ−20πiµS−1R−1+R(∂λS)e2πiµ−20πiµS−1R−1
e0e0+RS∂λe0e−20πiµS−1R−1−RSe0e−20πiµS−1(∂λS)S−1R−1
2πiµ2πiµ
−RSe0e−20πiµS−1R−1(∂λR)R−1
2πiµ
=1λ=−(∂λR)R−1−R(∂λS)S−1R−1+RS2πie2πiµ∂λµ−02πiµS−1R−1
0−2πie∂λµ
+R(∂λS)S−1R−1+(∂λR)R−1λ=1=0,(2.6.25)
whichmeansthatalsothethirdconditionoftheorem∗2.11issatisfied.Altogether,bytheorem2.11,Ψ
inducesalsointhecases>taCMC-immersionφonC.

18

rinoidsT3WehaveintroducedtrinoidsasCMC-immersionsT3→R3ofthethrice-puncturedtwo-sphereintoR3.
Withoutlossofgenerality,wewillassumethatthethreepointsremovedareS=(0,0,−1)T,P=(1,0,0)T
andN=(0,0,1)TinR3,i.e.T3=S2\{S,P,N}.However,duringthecourseofthisthesisitwillbe
sometimesconvenienttoconsideralternativemodelsforboththetrinoiddomainandthetargetspaceof
thetrinoid.Wethereforeidentifythethrice-puncturedtwo-spherewiththecomplexplaneCwiththe
twopoints0and1removed(or,equivalently,withtheextendedcomplexplaneCˆ:=C∪{∞}withthe
threepoints20,1and∞removed).Theidentifyingmapis3arestrictionofthewellknownstereographic
projectionS→Cˆ.WealsoidentifythetargetspaceRofthetrinoidwiththematrixLiealgebra
su(2)={12−ix−iz+y−ixiz−y;x,y,z∈R}.TheidentifyingmapR3→su(2)isdenotedbyJ.Using
3ittheiseasystandardtoverifyinnerthatprodJuctisonanRisometryand.Fdefiningurthermore,aninnerJprodefinesductaonLiesu(2)algebraby<A,B>homomorphism:=−2trace(betwABeen)
R3equippedwiththecrossproductandsu(2)equippedwiththeLiebracket[A,B]:=AB−BA.These
alternativemodelsofthetrinoiddomainandthetargetspaceofthetrinoidareexplainedindetailin
sections3.1and3.4,respectively.
Insections3.2and3.3,westudytheuniversalcoverofthetrinoiddomainaswellasthemonodromy
isactionreferredofitstoappfundamenendixtalA,groupwhereonwethegivuniveaersalbasiccoviner.troTheductionreadertowhotheisnotunderlyingfamiliartopwithologicaltheseconcepts,notions
basedonthebookofFulton[20].
Insections3.5to3.9weintroducethetrinoidpotentialη,whichproducestrinoidsviatheDPW-
memmethobd.ershaMoreovever,unitaryweexpmonolicitlydromycomputematriceasfamilywithofrespectsolutionstothetotheelemendifferentsoftialtheequationfundamen(2.tal4.1),groupwhoseof
.M3.1TrinoidsonthedomainM=C\{0,1}
TrinoidsareCMC-immersionsofthethrice-puncturedtwo-sphereT3intoR3.Asstatedearlier,we
canassumeTw.l.o.g.thatthethreeTpoints3removedfromthetwo-spherearelocatedatS=(0,0,−1)T,
P=(1,0,0)andN=(0,0,1)inR.ThusatrinoidisoriginallydefinedasaCMC-immersion
T3=S2\{S,P,N}→R3.However,wefinditmoreconvenienttointerpretatrinoidφasaCMC-
immersionofthet3wice-puncturedcomplexplane(or,equivalently,thethrice-puncturedextendedcomplex
plane),i.e.φ:M→R,where
M=C\{0,1}=Cˆ\{0,1,∞}.(3.1.1)
ThisispossiblebyidentifyingT3andMbythewellknownstereographicprojection
yp:S2→Cˆ,(x,y,z)T→1−xz+i1−zforz=1(3.1.2)
1=zfor∞ˆ2ˆ2thatmapping.definesAsap(S)=diffeomorphism0,p(P)=S1→andCp,(Ni.e.)a=∞,differenpintiableparticularbijectionalloSws→toCidenwithtifyT3differen=S2\tiable{S,inNv,Perse}
andCˆ\{0,1,∞}=M.
Thus,givenaCMC-immersionφ0:T3→R3,themappingφ:=φ0◦p−1definesaCMC-immersionof
MFintoromR3nowon,parametrizingweinterpretthesametrinoidssurface,asφ(M)=CMC-immersionsφ0(T3).M→R3.Furthermore,reparametrizinga
giventrinoidφ:M→R3(or,moreprecisely,theassociatedsurfaceφ(M))ifneeded,wewillassume
withoutlossofgeneralitythatφisconformal,i.e.thatthemetriconφ(M)inducedbyφisgivenby
ds2=eu(dx2+dy2)forsomerealvaluedfunctionu:M→R.(Inotherwords,weassumewithoutloss
ofgeneralitythatφ(M)isparametrizedinconformalcoordinates,whichisalwayspossible;cf.,e.g.,[1].)
Withrespecttotheseconsiderations,wegivethefollowing(adjusted)definitions:
Definition3.1.LetM=C\{0,1}and,forj=0,1,∞,zj=j∈Cˆ.
1.AconformalCMC-immersionφ:M→R3iscalledatrinoid(onM).
2.Letφ:M→R3beatrinoid.Anon-emptysubsetBj⊆φ(M)iscalledanannularendof
φlim(atzj)φ,(zif)=there∞.existsWithoutalosspuncturedofgeneralitneighby,orhoifonotdUjstatedofzjinotherwiseM,,sucwehtwillhatBjassume=φ(thatUj)Uandis
jzzj→openinMandthatUj∪{zj}issimplyconnectedinCˆ.
19

3.Letφ:M→R3beatrinoidandBjbeanannularendofφatzjwithBj=φ(Uj)foranappropriate
puncturedneighborhoodUjofzj3inM.Bjiscalleda(prop5erly)embeddedannularendofφ(at
zj),ifthemappingφ|Uj:Uj→Risa(proper)embedding.
Wecompletethissectionbyrecordingthefollowingresult:
Lemma3.2.LetM=C\{0,1}andφ:M→R3beatrinoidwiththreeembeddedannularends.Then,
Mwiththemetricinducedbyφiscomplete.
geometryRemark.W3.3.eTheassumeprotheofofreaderlemmais3.2familiarinvolvwithesthebasicnotionsconceptsofa(candomplete)resultsofmetricanalysisspaceandandthedifferennotiontial
ofa(geodesically)completeRiemannianmanifold.
Proofoflemma3.2.Byawellknownresultofdifferentialgeometry(“Hopf-Rinow”),itisenoughtoshow
thatM(interpretedasaRiemannianmanifold)isgeodesicallycomplete,i.e.thatatanypointp∈M
andforeachunitvectorvfromthetangentplaneTpMtoMatpthegeodesicγatpinthedirectionof
v,γ(t)=exp(tv),isdefinedforallt∈R.
Letp∈Mandv∈TpMwith|v|=1.Bytheorem1ofsection4-7andlemma1ofsection4-6of
W[6],ethereextendexiststhecurv>e0γsuc:hIthat→Mthetocurvtheeγ(t)maximal=exp(intv)tervaldefines(s0,ta0)geo⊆desicRsucinhMthatforγall:t(∈s0I,t0=)(−→,M).,
γ(t)=exp(tv),definesageodesicinM.
Considerasequencetn,n∈N,in(s0,t0)withlimn→∞tn=t0.Thecorrespondingsequenceγ(tn),
n∈N,inMpossessesanaccumulationpointp0inthecompactsupersetCˆofMandthusasubsequence,
whichconvergestop0.Denotingthissubsequencebyabuseofnotationagainbyγ(tn),n∈N,wehave
nlim→∞γ(tn)=p0(3.1.3)
Assumep0∈M.Then,bytheorem1aofsection4-7of[6],thereexist1,2>0suchthatatanypoint
qthe∈Mcurv,ewhicδ:h(−lies2,2inside)→theM,δ(spheret)=B1exp((p0tw)),ofdefinesradiusa1geoarounddesicp0in,Mandwithforδall(0)w=∈qT.qMBywith(3.1.3)|w|=and1
sincelimn→∞tn=t0,thereexistsn0∈Nsuchthatγ(tn0)∈M∩B1(p0)andt0−tn0<2.Thus,
Mthe.curvMoreoeδv0er,:(δ0−(0)2,=2)γ(→tnM),andδ0(t)δ0(=t)e=xp(γ(twt0+)tnwith)forw0all=tγ∈(tn(0−)2∈,2Tγ),(tni.e.0)Mδ0definesextendsathegeogeodesicdesicin
γ:(s0,t0)→Mbeyondt0to0ageodesicdefinedon0theinterval(s0,tn0+2)⊃(s0,t0),acontradiction
totheassumptionthat(s0,t0)definesthemaximalintervalinR,whereγisageodesic.Thusitremains
toconsiderp0∈Cˆ\M,i.e.p0∈{0,1,∞}.Then,by(3.1.3)andbythefactthatφdefinesanembedding
onasmallenoughpuncturedneighborhoodofp0withlimz→p0φ(z)=∞,weinferthat
nlim→∞φ(γ(tn))=∞.(3.1.4)
Thisimplieslimn→∞d(φ(p),φ(γ(tn)))=∞,whereddenotesthemetriconφ(M).Consequently,denoting
bydMtheinducedmetriconM,weobtain
∞=nlim→∞d(φ(p),φ(γ(tn)))=nlim→∞dM(p,γ(tn))=nlim→∞tn=t0,(3.1.5)
whichimpliesthatγ,γ(t)=exp(tv),definesageodesicontheinterval(s0,∞).
Consideringasequencesn,n∈N,in(s0,∞)withlimn→∞sn=s0,weinferbytheanalogous
argumentasabovethats0=−∞andconcludethatγ,γ(t)=exp(tv),actuallydefinesageodesicinM
ontheinterval(−∞,∞)=R,whichfinishestheproof.
3.2TheuniversalcoverM˜ofM
ItiswellknownthattheuniversalcoverM˜ofthetwice-puncturedcomplexplaneMcanbetakentobe
theupperhalfplane
M˜=H:={z=x+iy∈C;x,y∈R,y>0}.(3.2.1)
5AnembeddingU→Viscalledproper,ifinverseimagesofcompactsubsetsofVarecompact.

20

Thecorrespondingcoveringmap6isgivenby
+1z1π:M˜→M,π(z):=℘℘((21;;ZZzz++ZZ))−−℘(℘(2z;;ZZzz++ZZ)),(3.2.2)
22where℘denotestheWeierstrassfunction
111
℘(z;Zω1+Zω2):=z2+0=ω∈Zω1+Zω2(z−ω)2−ω2,(3.2.3)
ω1whichmeromorphicisdefinedforfunctionz∈onCCandwithω1,ω2second∈C\order{0}poleswithatω2the/∈Rp.oinAstsaofthefunctionperioofditslatticefirstvariablez,℘isa
Ω=Zω1+Zω2.(3.2.4)
Moreprecisely,℘isanelliptic(i.e.doubly-periodic)functionwithrespecttoΩandthussatisfies
℘(z+ω;Ω)=℘(z;Ω)forallz∈Candallω∈Ω.(3.2.5)
Moreover,consideringthedefiningequation(3.2.3),weobtain
℘(−z;Zω1+Zω2)=℘(z;Zω1+Zω2),(3.2.6)
℘(µz;Zµω1+Zµω2)=µ−2℘(z;Zω1+Zω2)forallµ∈C\{0}(3.2.7)
℘(z¯;Zω¯1+Zω¯2)=℘(z;Zω1+Zω2).(3.2.8)
AhaveZdetailedω˜+Zstudy=Zofωthe+Zforfunctionallω˜℘∈canCbewithfound,ω˜−ωe.g.,∈inZ[24].andInthereforethespecialcaseω2=1weadditionally
11111℘(z;Zω˜1+Z)=℘(z;Zω1+Z)forallω˜1∈Cwithω˜1−ω1∈Z.(3.2.9)
Thefollowinglemmarecordssomeusefulpropertiesofthecoveringmapπ:M˜→M.
Lemma3.4.Thecoveringmapπ:M˜→Masgivenin(3.2.2)satisfies
1π(z+1)=π(z),(3.2.10)
π(−1)=1−π(z),(3.2.11)
zπ(−z¯)=π(z).(3.2.12)
Remark3.5.Notethatforz∈M˜alsoz+1,−z1and−z¯areelementsofM˜.Thereforetheleft-hand
sidesoftheequationsstatedintheabovelemmaarewelldefined.
ProofofLemma3.4.Westartwiththeproofof(3.2.10).Usingequations(3.2.9)and(3.2.5)weobtain
11℘(2;Z(z+1)+Z)=℘(2;Zz+Z),(3.2.13)
℘((z+1)+1;Z(z+1)+Z)=℘(z+1;Zz+Z)=℘(z;Zz+Z),(3.2.14)
222℘(z2+1;Z(z+1)+Z)=℘(z2+1;Zz+Z)(3.2.15)
usthand℘(21;Z(z+1)+Z)−℘((z2+1)+1;Z(z+1)+Z)℘(21;Zz+Z)−℘(2z;Zz+Z)1
π(z+1)=℘(21;Z(z+1)+Z)−℘(z2+1;Z(z+1)+Z)=℘(21;Zz+Z)−℘(z2+1;Zz+Z)=π(z).
(3.2.16)6WehaveslightlymodifiedthecoveringmapM˜→M,z→℘℘((z21;;ZZzz++ZZ))−−℘℘((zz2+1+1;;ZZzz++ZZ))giveninchapterI,§4of[24]by
composingitwiththeMoebiustransformationM→M,z→z−z1.2Thiswayw2eensurethattheboundariesofthesingle
1)sheetsto−∞ofour(resp.chosen+∞)alongtesselationtheofM˜negativareemapp(resp.edpbyositivthee)rcoeavleringaxis.mapπontothe“cuts”inM=Cextendingfrom0(resp.
21

Fortheproofof(3.2.11)weapply(3.2.7),(3.2.5)andtheidentityofsetsZ=−Ztoobtain
℘(21;Z(−z1)+Z)=z2℘(2z;Z(−1)+Zz)=z2℘(2z;Zz+Z),(3.2.17)
℘(−z1+1;Z(−1)+Z)=z2℘(−1+z;Z(−1)+Zz)=z2℘(z+1−1;Zz+Z)=z2℘(z+1;Zz+Z),
2z222(3.2.18)
1℘(−2z;Z(−z1)+Z)=z2℘(−21;Z(−1)+Zz)=z2℘(21−1;Zz+Z)=z2℘(21;Zz+Z).(3.2.19)
impliesThis11−1+11zz+1
π(−1)=℘(2;Z(−z)+Z)−℘(z21;Z(−z)+Z)=℘(2z;Zz+Z)−℘(21;Zz+Z)=1−π(z).
z℘(21;Z(−z1)+Z)−℘(−2z;Z(−z1)+Z)℘(2;Zz+Z)−℘(2;Zz+Z)
(3.2.20)Itremainstoprove(3.2.12).Usingequations(3.2.8)and(3.2.5)weobtain
℘(1;Zz¯+Z)=℘(1;Zz+Z),(3.2.21)
22℘(12−z¯;Zz¯+Z)=℘(12−z;Zz+Z)=℘(12+z−z;Zz+Z)=℘(12+z;Zz+Z),(3.2.22)
℘(−z¯;Zz¯+Z)=℘(−z;Zz+Z)=℘(z−z;Zz+Z)=℘(z;Zz+Z).(3.2.23)
2222Therefore,π(−z¯)=℘(21;Z¯z+Z)−℘(12−z¯;Zz¯+Z)=℘(21;Zz+Z)−℘(1+2z;Zz+Z)=π(z).(3.2.24)
℘(21;Zz¯+Z)−℘(−2z¯;Zz¯+Z)℘(21;Zz+Z)−℘(2z;Zz+Z)

Asadirectconsequenceoftheabovelemmaweobtain
1π(z+2)=π(z+1)=π(z),
z−2z−111
π(2z+1)=1−π(z)=1−π(−2−z)=1−π(−z)=π(z).
Thisshowsthatthetwomappings

satisfy

˜U:M˜→M˜,z→z+2,
zS˜:M˜→M˜,z→
1+z2

(3.2.24)

(3.2.25)(3.2.26)

(3.2.27)(3.2.28)

π◦U˜=πand(3.2.29)
π◦S˜=π,(3.2.30)
respectively.As,moreover,U˜andS˜arehomeomorphismsofM˜(withinversemappingsU˜−1:z→z−2
andS˜−1:z→−2zz+1),U˜andS˜definetwocoveringtransformationsonM˜.Infact,itturnsoutthatthe
wholeautomorphismgroupofπ(cf.appendixA,definitionA.7)isgeneratedbyU˜andS˜(cf.chapter
IV,5of[34]).Figure3.1belowshowsatesselationofM˜withrespecttothesheetFgivenby
F={z=x+iy∈M˜;−1≤x<1and|z+1|≥1and|z−1|>1}.(3.2.31)
2222Figure3.2showsinmoredetailwhereinMthedifferentpartsofFaremappedbyπ.(Thiscanbe
validatedbyadirectcomputation.)

22

Figure3.1:AtesselationoftheupperhalfplaneM˜=HwithrespecttoF.

Figure3.2:CorrespondencebetweenthemainsheetFofM˜andMviaπ|F
Remark3.6.Naturally,theuniversalcoveringπdefinesalocalhomeomorphismM˜→M.Notethat,
moreover,πisholomorphiconM˜(cf.chapterI,§4of[24]),whichimpliesthatπactuallydefinesa
conformalmappingM˜→M(cf.3chapterVI,§1of[4]).Consequently,toagiven˜trinoidφ3:M→R3,i.e.
aψisconformalCMC-immersionifandofMonlyiniftoφRis,correspconformal.ondsI.e.,atoanyCMC-immersionconformalψ:=φ◦CMC-immersionπofMinMto→RR.3Fcorrespurthermore,onds
aconformalCMC-immersionM˜→R3.
Recallthat,conversely,agivenCMC-immersionψ:M˜→R3“descends”onlytoaCMC-immersion
φ:M→R3,ifcertainconditionsaremet(cf.theorem2.11).However,ifthisisthecase,andif,in
addition,ψisconformal,thenφwillbeconformalaswell.3
ing”Altogether,conformalweinferCMC-immersionsthattheM˜conformal→R3.CMAsC-statedimmersionsearlier,Mwe→canRcorrespwithoutondlossviaofπtogeneralitthey“descend-restrict
3ourrestrictstudyofourselvestrinoidstottheothestudystudyofcofconformalonformalCMC-immersionsCMC-immersionsM˜M→R→3.R.ThCus,onsweequenwilltly,tacitlywewillassumealso
fromnowonthatanyCMC-immersionψ:M˜→R3producedbytheloopgroupmethodhasalready
beenreparametrizedintoaconformalCMC-immersionofM˜intoR3.
3.3ThefundamentalgroupΓofManditsmonodromyactiononM˜
ˆInthispath-connected,section,wΓeisinuptrotoducetheisomorphismsfundamenindeptalendengrouptofΓothefcMhoice=Cof\a{0,base1,p∞}oin.tin(RecMall.)that,Moreoasver,Mwise
explicitlyconstructthecoveringtransformationsonM˜correspondingtothegeneratingelementsofΓ.
TheunderlyingideasareprovidedinappendixA.
WefirstconsiderthefundamentalgroupofthetrinoiddomainT3=S2\{S,P,N}.AsT3ispath-
connected,itsfundamentalgroupisuptoisomorphismsindependentofthechoiceofabasepointinT3.

23

Wechoosethebasepointtobe(54,0,−53)∈T3.ThefundamentalgroupofT3isthengeneratedbythe
equivalenceclassesoftwoloopsγSandγPbasedat(54,0,−53),whereγS(resp.γP)surroundsexactly
thatonceathelooppoinγtinST(resp.surrP)oundswithoroutenclosesenclosingagivPenandpoinNtin(resp.S2,SifandthispN)oinattliestheonsamethetime.righthandTherebyside,wewhilesay
3“walkingalong”theloopγfromγ(0)toγ(1)onthe“outside”ofS2.Notethatan−y1loop−1surrounding
exactlyoncethepointN(andneitherSnorP)ishomotopictotheloopproductγP∙γS.
Byapplyingthestereographicprojectionpdefinedin(3.1.2)wecantranslatethegeneratingelements
ofthefundamentalgroupofS2\{S,P,N}intothecorrespondinggeneratingelementsofthefundamental
groupofM1=Cˆ\{40,1,∞}3:TheloopγS(resp.γP)ismappedbypontoaloopγ0(resp.γ1)inM,which
isbasedat=p((,0,−))∈M.Notethatthestereographicprojection“unfolds”thethrice-punctured
twosphere2ontothe5t5wice-puncturedcomplexplane,relatingthe“outside”ofS2tothe“lowerside”of
Cˆ.Furthermore,ppreservesorientation.Thus,asγS(resp.γP)keepsthepointS(resp.P)2onitsright
handsidewhentracedfromγS(0)toγS(1)(resp.fromγP(0)toγP(1))onthe“outside”ofS,γ0(resp.
γ1)keepsthepoint0=p(S)(resp.1=p(P))onitsrighthandsideaswellwhentracedfromγ0(0)to
γ0(1)(resp.fromγ1(0)toγ1(1))onthe“lowerside”ofCˆ.Thismeansthat-viewed“fromunderneath”
ˆpCoin-γt0of(resp.viewγb1y)loenclosesokingatthethepointextended0(resp.complex1)clocplanekwiseCˆinM“from.abovConsequene”,γtly,(resp.takingγ)theenclosesmorethepfamiliaroint
100(resp.1)counter-clockwiseinM,keepingitonitslefthandsidewhileevolvingfromγ0(0)toγ0(1)
(resp.fromγ1(0)toγ1(1)).Fromnowon,wesaythataloopγinMsurroundsorenclosesagivenpoint
inCˆ,ifthispointliesonthelefthandsidewhile“walkingalong”γfromγ(0)toγ(1)onthe“upperside”
ˆofandC.NNatur(resp.allyS,andasγNS)at(resp.theγP)sameenclosestime,γth0ep(resp.ointγS1)(resp.enclosesP)intheTp3ointexactly0(resp.once1)wiinthoutMexactlyenclosingonceP
without−1enclosing−11=p(P)and∞=p(N1)(resp.0=p(S)and∞=p(N))atthesametime.Finally,
γ∞:=γ1∙γ0definesaloop(basedat2)surroundingthepoint∞exactlyoncewithoutenclosing0
1.andAltogether,thefundamentalgroupΓofMisgeneratedbythehomotopyequivalenceclassesofγ0
:γand1Γ=<[γ0],[γ1]>.(3.3.1)
Next,byapplyingtheoremA.14,weconstructthecoveringtransformationsγ˜0,γ˜1,γ˜∞:M˜→M˜
γ∞)correspinducondingesthetothesamelocoopsvγering0,γ1andtransformationγ∞inM.γ˜0Note(resp.that,γ˜1,asrespan.yγ˜∞lo),opithisomotoenoughpictotoγ0define(resp.γ0,γ1γ,1resp.and
γonly“qualitatively”(asdoneabove).
∞AsΓisbasedatthepointx=21∈M,weneedtochooseapointy∈M˜,suchthatπ(y)=x.Let
(3.3.2)i.=yUsingequation(3.2.11),weobservethatysatisfies
π(y)=π(−y1)=1−π(y)(3.3.3)
andthusπ(y)=21=x,asdesired.Now,followingtheproceduredescribedinsectionA.4,wedenoteby
γ˜j(y)theendpointoftheuniqueliftoftheloopγjtoapathinM˜startingaty.Keepinginmindthat
thepointγ˜j(y)willbethesameforanyloophomotopictoγj,andbringinginourdetailedknowledge
aboutcorrespondingdomainsinM˜andM(i.e.domainshomeomorphicwithrespecttoπ),wecanforgo
anyfurthertechnicalcalculationsanddeterminethevaluesγ˜j(y)byjustlookingatthefiguresbelow:
Astheloopγ0(asgiveninthefigure3.3)runsfromγ0(0)toγ0(1),ittakescoursethroughthe
subcorrespdomainsondingT5,subT4,domainsT9andinTM˜8,inendingM.atConsequenanothertly,preimageitsliftofx,startingnamelyatatythetakpesointScourse˜−1(y)through(cf.figurethe
3.1).Thisimpliesγ˜0(y)=S˜−1(y).ByuseoftheoremA.8,weconcludethatactuallyγ˜0=S˜−1onM˜.
Analogously,bytracingtheloopsγ1(resp.γ∞)inthefigure3.4(resp.figure3.5)abovethroughT7,T12,
T1endingandatT6U˜(((resp.y))T6,(resp.T1,atT12U˜,−T17(,S˜(Ty8,))).T9,FT4romandthis,T5)wweeconcobtainludetheasabocorrespveonthatdingγ˜=liftsU˜andstartingγ˜at=yU˜−1andS˜
onM˜.1∞
Summarizingourpreviousconsiderations,theelements[γ0],[γ1]and[γ∞]inΓgiverisetothefollowing

24

Figure3.3:Theloopγ0onMandits(qualitative)liftonM˜connectingyandS˜−1(y)=:γ˜0(y).

Figure3.4:Theloopγ1onMandits(qualitative)liftonM˜connectingyandU˜(y)=:γ˜1(y).

Figure3.5:Theloopγ∞onMandits(qualitative)liftonM˜connectingyandU˜−1S˜(y)=:γ˜∞(y).
coveringtransformationsγ˜0,γ˜1andγ˜∞,respectively:
γ˜0:M˜→M˜,γ˜0(z)=S˜−1(z)=z,(3.3.4)
1+z2−γ˜1:M˜→M˜,γ˜1(z)=U˜(z)=z+2,(3.3.5)
γ˜∞:M˜→M˜,γ˜∞(z)=U˜−1S˜(z)=−3z−2.(3.3.6)
1+z2Notethat,since[γ0]and[γ1]generatethefundamentalgroupΓofM,weinferbytheoremA.14that
γ˜0andγ˜1generatetheautomorphismgroupAut(M˜/M)ofcoveringtransformationsM˜→M˜.More
precisely,wecanstatethefollowing
3.7.LemmaAut(M˜/M)=<γ˜0,γ˜1>={γ˜:M˜→M˜;γ˜biholomorphic,π◦γ˜=π}(3.3.7)
25

Proof.Asindicatedbefore,theidentityAut(M˜/M)=<γ˜0,γ˜1>isadirectconsequenceof(3.3.1)and
theoremA.14.Moreover,inviewofdefinitionA.7andusingthefactthatγ˜0andγ˜1definebiholomorphic
mappingsM˜→M˜,weinferthatAut(M˜/M)={γ˜:M˜→M˜;γ˜biholomorphic,π◦γ˜=π},whichfinishes
of.protheConsidering,again,asolutionΨtothedifferentialequation(2.4.1)asinsection2.5anddenotingthe
monodromymatricesofΨwithrespectto[γj],j=0,1,∞,byMj(λ):=M(γj,λ),j=0,1,∞,wehave
(2.5.1)ybΨ(γ˜0(z),λ)=M0(λ)Ψ(z,λ)(3.3.8)
Ψ(γ˜1(z),λ)=M1(λ)Ψ(z,λ)(3.3.9)
Ψ(γ˜∞(z),λ)=M∞(λ)Ψ(z,λ).(3.3.10)
Asγ˜0◦γ˜1◦γ˜∞istheidentitymappingonM˜,wehavefurthermore
Ψ(z,λ)=Ψ((γ˜0◦γ˜1◦γ˜∞)(z),λ)=M0(λ)M1(λ)M∞(λ)Ψ(z,λ),(3.3.11)
seimplihwhicM0(λ)M1(λ)M∞(λ)=I.(3.3.12)
3.4Thesu(2)modelofR3
Afterthestudy3ofthetrinoiddomainintheprevioussectionswenowhave1acloserlookatthetargetspace
oftheaunivtrinoid,ersalRco.verUsingM˜ofthethelooptrinoidgroupdomainmethoMd,bwyeevobtainaluatingfortheanyλ0Sym-Bob∈SaenkoformCMC-immersionula(2.4.3)atdefinedλ=λon0.
However,weobservethatforagivenextendedframeF∈ΛSU(2)σthecorrespondingCMC-immersion
SymBob(F)|λ=λ0=−21H(∂∂tF∙F−1+2iFσ3F−1)|λ=λ0(3.4.1)
actuallydefinesamappingM˜→su(2)fromM˜intothematrixLieAlgebra
su(2)={−ix+ziyx−−ziy;(x,y,z)∈R3},(3.4.2)
2whichistheLiealgebraofthematrixgroupSU(2).Toobtain3amappingM˜→R3(asdesired),wehave
toidentifysu(2)withthe3-dimensionalEuclideanspaceR.Thecorrespondingidentifyingmapisgiven
byx
2J:R3→su(2),y→1−ix−iz+y−ixiz−y.(3.4.3)
zDenotingby“×”the3crossproductonR3andby“[∙,∙]”theLiebracketonsu(2),wehaveforanytwo
elementsr1,r2ofRtheidentity
J(r1×r2)=[Jr1,Jr2].(3.4.4)
Duetothisequation,JdefinesanisomorphismbetweentheLieAlgebraR3equippedwiththecross
productandtheLieAlgebrasu(2)equippedwiththeLiebracket.Keepingthisinmind,wecannow
statei∂1SymBob(F)|λ=λ0=−2H(∂tF∙F−1+2Fσ3F−1)|λ=λ0=J(ψ)(3.4.5)
whereψdenotesthedesiredCMC-immersionM˜→3R3.
groupTheSO(3)ofautomorphismallorthogonalgroup3of×the3-matricesLieAlgebrawithRdeterminan(equippted+1(cf.with[10],thecrosssectionproA.3).duct)Theisgivenisomorphismbythe
Jprovidesaone-to-onecorrespondencebetweenSO(3)andtheautomorphismgroupAut(su(2))ofsu(2):
Aut(su(2))=JSO(3)J−1,(3.4.6)
i.e.anyautomorphismU∈SO(3)ofR3inducesanautomorphismJ◦U◦J−1ofsu(2)andviceversa.
Moreover,asanyautomorphismofsu(2)canberealizedbyconjugationwithaunitarymatrixP∈SU(2)

26

ofdeterminant1,whichisuniquelydetermineduptosign(again,c.f.[10],sectionA.3),wecanwritefor
su(2)X∈(J◦U◦J−1)(X)=PXP−1.(3.4.7)
3J◦SimilarlyV◦,J−giv1ofenansu(2),thereautomorphismexistsVa∈unitaryO(3)\SO(3)matrixofPR∈alongSU(2)ofwithdetethermincorrespant1,ondingwhichisautomorphismuniquely
determineduptosign,suchthatforX∈su(2)
(J◦V◦J−1)(X)=−PXP−1.(3.4.8)
001100Thisisaconsequenceofthefollowing:ConsidertheautomorphismT=010∈O(3)\SO(3)of
−R3,whichinducestheautomorphismJ◦T◦J−1ofsu(2)givenby
(J◦T◦J−1)(X)=−i0X−i0=:Y.(3.4.9)
i0i0−AsV◦T∈SO(3),thereexistsPˆ∈SU(2),suchthat(J◦V◦T◦J−1)(X)=PˆXPˆ−1forX∈su(2).
SettingP:=Pˆ−i0∈SU(2),thisimplies
i0(J◦V◦J−1)(Y)=(J◦V◦T◦J−1)(X)=PˆXPˆ−1=−PYP−1,(3.4.10)
forallY∈su(2),whichproves(3.4.8).
3.5Thetrinoidpotential
Inannthisularendssection,viawetheinlotroopducegroupaclassmethoofd.pFotenollotials,wing[17],whichwewilldefineproducetheseptrinoidsotentialswithontheproperlytrinoidembdomainedded
M=Cˆ\{0,1,∞}=M˜/Γ(ratherthanontheuniversalcoverM˜=HofM).Fromeachsuchpotential
ηwecanobtaina˜holomorphicpotentialη˜onM˜bycarryingoutthepullbackconstructioninducedby
thecoveringπ:M→M,i.e.∗
η˜=πη.(3.5.1)
Inviewof2.3,η˜isinvariantundertheactionofthefundamentalgroupΓonM˜andthusallowsforthe
applicationoftheorem2.11.
Whilekeepinginmindthenecessityofthepullbackconstructiontoobtainthe“true”trinoidpotential
η˜ing(forpotenusetialwithηontheMlo.opgroupAccordinglymetho,d),insteadweofrestrictsolvingourequationconsiderations(2.4.1),fromwenoturnwontotothethecorrespcorrespondingond-
differentialequationonM:
dΦ=Φη.(3.5.2)
NotethatanysolutionΦto(3.5.2)naturallyinducesthepullbacksolutionΨ=π∗Φto(2.4.1),as
dΨ=d(π∗Φ)=π∗(dΦ)=(π∗Φ)(π∗η)=Ψη˜.(3.5.3)
Remark3.8.Thepotentialηweconsidercomesalongwiththreesingularitiesatz0=0,z1=1and
z∞=∞.Thesesingularitiescarryovertothesolutiontothedifferentialequation(3.5.2).Thus,there
aexistssolutionnoΦ.Tholomorphicracingitsolutionalongtoa(3.5.closed2),curvwhicehγis:w[0ell,1]→definedM,globallywhichisonMbased.at(Suppxose=γthere(0)=γexists(1)sucandh
awellsurroundsknownoneofresulttheofcomplexsingularitiesofanalysis.Φ,oneBut,asobtainsγ(0)on=γreturn(1),tothisxameansdifferenthattΦvisaluemΦ(γultiply(1))de=finΦ(edγat(0))x,-
acontradiction.)
aConsequensimplytly,connectedinordersubtodomainobtainofaMw.ellTothdefinedisendweholomorphicintroducesolutioncertainΦto“cuts”,(3.5.2),i.e.weneehalf-lines,dtowhicrestricthwtoe
excludefromthedomain.Moreprecisely,wecutMalongtherealaxisfrom0to−∞andfrom1to+∞.
Thus,insteadofM,weconsiderthesimplyconnecteddomain
D=C\{x∈R;x≤0orx≥1}.(3.5.4)
OnD,aholomorphicsolutiontoequation(3.5.2)iswelldefined.
ToagivensolutionΦto(3.5.2)onDcorrespondsasolutionΨtoequation(2.4.1)viathepullback
27

˜∗construction:homeomorphicΨ(via=ππ)Φ.toDBy.Hothiswever,construction,bycontinΨuingwillΦbeatfirstholomorphicallydefinedon“acrossasubstheetFcuts”0ofinM,M,whicwehcanis
accordingly(again,viaπ)continueΨbeyondF0inM˜.(EachtimewecrossoneofthecutsinM,we
thesethinkofmeanens,teringweaobtain“newacopy”holomorphicofDandsolutionthusΨcircumtovenequationtthe(2.4.1),problemofwhicΦhisnotbdefineingedwellgloballydefined.)onM˜.By
Wenowexplicitlyintroducethepotentialη,whichwewillusethroughoutthiswork.Asmentioned
earlier,ηcomesalongwiththreesingularitiesatz0=0,z1=1andz∞=∞.Thesesingularities
willparametrizingcarryoverthetothesurfacesolutionandthtousthewilldifferengeneratetialtheequationthree(3.5.2)trinoidasends.wellFasollotowingthesectioninduced3.1ofimmersion[17],wφe
mayrestricttothecasewhereηisoff-diagonal,thatisoftheform
η=τ(z0,λ)ν(z0,λ)dz,(3.5.5)
where,fornow,ν∗andτdenotesomeholomorphicfunctionsinz∈M=Cˆ\{z0,z1,z∞}whichalso
dependonλ∈C.
Wewouldliketoconstructtrinoidswithproperlyembeddedannularends.Accordingto[25],these
endssection2.6.asymptoticallyTherefore,showwethefurtherbehaviourassumeofthattheunduloidalpotentialDelaunaηyneareacsurfaces,hwhicsingularithhayvzebeenadoptsstusomediedofin
jthepropertiesofthecorresponding(unduloidal)Delaunaypotential
1z−zjDjdz(3.5.6)
involvingtheoff-diagonalDelaunaymatrix
Dj=X0jX0j,(3.5.7)
whereXj=sjλ−1+tjλ,Xj=sjλ+tjλ−1,(3.5.8)
sj∈[41,12),sj+tj=21.(3.5.9)
potenRemarktialgiv3.9.eninNotesectionthatthe2.6(definedDelaunayonpotenthetialunivgiversalencoabvoerveofCdefines\{zjthe})tottranslationhespaceofCthe\{zj}Delaunaitself.y
Therefore,˜thisalternateversionoftheDelaunaypotentialrelatestothepotentialη,whichisdefinedon
M,notonM.
u=Remark1and3.10.considerWhenthepdealingotentialwithη(u,theλ)nearsingularittheyz∞singularit=∞,ywue∞in=tro0duceand,thecoaccordinglyordinate,thetransformation(unduloidal)
Delaunazypotential
1D∞du(3.5.10)
uinvolvingtheoff-diagonalDelaunaymatrixD∞givenin(3.5.7).
Remark3.11.Ingeneral,unduloidalDelaunaysurfacesareobtainedviatheloopgroop1methodfrom
a2.13andholomorphic3.9).pHootenwevtialer,ofittheturnsformout(3.5.6)that(forwitheachjparameters∈{0,1s,j,t∞}j)>0ourpotesatisfyingntialsjη+intjtro=2duced(cf.inremarks(3.5.5)
isgaugeequivalentto1a“perturbed”Delaunaypotentialoftheform(3.5.6)withparameterssj,tj>0
[17],satisfyingallsj+CMC-surfacestj=2andwiththeproperlyfurtherembrestreddedictionannsjular≥tjends(cf.whichremarkcanbe3.31).obtainedConsequenfromtly,holomorpfollowinghic
potentialsoftheform(3.5.5)showtheasymptoticbehaviourofunduloidalDelaunaysurfacesgenerated
fromholomorphicpotentialsoftheform(3.5.6)with
sj,tj>0,sj+tj=1,sj≥tj,(3.5.11)
2,tlyalenequivor,sj∈[41,21),sj+tj=21.(3.5.12)
Therefore,weconsiderrightawayonlyDelaunaypotentialsassociatedwithparameterssj,tjsatisfying
(3.5.12),i.e.inparticularsj≥tj.
28

tobeFirstregularofall,assingulartheities.singularitiesMoreofpreciselyDelauna,theyypwillotenbetialsareregularregular,singularwepoinrequiretsofallasingusecondlaritiesorderzjofscalarη
ODEassociatedwiththedifferentialequation(3.5.2)inthesenseofthefollowingstraightforwardlemma.
Lemma3.12.EverysolutionΦofthedifferentialequation(3.5.2)canbewrittenintheform
y1Φ=yν2yy1,(3.5.13)
2νwherey1,y2isafundamentalsystemofthedifferentialequation
y−νy−ντy=0.(3.5.14)
νWerequirethatz0,z1,z∞areregularsingularpointsof(3.5.14),i.e.werequirethatequation(3.5.14)
iscanofhaFveuchsianatypsingularite(cf.,ywhice.g.,h,chohapterwev7er,ofdo[2])esnotwithshothreewupinsingularthepoinsolutions.ts.InSucgeneral,haaFucsingularithsianyisequationcalled
haappvearfewenterthansingularitythree.Inpropourerlycaseemwbedoeddednotannwuanlartanyends.Fapparenromt[2]andsingularities,sectionssince3.3andotherwise3.5ofwe[17]wouldwe
obtainthatthethreeendsat0,1and∞arenon-apparentregularsingularpointsifandonlyif
ν(z,λ)=λ−1z−a0(z−1)−a1,(3.5.15)
τ(z,λ)=−λza0(z−1)a1b0z(2λ)+(zb1−(λ1))2+c0z(λ)+zc1−(λ1),(3.5.16)
forsomeintegersa0,a1,a∞andsomeevenfunctionsb0,b1,b∞,c0,c1inλ∈C∗satisfying
a0+a1+a∞=2,(3.5.17)
b0(λ)+b1(λ)+0∙c0(λ)+1∙c1(λ)=b∞(λ),(3.5.18)
c0(λ)+c1(λ)=0.(3.5.19)
ThepotentialηisdefinedonM.Therefore,itspullbackη˜totheuniversalcoverM˜isinvariantunder
zthe∞,corespveringectively,intransformationsM.γ˜According0,γ˜1toandγ˜∞section,whic3.3,hthecorresppulledondtobacksurroundingsolutionΨthe:=π∗Φsingularitiesto(2.4.1)z0,z1picandks
upamonodromymatrixMj(λ)underthecoveringtransformationγ˜j.Following[17],thesemonodromy
matricescanalsobecomputeddirectlyonM:“Cutting”M=Cˆ\{0,1,∞}alongtherealaxisfrom0
to−∞andfrom1to+∞,weobtainasimplyconnectedsubdomain
D=C\{x∈R;x≤0orx≥1}(3.5.20)
toofM,extendonawhicgivhenwecansolutiongloballyΦtosolv(3.5.2),ethewhicdifferenhistialdefinedequationonD,to(3.5.2)M,(cf.onehasremarkto3.8).“crossHothewever,cuts”,inwhicorderh
resultsinachangeofthestartingsolution.Moreprecisely,oneobservesthefollowing:Whenextendinga
1alongstartingthelosolutionopγΦtobased(3.5.2),atx,whicwhichhisdefinedenclosesontheaneighsingularborhoityodz,ofonethepoinobtaintxs=“on2inDreturn”,holomotoxrphicallyanother
jjsolutionΦ,whichdiffersfromΦbyaz-independentmatrix.ThischangeinΦwhensurrounding
zjcorrespnewondsexactlytothechangeinΨ=π∗Φunderthecoveringtranformationγ˜jinducedby[γj],
i.e.thematrixrepresentingthechangeinΦwhensurroundingzjcoincideswiththemonodromymatrix
Mj(λ)pickedupbyΨunderγ˜j.So,accordingto
Ψ(γ˜j(z),λ)=Mj(λ)Ψ(z,λ)(3.5.21)
writeewΦ(z,γj,λ)=Mj(λ)Φ(z,λ),(3.5.22)
whereΦ(z,γj,λ)denotesthevalueΦnew(z,λ)ofthemodifiedsolutionΦnewobtainedbyextendingΦ
holomorphicallyalongtheloopγj.Inviewof(3.5.22),wewillsometimesrefertothemonodromymatrix
Mj(λ)ofΨwithrespectto[γj]alsoasmonodromymatrixofΦwithrespecttotheloopγjinM.
Byuseoflemma3.12,onecanexplicitlycomputeuptoconjugationthemonodromymatricesof
asolutionΦto(3.5.2)bystudyingthebehaviourof(i.e.thechangein)thefundamentalsystemsof
thedifferentialequation(3.5.14)whensurroundingthesingularitiesz0,z1andz∞.Inparticular,the
29

eigenvaluesofthemonodromymatricesareknown.Thisisdoneinsection3.4of[17],andwereferthere
details.moreforisOnesomehoshwouldrelatedexpecttothethatthemonomonodromydrommatrixymatrixoftheMjofDelaunaΦy(andsurfaceΨ)wcorresphichondisingthetotheasymptoticsingularitshapyezofj
themonoenddromofythematrixtrinoidofwtheeaimcorresforpatondingzj.ItisDelaunaythereforesurfacenatural(andtothusassumepossesthatsesMthejissameconeigenjugatevtoalues).the
Actually,onecanprovethatthisisnecessarilythecase[8].Referringtosection3.6of[17],thisis
equivalenttorequiringforeachj∈{0,1,∞}
bj(λ)=1(1−aj)2−µj2,(3.5.23)
4whereµj=XjXj=41+wj(λ−λ−1)2,wj=sjtj(3.5.24)
andother±µjusefulareproptheertieseigenvofaluesµ,ofinDappj.endixTheB.relationbetweenµjandwjin(3.5.24)isproved,alongwith
jRemark3.13.Ifsj=tj,µjdefinesbylemmaB.1aholomorphicfunctionofλonthecutplaneC∗\W1,j,
whereW1,j={λ∈C∗;(λ)=0and(λ)∈(−∞,−sj]∪[−tj,tj]∪[sj,+∞)}.(3.5.25)
tjsjsjtj
Moreover,µjiswelldefinedandcontinuousontheslightlylargersetC∗\W˜1,j,where
W˜1,j=W1,j\{±istj,±itsj}.(3.5.26)
jjInof)theparticular,unitµcirclejSdefines1inaC∗con.Iftinsjuous=tjand=41,µjholomorphicdefinesbymappinglemmaon(aB.1asufficienholomorphictlysmallopfunctionenneighofλb∈orhoC∗o.d
Inanycase,themappingµj2canbeholomorphicallyextendedtoC∗:
(µj(λ))2=Xj(λ)Xj(λ)=41+wj(λ−λ−1)2.(3.5.27)
Consequently,alsothefunctionsbjdefinedin(3.5.23)areholomorphicforλ∈C∗.
BythechoiceofD0,D1,D∞andsomeintegersa0,a1,a∞satisfying(3.5.17)thefunctionsbjand
cjaregivenbyequations(3.5.23),(3.5.18)and(3.5.19)explicitly,wherebyηisdeterminedcompletely.
theWhilecwhoiceecanoftheassumeDwillw.l.o.g.determinea0=0,a1whether=0theandassoa∞ciate=2dp(weotencarrytialηthiswilloutgiveriseexplicitlytoainsection“descending”3.6),
j1CMC-immersionψinthesenseoftheorem2.11.Inordertoensurethis,wefurtherneedtorequirefor
λ∈S0≤cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))≤1.(3.5.28)
sin(2πµ0)sin(2πµ1)
Equation(3.5.28)willbereferredtoastheunitarizabilitycondition,asitisequivalentwiththeexistence
ofasolutionΨto(2.4.1),whichhasunitarymonodromymatricesatthesingularitiesatz0,z1,andz∞.
aInnewotherwsolutionords,with(3.5.28)unitaryholdsifmonoanddromonlyyifanmatrices.ysolutionof(2.4.1)canbe“dressed”(cf.section2.4)into
Altogether,by[17],theorem5.4.1andcorollary5.4.2,and[26],theorems3.5and5.9,wehaveinfact
1theTheoremform(3.5.5)3.14.LassoetDciate0,dD1with,Dthe∞begivenDelaunayDelaunaymatricmatriceses.satisfyingAssume(3.5.28)thatηforsatisfiesallλe∈Squations.Letηb(3.5.15)eof
to(3.5.19)and(3.5.23).Then,ηyieldsforλ=1atrinoidwithproperlyembeddedannularendsafter
someappropriater-dressing.
Remark3.15.Actually,theorem5.4.1andcorollary5.4.2of[17]onlyensurethatthepotentialηyields
3.5forλand=15.9aoftrinoi[26],demwithbembeddedneddeessdofanntheularannendularsafterendssomeimpliespropappropriateeremrb-dressing.eddedness.However,bytheorems
30

Remark3.16.Wenotethattheunitarizabilitycondition(3.5.28)doesnothold(onS1)ifatleasttwo
ofthethreeDelaunaymatricesD0,D1,D∞areassociatedwithparameterssj,tjsatisfyingsj=tj=41:
Supposesj=tj=41foratleasttwoj∈{0,1,∞}.Thecorrespondingmappingsµjthensatisfy
µj(i)=1(i−i)=0.(3.5.29)
4

expressiontheconsiderw,Nocos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))(3.5.30)
sin(2πµ0)sin(2πµ1)
from(3.5.28).Sinceµj(i)=0foratleasttwoj∈{0,1,∞},thenumeratorof(3.5.30)simplifies
intocos2(πµk(i)),wherek∈{0,1,∞}denotestheindexforwhichnotnecessarilysk=tk=41holds.
11Nevertheless,wehave
µk(i)=4−4wk∈[0,2),(3.5.31)
wherewehaveusedthatwk∈(0,1](cf.lemmaB.4).Thisimpliescos2(πµk(i))∈(0,1],i.e.the
numeratorof(3.5.30)takesapositive16realvalueatλ=i.Incontrast,sinceµj(i)=0foratleasttwo
j∈{0,1,∞},thedenominatorof(3.5.30)equals0atλ=i.Consequently,(3.5.28)doesnotholdat
.i=λAltogether,werecordfortherfollowingconsiderationsthatinordertoconstructa(conformal)CMC-
immersionM→R3onM=C\{0,1},weneedtostartwithapotentialηoftheform(3.5.5),such
thatsj=tjforatleasttwoj∈{0,1,∞},wheresj,tjdenotetheparametersoftheDelaunaymatrices
λD0∈,SD11.,D∞Re-indexingassociated7thewithηDelauna.Othyerwise,matricestheD,D,unitarizabilitDifynecessarycondition,we(3.5.28)assumeisnotfromnowsatisfiableonforwithoutall
lossofgeneralitythat01∞
s0=t0,s1=t1.(3.5.32)
Definition3.17.LetM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversalcoveringdefinedin
(3.2.2).LetD0,D1andD∞beDelaunaymatriceswitheigenvalues±µ0,±µ1and±µ∞,respectively,
whichsatisfy(3.5.28)forallλ∈S1.Apotentialηoftheform
0ν(z,λ)
η=τ(z,λ)0dz,(3.5.33)

where

ν(z,λ)=λ−1z−a0(z−1)−a1,(3.5.34)
τ(z,λ)=−λza0(z−1)a1b0(2λ)+b1(λ)2+c0(λ)+c1(λ)(3.5.35)
z(z−1)zz−1
forsomeintegersa0,a1,a∞andsomeevenfunctionsb0,b1,b∞,c0,c1ofλ∈C∗satisfying
a0+a1+a∞=2,(3.5.36)
bj(λ)=1(1−aj)2−µj2,(3.5.37)
4b0(λ)+b1(λ)+c1(λ)=b∞(λ),(3.5.38)
c0(λ)+c1(λ)=0(3.5.39)
willbecalledatrinoidpotential(onM).Notethatsuchpotentialsareholomorphicforz∈Mandfor
.∗Cλ∈GivenatrinoidpotentialηonM,weobtainapotentialη˜onM˜bythepullbackconstructioninduced
,πybη˜=π∗η,(3.5.40)
whichisholomorphicforz∈M˜andforλ∈C∗andwillbecalledatrinoidpotential(onM˜).
7Actually,re-indexingtheDelaunayˆmatricesD0,D1,D∞correspondstoreplacingthepotentialηbyγ∗η,whereγ
denotessomeMoebiustransformationonC.

31

structedRemarkvia3.18.theloItopisgroupclaimedmethoin[8]dfthatrompallotentrinoidstialsoftwithhefpropormerly(3.em5.33)b(eddedor,moreannularpreciselyends,ofcanthebeformcon-
(3.5.40)).Startingwithatrinoidpotentialηandtheassociatedpullbackpotentialη˜,anysolutionΨtoˆ(2.4.1),
bywhichtheoreminturn3.14,willcanprobeducedressedatrinbyoidsomewithpropappropriatelyerlyemcbeddhosenedannmatrixularTe=nTds,(λ)indefinedtoaonMsolution.TheΨ=TsameΨ,
ˆIfΨmatrix(resp.TΦ)tranformspickstheupthecorrespmonoondingdromysolutionmatrixΦMtoj(λthe)aroundequationthe(3.5.2)singularitintoyzanothj,eΨˆr(resp.solutionˆΦ)Φhas=TtheΦ.
monodromymatrixMˆj(λ)=T(λ)Mj(λ)(T(λ))−1atzj(cf.section2.5).Thesemonodromymatrices
necessarilysatisfytheconditionsoftheorem2.11.Inparticular,theyareunitaryonS1.Thus,inorder
tofindasolutionΨˆyieldingatrinoidwithproperlyembeddedannularendsonMinthesenseoftheorem
2.11,wewillperformthefollowingtwosteps:
1.W3.7eandcompute3.8).aNotesolutionthatΦΨ=toπ∗Φ(3.5.2)definwithesamonosolutiondromtoy(2.4.1)matricesandMjpatzossessesj,j=the0,1,same∞(seemonodromsectionsy
Φ.asmatricesˆ2.TW(eλ)Mj(determineλ)(T(λall))p−1ossiblesatisfythedressingconditionsmatricesofTsuctheoremhthat2.11.theThen,“dressedthemonocorrespdromyondingmatrices”newMsolutionj=
Ψˆ=TΨto2.4.1producesviatheloopgroupmethoda(conformal)CMC-immersionM˜→R3,
which,bytheorem2.11,descendstoa(conformal)CMC-immersionM→R3.
Mˆ0RemarkandMˆ1,3.19.respNoteectivelythat,,sincegeneratethetheelemenfundaments[γ0]tal,[γ1]group∈ΓΓ,itcorrespisenoughondingtotovtheerifymonopthethreedromyconditionsmatrices
oftheorem2.11onlyforMˆ0andMˆ1.
Remark3.20.Startingwithatrinoidpotentialη,theassociatedpullbackpotentialη˜andasolutionΨ
ˆtothedifferentialequation(2.4.1),wecomputeallpossibledressingmatricesTyieldinganew˜solution
Ψ=theoremTΨto2.11,(2.4.1),descendswhictohaproducesCMC-immersion(viatheloφoponMgroup.Thesemethopd)aossibledressingCMC-immersionmatrψicesoninMpartithat,cublary
encompassthosedressingmatrices,whichactuallyinducetrinoidswithproperlyembeddedannularends.

3.6Thestandardizedtrinoidpotential
Recallthetrinoidpotentialηasintroducedinsection3.5:
η=τ(z0,λ)ν(z0,λ)dz.(3.6.1)
Asstatedbefore,thechoiceofthreeoff-diagonalDelaunaymatricesD0,D1,D∞witheigenvalues±µ0,
±willµ1,±µdetermine∞,resptheectivpotenely,tialtogetherηwithcompletelythe.cThehoicefofunctionsthreeνinandtegersτaare0,agiv1,ena∞bysatisfyinga0+a1+a∞=2

ν(z,λ)=λ−1z−a0(z−1)−a1,(3.6.2)
τ(z,λ)=−λza0(z−1)a1b0(λ)+b1(λ)+c0(λ)+c1(λ),(3.6.3)
z2(z−1)2zz−1
whereb0,b1,b∞,c0,c1areobtainedfrom
1bj(λ)=4(1−aj)2−µj2forj=0,1,∞,(3.6.4)
b0(λ)+b1(λ)+0∙c0(λ)+1∙c1(λ)=b∞(λ),(3.6.5)
c0(λ)+c1(λ)=0.(3.6.6)
differenThetpurptrinoidosepofotenthistialssectioncorrespistoshoondingwtothat-differenforatpfixedossiblechoicecofhoicesofDelaunainytegersmatricesa0,a1D,0,a∞D1,areD∞-gauge-the
equivalent,i.e.relatedbycertaingaugetransformations(cf.section2.4),andthereforewillproducethe
ThesamecorrespsurfaceviaondingthetrinoidpDPW-methootentiald.willThisbeimpliescalledthethatwestandarcandizedassumetrinoidw.l.o.g.paotential0=.0,a1=0,a∞=2.

32

Theorem3.21.LetD0,D1,D∞beDelaunaymatriceswitheigenvalues±µ0,±µ1,±µ∞,respectively.
Leta,a,abesomeintegerssatisfyinga+a+a=2andηdenotethetrinoidpotentialdetermined
byµ00,µ11,µ∞∞anda0,a1,a∞asabove.F0urthermor1e,∞letaˆ0=0,aˆ1=0,aˆ∞=2andηˆdenotethetrinoid
potentialcorrespondingtoµ0,µ1,µ∞andaˆ0,aˆ1,aˆ∞.Thenwehave
η#l=ηˆ,(3.6.7)
wherel=g0g1h0h1and
−a0−a1
g0=z02za020,g1=(z−01)2(z−01)a21,h0=a01λ10,h1=a11λ10.(3.6.8)
2z2(z−1)
Remark3.22.Notethatg0,g1,h0andh1arewelldefinedonlyonasimplyconnectedsubdomainof
M,e.g.onthecutdomainDintroducedinremark3.8.Keepingthisinmind,thefollowingproofofthe
aboargumenvets,theoremwecaatnfirstafterwonlyardsholdsextendforzthis∈D,resulti.e.toweMproandvetηh#usl=ηˆobtain,foraallsz∈claimed,D.Hηo#wlev=er,ηˆbforyallconztin∈uitMy.
Proofoftheorem3.21.Abbreviating
Q=Q(z,λ)=b0(λ)+b1(λ)+c0(λ)+c1(λ)(3.6.9)
z2(z−1)2zz−1
forthemomentandrecallingthegaugeequation
η#g=g−1ηg+g−1dg,(3.6.10)
evhaewη#(g0g1h0h1)=(η#g0)#(g1h0h1)=−a20z−1a1λ−1(az0−−11)−a1dz#(g1h0h1)
−λ(z−1)Qz
a0−1a1−1−12
=−2z−−2λQ(z−1)a0z−1+λa1(z−1)−1dz#(h0h1)
222−a21(z−1)−1λ−10λ−1
=−λQ+λ2za(z0a−11)+λ4az02−λ2az02a21(z−1)−1dz#h1=−λQˆ0dz,(3.6.11)

where22Qˆ=Q−a0a1−a0+a0−a1+a1
2z(z−1)4z22z24(z−1)22(z−1)2
1=4b0−a02+2a0+4b1−a21+2a1+c1−2a0a1.(3.6.12)
4z24(z−1)2z(z−1)
As1(4bj−aj2+2aj)=bˆj(3.6.13)
4and(usinga0+a1+a∞=2)
111111
c1−2a0a1=b∞−b0−b1−2a0a1=−4+µ02+µ12−µ2∞−2a∞+2(a0+a1)+4a2∞
11−(a02+a12+2a0a1)=−+µ02+µ12−µ2∞=bˆ∞−bˆ0−bˆ1=cˆ1,(3.6.14)
44thatinferewbˆ0bˆ1cˆ1bˆ0bˆ1−cˆ1cˆ1
Qˆ=z2+(z−1)2+z(z−1)=z2+(z−1)2+z+z−1(3.6.15)
andthus,asclaimed,η#l=ηˆ.
Inviewoftheorem(3.21),wewillfromnowonrestrictourstudywithoutlossofgeneralitytotrinoid
potentialswitha0=a1=0anda∞=2,explicitlyintroducedinthefollowingdefinition.

33

Definition3.23.LetM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversalcoveringdefinedin
(3.2.2).LetD0,D1andD∞be1Delaunaymatriceswitheigenvalues±µ0,±µ1and±µ∞,respectively,
whichsatisfy(3.5.28)forallλ∈S.Apotentialηoftheform
η=τ(z0,λ)ν(z0,λ)dz,(3.6.16)
where

ν(z,λ)=λ−1,(3.6.17)
τ(z,λ)=−λb0(2λ)+b1(λ)2+c0(λ)+c1(λ)(3.6.18)
z(z−1)zz−1
forsomeevenfunctionsb0,b1,b∞,c0,c1ofλ∈C∗satisfying
bj(λ)=1−µj2,(3.6.19)
4b0(λ)+b1(λ)+c1(λ)=b∞(λ),(3.6.20)
c0(λ)+c1(λ)=0(3.6.21)
willbecalledastandardizedtrinoidpotential(onM).Notethatsuchpotentialsareholomorphicfor
z∈Mandforλ∈C∗.
GivenastandardizedtrinoidpotentialηonM,weobtainapotentialη˜onM˜bythepullback
constructioninducedbyπ,
η˜=π∗η,(3.6.22)
whichisholomorphicforz∈M˜andforλ∈C∗andwillbecalledastandardizedtrinoidpotential(on
˜.)M3.7TheFuchsianODE
Weconsiderastandardizedtrinoidpotentialηoftheform(3.6.16).Inordertosolve(3.5.2)wetakea
closerlookattheFuchsiandifferentialequation(3.5.14),whichreadsmoreexplicitlyas
b0b1c0c1
y+z2+(z−1)2+z+z−1y=0.(3.7.1)
Thecorrespondingindicialequationsaroundthesingularitiesz0=0,z1=1andz∞=∞aregivenby
w(w−1)+bjforj=0,1,∞,respectively(cf.section7.2of[2]),andpossesstheroots
rj,±=211±1−4bj=21(1±2µj)forj=0,1,(3.7.2)
r∞,±=1−1±1−4bj=1(−1±2µj),(3.7.3)
22wherewehavesimplifiedbyusing(3.6.19).
Remark3.24.Inviewofremark3.13,µjdefinesaholomorphicmappingforλ∈C∗\W1,j,where
{λ∈C∗;(λ)=0and(λ)∈(−∞,−tsjj]∪[−stjj,stjj]∪[tsjj,+∞)}ifsj=tj
W1,j=∅ifsj=tj
(3.7.4)Consequently,thefunctionsrj,±,j=0,1,∞,dependholomorphicallyonλ∈C∗\W1,j.Fromnowon,
werestrictourconsiderationstothedomainC∗\W1,where
W1=W1,0∪W1,1∪W1,∞.(3.7.5)
OnC∗\W1,allthemappingsµjandrj,±areholomorphicinλ.

34

Definingrj:=rj,+andsubstituting
y=zr0(z−1)r1w,(3.7.6)
equation(3.7.1)translatesintothehypergeometricdifferentialequation
w+−γ+(1+α+β)zw+αβw=0,(3.7.7)
z(z−1)z(z−1)
whereα=r0,++r1,++r∞,+=21+µ0+µ1+µ∞,(3.7.8)
1β=r0,++r1,++r∞,−=2+µ0+µ1−µ∞,(3.7.9)
γ=1+r0,+−r0,−=1+2µ0.(3.7.10)
areinThetegers.theoryInofhviewypofergeometricequationsfunctions(3.7.8),(3.7.9)requiresandaspecial(3.7.10),discusstheseioncasesforthecanbcaseseavwhereoidedγbyorα+excludingβ−γ
allvaluesofλ∈C∗fromourconsiderations,forwhicheitherµ0(λ)orµ1(λ)isahalf-integer.Wedenote
thesubsetofλ∈C∗,forwhicheitherµ0(λ)orµ1(λ)isahalf-integer,byW2:
W2={λ∈C∗;µ0(λ)∈1Zorµ1(λ)∈1Z}.(3.7.11)
22WIn2formsparticular,abydiscretelemmasubsetB.3,ofwCe∗,havewhic1h∈doWes2.notpStandardossessananalysisyaccumoftheulationfunctionspointsµ0onandS1.µ1Tyieldsakinginthatto
accountremark3.24,werestrictourfurthercalculationstotheλ-domainC∗\(W1∪W2),i.e.inwhat
followswewillonlyuseλ∈C∗\(W1∪W2).
Assumingγ∈/Zandα+β−γ∈/Z,whichisthecaseforallλ∈C∗\(W1∪W2),therearethe
followingnaturalfundamentalsystemswj1,wj2of(3.7.7)atzj,j=0,1(cf.chapter8of[2]):
w01=F(α,β,γ;z),(3.7.12)
w02=z1−γF(α−γ+1,β−γ+1,2−γ;z),(3.7.13)
w11=F(α,β,α+β−γ+1;1−z),(3.7.14)
w12=(1−z)γ−α−βF(γ−β,γ−α,γ−α−β+1;1−z).(3.7.15)
whereFdenotesthehypergeometricseries
∞F(α,β,γ;z)=α(α+1)∙∙∙(α+n−1)β(β+1)∙∙∙(β+n−1)zn.(3.7.16)
n=0γ(γ+1)∙∙∙(γ+n−1)n!
InRemarkparticular,3.25.wewRecallanttothatstudywearetheinmonoteresteddrominyfindingmatricesasolassoutionciatedΦtowiththeΦ.Indifferenviewtialofequationremark3.19(3.5.2).it
sufficestorestrictourconsiderationstothemonodromymatricescorrespondingtothesingularitiesz0=0
andz1=1,sincetheygeneratethemonodromygroup.Consequently,ourdiscussionofasolutionΦto
in(3.5.2)thiswcanorkbeonlyrestrictedfundamentotalthesystemsanalysistooftheitsbehaequationviour(3.7.7)nearz0definandedz1.nearItz0thereforeandz1,sufficrespesectivtoely.consider
Bydefinitionofthehypergeometricseries,thesolutionsw01,w02(resp.w11,w12)to(3.7.7)givenin
(3.7.12)and(3.7.13)(resp.(3.7.14)and(3.7.15))are,atfirst,defined(withrespecttoz)ontheopen
bediscofextendedradius1aroundholomorphicallyz0=to0the(resp.singlez1=cut1).Hocomplexwever,planebyC[2],\{wx01∈Rand;xw≥021}(resp.(resp.wto11theandwsingle12)cucant
complexplaneC\{x∈R;x≤0}).Consequently,allwjkarewelldefinedonthedoublecutcomplex
planeD=C\{x∈R;x≤0orx≥1}.Moreover,accordingto[2],p.235,thefollowingrelationshold
:onDw01=κ1101w11+κ1201w12,(3.7.17)
w02=κ1102w11+κ1202w12,(3.7.18)

35

where

01Γ(γ)Γ(γ−α−β)
κ11=Γ(γ−α)Γ(γ−β),(3.7.19)
κ01=Γ(γ)Γ(α+β−γ),(3.7.20)
12Γ(α)Γ(β)
κ02=Γ(γ−α−β)Γ(2−γ),(3.7.21)
11Γ(1−α)Γ(1−β)
02Γ(α+β−γ)Γ(2−γ)
κ12=Γ(α−γ+1)Γ(β−γ+1),(3.7.22)
andΓdenotestheGammafunctionΓ(z)=0∞e−ttz−1dt.
Remark3.26.TheGammafunctionΓisoriginallydefinedonthecomplexhalfplane{λ∈C;(λ)>0}.
Ho−wever,ΓcanbeholomorphicallyextendedtothecomplexplaneC−excludingthenon-positiveintegers
Z0.connectionThcoroughoutefficienthistswκ0iork,,i,jwe∈in{0,1terpret},areΓwasellbeingdefineddefinedandonCholomorphic\Z0.onIntheirordertodomainensureofthatdefinition,the
j1weeliminateanyλ∈C∗fromourconsiderations,forwhichtheargumentofanyoftheGammafunctions
occurringintheκ0itakesvaluesinZ−.Denotingthesetoftheseλ-valuesbyW,werestrictourstudy
totheλ-domainC1∗j\(W1∪W2∪W30).AsW2before,W3isalsoadiscretesubset3ofC∗,whichdoesnot
possessanyaccumulationpointsonS1.Allλ-dependentfunctionsintroduceduptothispoint,including
themappingsκ10ji,i,j∈{0,1},areholomorpicforλ∈C∗\(W1∪W2∪W3).
From(3.7.12)to(3.7.15)togetherwith(3.7.6)weobtainfundamentalsystemsyj1,yj2aroundzj
solvingtheFuchsianequation(3.7.1):
y01=zr0(1−z)r1F(α,β,γ;z),(3.7.23)
y02=zr0+1−γ(1−z)r1F(α−γ+1,β−γ+1,2−γ;z),(3.7.24)
y11=zr0(1−z)r1F(α,β,α+β−γ+1;1−z),(3.7.25)
y12=zr0(1−z)r1+γ−α−βF(γ−β,γ−α,γ−α−β+1;1−z).(3.7.26)
Notethatrelations(3.7.17)and(3.7.18)relatingthefundamentalsystemw01,w02tow11,w12are
equivalenttotherelations
y01=κ0111y11+κ1201y12,(3.7.27)
y02=κ1102y11+κ1202y12,(3.7.28)
forthefundamentalsystemsy,yandy,y.
Likethesolutionswto01(3.7.7),02the11solutions12yto(3.7.1)arewelldefinedandholomorphic(with
respecttoz∈M)onthejkdoublecutcomplexplanejDk=C\{x∈R;x≤0orx≥1}.
3.8SolvingdΦ=Φη
knoThewn,follothatwingtheseconsiderationsfunctionsareinvolvsingulareatlogarithmsz=0andandcansquarethusrobeotsofdefinedthecomplexholomorphicallyvariableonlyz.forItviswaluesell
ofzfromthecomplexplanecutfromz=0to∞alonganarbitraryhalf-line.Forourpurposes,itis
convenienttoconsiderboththecomplexplanecutfrom0to∞alongthenegativerealaxis,
C>0=C\{x∈R;x≤0}={z=reiϕ∈C;r∈R+,ϕ∈(−π,π)},(3.8.1)
andthecomplexplanecutfrom0to∞alongthepositiverealaxis,
C<0=C\{x∈R;x≥0}={z=reiϕ∈C;r∈R+,ϕ∈(0,2π)}.(3.8.2)
bWye√.denoteByconthetrast,wholomorphicedenotenaturalthelogarithmholomorphiconnatuC>0ralbylnlogarithmandtheonhC<0byolomorphicln∗andsquartheerootholomorphiconC>0
squarerootonC<0by√∗.Thedefinitionbelowliststheexplicitmappings,whichweusethroughout
thiswork,involvingtherealnaturallogarithmlnR:R+→R.
36

Definition3.27.TheholomorphiclogarithmlnonC>0isdefinedby
ln:C>0→C>0,z→ln(z)=lnR(r)+iϕ,(3.8.3)
wherez=reiϕwithr∈R+andϕ∈(−π,π).Theholomorphicsquareroot√onC>0isdefinedby
√:C>0→C>0,z→√z=e21ln(z).(3.8.4)
Theholomorphiclogarithmln∗onC<0isdefinedby
ln∗:C<0→C<0,z→ln∗(z)=lnR(r)+iϕ,(3.8.5)
wherez=reiϕwithr∈R+andϕ∈(0,2π).Theholomorphicsquareroot√∗onC<0isdefinedby
√∗:C<0→C<0,z→√∗z=e21ln∗(z).(3.8.6)
Lemma3.28.Letz=reiϕ∈C<0,withr∈R+andϕ∈(0,2π).Then,wehave
√∗z=i√−z.(3.8.7)
Proof.Sincez=reiϕ∈C<0withr∈R+andϕ∈(0,2π),weinferthat−z=−reiϕ=rei(ϕ−π)∈C>0.
Usingthis,theclaimisadirectconsequenceoftheabovedefinition:
√∗z=e21ln∗(z)=e21(lnR(r)+iϕ)=e21iπ∙e21(lnR(r)+i(ϕ−π))=ie21ln(rei(ϕ−π))=ie21ln(−z)=i√−z.(3.8.8)
Withthesepreparationsmade,weresumeourstudyofthedifferentialequation
dΦ=Φη.(3.8.9)
DelaunaRecallingythematricesresultDof0,D1section,D∞3.6,withbyeigensettingvaluesa0=±aµ10,=±0µ1,and±µa∞∞,=resp2ectivandcelyho,weosingconsiderthreew.l.o.g.off-diagonalthe
standardizedtrinoidpotential
η=τ(z0,λ)ν(z0,λ)dz,(3.8.10)
whereν(z,λ)=λ−1,(3.8.11)
τ(z,λ)=−λb0(2λ)+b1(λ)2+c0(λ)+c1(λ),(3.8.12)
z(z−1)zz−1
andb0,b1,b∞,c0,c1areobtainedfrom
bj(λ)=41−µj2forj=0,1,∞,(3.8.13)
b0(λ)+b1(λ)+0∙c0(λ)+1∙c1(λ)=b∞(λ),(3.8.14)
c0(λ)+c1(λ)=0.(3.8.15)
Inordertosolvethedifferentialequation(3.8.9)wewanttounderstanditssolutionsnearthesingularities
ofηconsider,.Forforthisjit=0,will1,theturnoutgaugedtopbeotenhelpfultial(cf.tousesectionspecial2.4)formsofthepotentialandthesolutions.We
ηˆj=V+−,j1ηV+,j+V+−,j1dV+,j,(3.8.16)
where√√−1
V+,0=−λ√zz∙−1∙λX√0λX0−1√z−1∙0√λX0,(3.8.17)
2√√−1
V+,1=λii√11−−zz−∙1∙√λXλX1−1−i√1−z0−1∙√λX.(3.8.18)
11237

Remark3.29.ThegaugematrixV+,j,j=0,1,isgiveninsection5.2of[17]as
√z−zj∙λXj−10
V+,j=−2λ√z−zj−1∙λXj−1√z−zj−1∙λXj,(3.8.19)
1√√−ahwherealf-linetheLjsquareinCro,otsinextendingvolvingfromz,zi.e.jtoz∞−,zjandandrestrictz−zthej,menaretionednotsquareexplicitlyrootsdefinedtoCun\tilLj.weFcorhoourose
purposes,itisconvenienttothinkofV+,0asbeingdefined(inz)onC\{x∈R;x≤0}andof√V+,1as
being√defined−1(inz)onC√\{x∈R;x≥√1}.Th−1us,weinterprettheundeterminedsquarerootsz−zj
and√∗z−zjof√∗[17]as−1z−z0andz−z0inthesenseofdefinition3.27inthecasej=0,but
as(3.8.17)z−z1and,andinviewz−ofz1lemmainthe3.28,senseequationofdefinition(3.8.18),3.27wherein-thenow-casethejo=1.ccurringThus,squareweroobtainotsinvequationolving
zaredefinedaccordingtodefinition3.27.
Remark3.30.Recallthattheλ-dependentfunctionsXjandXjaredefinedbyXj(λ)=sjλ−1+tjλand
Xj(λ)=sjλ+tjλ−1,respectivelywithpositiverealparameterssj,tjsatisfyingsj≥tjandsj+tj=21.
Moreover,weassumeforj=0,1thatsj=tj,i.e.sj>tj(cf.remark3.16).
WenotethatunderthisassumptionthesquarerootsλXjandλXj−1,j=0,1,occurringinV+,0
andV+,1areholomorphicatλ=0.(Seesection5.2of[17]fordetails.)Therefore,thematricesV+,0and
V+,1areelementsoftheloopgroupΛr+SL(2,C)σforsomer∈(0,1],thusensuringthatthepotentialηˆj
definedinequation(3.8.16)actuallydefinesagaugedversionofthestartingpotentialη.
Remark3.31.Bysection5.2of[17],thegaugedpotentialsηˆj,j=0,1,definedin(3.8.16)areperturbed
versionsofthecorresponding(unduloidal)Delaunaypotentialsz−1zjDjdzdefinedin(3.5.6),wherethe
off-diagonalDelaunaymatricesDjgivenin(3.5.7)involveparameterssjandtjwhichinparticularsatisfy
sj≥tj.+
potenWetialnoteη(u,λwithout)pro(obtainedof,thatfromtheretheexistsstartingalsopotenforjtial=η∞(za,λ)matrixbyV+,applying∞∈ΛthercoSL(2,Cordinate)σwhichtransformationgaugesthe
u=z1,cf.remark3.10)intoaperturbedversionofthe(unduloidal)Delaunaypotentialu1D∞du,where
theDelaunaymatrixD∞isgivenin(3.5.7)andinvolvesparameterss∞andt∞,whichinparticular
satisfys∞≥t∞.
Bysections4.2and4.3of[17],thereexistsforthegaugedpotentialηˆjanEDP-representation8Φˆj∗,
whichisholomorphic(inz)onacutdiscDj∗aroundzjandsatisfiesdΦˆj∗=Φˆj∗ηˆjthere.WewillcallΦˆj∗
short.forEDP-solutionanΦˆj∗=eln(z−zj)Dj∙Pj,(3.8.20)
wherePj=I+(z−zj)Pj,1+(z−zj)2Pj,2+...isholomorphicatz=zj,P(z=zj)=IandPjisuniquely
determinedbythesepropertiesandthefactthat(3.8.20)solves(3.8.9)(cf.[5]).ThecutdiscsDj∗,where
therespectivesolutionˆΦj∗isdefined,aregivenby
D0∗={z∈C;|z|<0∗}\{x∈R;x≤0},(3.8.21)
D∗1={z∈C;|z−1|<1∗}\{x∈R;x≤1},(3.8.22)
where∗and∗denotesufficientlysmallpositiverealnumbers.
For0our1concerns,itwillbemoreconvenienttoworkwiththefollowinglocalsolutionsΦˆjtodΦˆj=Φˆjηˆj
aroundzj,slightlymodifyingΦˆj∗inthecasej=1:
Φˆ0=eln(z)D0∙P0,(3.8.23)
Φˆ1=eln(1−z)D1∙P1.(3.8.24)
Weprovethefollowinglemma:
Lemma3.32.ThemappingsΦˆ0andΦˆ1asdefinedabovesolve
dΦˆj=Φˆjηˆj(3.8.25)
8TheexpressionEDP-representationisanabbreviationforexponential-Delaunay-powerseries-representation.

38

(3.8.25)

holomorphicallyonacutdiscD0aroundz0=0andonacutdiscD1aroundz1=1,respectively,where
D0={z∈C;|z|<0}\{x∈R;x≤0},(3.8.26)
D1={z∈C;|z−1|<1}\{x∈R;x≥1},(3.8.27)
forsufficientlysmallj>0.
Proof.Westartwiththecasej=0.Forsome0>0,P0ln(zis)D0defined(andholomorphic)∗on{z∈C;|z|<
0}.Bydefinitionofthecomplexnaturallogarithm,eisholomorphiconC\{x∈R;x≤0}.
Together,Φˆ0isholomorphiconD0.Moreover,asΦˆ0=Φˆ0∗,itisclearthatΦˆ0solves(3.8.25)onD0.
forWsomeeturn1>to0,thewhilecaseejln(1=−z1.)D1isSimilarasholomorphicbefore,Pon1Cis∗\{definedx∈R;(andx≥1}.holomorphic)Thus,Φˆ1onis{z∈Cholomorphic;|z−1|on<D11}.
sinceer,vMoreoln(1−z)−ln(z−1)=+iπforz∈Cwith(z)<0(3.8.28)
−iπforz∈Cwith(z)>0
weinferthatforz∈C∗\R
eln(1−z)D1e−ln(z−1)D1=e(ln(1−z)−ln(z−1))D1=eα(z)iπD1,(3.8.29)
whereα(z)=+1if(z)<0andα(z)=−1if(z)>0.Thus,forz∈D1∩(C∗\R),weobtain
Φˆ1=eα(z)iπD1eln(z−1)D1P1=eiπD1Φˆ1∗if(z)<0(3.8.30)
e−iπD1Φˆ1∗if(z)>0
Consequently,usingthefactthatΦˆ1∗solvesdΦˆ1∗=Φˆ1∗ηˆ1onD1∗,wehaveforz∈D1with(z)<0that
dΦˆ1=eiπD1dΦˆ1∗=eiπD1Φˆ1∗ηˆ1=Φˆ1ηˆ1(3.8.31)
and,analogously,forz∈D1with(z)>0that
dΦˆ1=e−iπD1dΦˆ1∗=e−iπD1Φˆ1∗ηˆ1=Φˆ1ηˆ1.(3.8.32)
Together,Φˆ1solves(3.8.25)forz∈D1∩(C∗\R).Bycontinuity,Φˆ1solves(3.8.25)forz∈D1,which
of.prothefinishesByΦˆ0andΦˆ1,wehavefoundtwolocalsolutionstoequation(3.8.25)onacutdiscD0aroundz0=0
andonacutdiscD1aroundz1=1,respectively.Fromthese,weobtainlocalsolutionsΦj,j=0,1,to
theoriginaldifferentialequation(3.8.9)bysetting
Φ0=Φˆ0V+−,10=eln(z)D0P0V+−,01,(3.8.33)
Φ1=Φˆ1V+−,11=eln(1−z)D1P1V+−,11,(3.8.34)
Remark3.33.Bydefinition,V+,0(resp.V+,1)isholomorphic(inz)onC\{x∈R;x≤0}(resp.on
C\{x∈R;x≥1}).Moreover,Φˆjisholomorphic(inz)onDj.Consequently,alsoΦjisholomorphic(in
z)onDj.
zj,Thewhichfolloisclosewinglemmaenoughtoprozjvides,sucthehthatmonoγjdromonlyymatrixenclosesoftheΦjwithsingularitrespyzectj.totheW.l.o.g.,loopweγjsetinMaround
γ0(t)=1eit,−π<t<π,(3.8.35)
21γ1(t)=1+2eit,0<t<2π.(3.8.36)
Remark3.34.Notethattheloopsγ0andγ1asdefinedabovepassthroughthepoint21∈M,as
presumedinourconsiderationsofthemonodromymatricesofΦjinsection3.5.
Lemma3.35.Withγ0andγ1definedby(3.8.35)and(3.8.36),respectively,thesolutionsΦjto(3.8.9)
givenin(3.8.33)and(3.8.34)satisfy
Φj(z,γj,λ)=Mj∗(λ)Φj(z,λ),(3.8.37)
ewherMj∗(λ)=−e2πiDj.(3.8.38)
39

Remark3.36.Notethatin(3.5.22)themonodromymatrixofasolutionΦto(3.8.9)withrespectto
thedenotelooptheγjinmonoMdromaroundymatrixzjisofthedenotedbsolutionyMjΦ(jλ).toTo(3.8.9)avoidwithrespinconsistencyecttothewithloopourγjinnotation,Mwaroundethzusj
byMj∗(λ).)
Proofoflemma3.35.Asstatedbefore,extendingthesolutionΦjto(3.8.9),whichisdefinedonthecut
domainDjaroundthesingularityzjofη,holomorphicallyalongaclosedloopγj,whichencloseszj,across
thecut∗resultsinachangeofthestartingsolution.Thischangeisexpressedinformofthemonodromy
matrixMj(λ)ofΦjwithrespecttotheloopγj(cf.(3.5.22)):
Φj(z,γj,λ)=Mj∗(λ)Φj(z,λ).(3.8.39)
∗cutIninDjorder,oncetocomputeapproacthehingthemonocutdromfromybmatrixelowMandj(λ),onceweinfromvabestigateove.thWeebehastartviourwithoftheΦj|γjcasenearj=the0.
SincethecutinD0isgivenby(−0,0]⊆Randγ0(t)=21eitwitht∈(−π,π),M0∗(λ)representsthe
transitionfromlimt→−πΦ0(γ0(t))tolimt→πΦ0(γ0(t)).Thus,considerthefollowingrelations.
First,wehave
t→limπeln(γ0(t))D0=e(lnR(21)+πi)D0=e2πiD0e(lnR(21)−πi)D0=e2πiD0∙t→−limπeln(γ0(t))D0.(3.8.40)
asurthermore,Flimγ0(t)±1=lime±21ln(21eit)=e±21(lnR(21)+πi)=e±πie±21(lnR(21)−πi)
t→πt→π
=(−1)∙t→−limπe±21ln(21eit)=(−1)∙t→−limπγ0(t)±1,(3.8.41)
weinferthat−1−1
t→limπV+,0(γ0(t),λ)=(−1)∙t→−limπV+,0(γ0(t),λ).(3.8.42)
Finally,sinceP0isholomorphic(inz)aroundz0,wehave
t→limπP0(γ0(t),λ)=t→−limπP0(γ0(t),λ).(3.8.43)
Altogether,weconcludethat
t→limπΦ0(γ0(t),λ)=t→limπeln(γ0(t))D0P0(γ0(t),λ)V+−,01(γ0(t),λ)=−e2πiD0∙t→−limπΦ0(γ0(t),λ)(3.8.44)
,ytlconsequenand,M0∗(λ)=−e2πiD0.(3.8.45)
Weturntothecasej=1.SincethecutinD1isgivenby[1,1+1)⊆Randγ1(t)=1+21eit
witht∈(0,2π),M11∗(itλ)represen1i(tts−π)thetransition1fromlimt→0Φ1(γ1(t))tolimt→2πΦ1(γ1(t)).Since
ln(1−γ1(t))=ln(−2e)=ln(2e)=lnR(2)+i(t−π),wehave
limeln(1−γ1(t))D1=e(lnR(21)+πi)D1=e2πiD1e(lnR(21)−πi)D1=e2πiD1∙limeln(1−γ1(t))D1.(3.8.46)
t→2πt→0
aser,vMoreo11±1±1±1
t→lim2π1−γ1(t)=t→lim2π−2eit=t→lim2π2ei(t−π)
=lime±21ln(21ei(t−π))=e±21(lnR(21)+πi)=e±πie±21(lnR(21)−πi)
π2t→=(−1)∙lime±21ln(21ei(t−π))=(−1)∙lim1−γ1(t)±1,(3.8.47)
t→0t→0
thatinferewt→lim2πV+−,11(γ1(t),λ)=(−1)∙t→lim0V+−,11(γ1(t),λ).(3.8.48)
Finally,asP1isholomorphic(inz)aroundz1,wehave
t→lim2πP1(γ1(t),λ)=t→lim0P1(γ1(t),λ).(3.8.49)
40

thatconcludeewAltogether,t→lim2πΦ1(γ1(t),λ)=t→lim2πeln(1−γ1(t))D1P1(γ1(t),λ)V+−,11(γ1(t),λ)=−e2πiD1∙t→lim0Φ1(γ1(t),λ)(3.8.50)
,ytlconsequenand,M1∗(λ)=−e2πiD1.(3.8.51)

Bylemma3.12,Φjmaybedescribedintermsofanappropriatefundamentalsystemsolving(3.5.14)
aroundzj,whichitselfmaybeexpressedintermsofthefundamentalsystemyj1,yj2giveninequations
(3.7.23)to(3.7.26).Thatis,wemaywriteforj=0,1
αjyj1ν+βjyj2αjyj1+βjyj2
Φj=δjyj1ν+jyj2δjyj1+jyj2,(3.8.52)
whereαj,βj,δj,jdenotez-independentfunctionsofλ.Itturnsoutthat,byevaluatingin(3.8.33)
(resp.in(3.8.34))thepropertiesofbothPjandthefundamentalsystemyj1,yj2atzj,especiallythe
holomorphicityonacutdiscaroundzj,theconnectioncoefficientsαj,βj,δj,jcanbecomputed
explicitly:Lemma3.37.Letj∈{0,1}.Theconnectioncoefficientsαj,βj,δj,joccuringin(3.8.52)aregivenby
λXµ2jjαj=−βj=(i)jXj,(3.8.53)
1jδj=j=(i)2λXj.(3.8.54)
Proof.TheproofofthislemmaisquitetechnicalandthereforepostponeduntilappendixC.
Recallthaty01,y02,y11,y12maybeextendedholomorphically(inz)toD,thecomplexplaneexcluding
two“cuts”from0to−∞andfrom1to+∞,asintroducedin(3.5.20).By(3.8.52),alsoΦ0andΦ1
canbeextendedholomorphically(inz)toD.DenotingtheextensionsagainbyΦ0andΦ1,respectively,
weobtaintwosolutionsto(3.8.9)-nowdefinedgloballyforzfromthesimplyconnected,“double-cut”
complexplaneD-whichwillonlydifferbyamatrixA=A(λ),whichisindependentofz.Thatis,we
evhaΦ0=A(λ)Φ1.(3.8.55)
ThematrixAcanbeexplicitlycomputed:
Lemma3.38.√01−101
A=−i√µ1R0Sλκκ1102λκ02κ12S−1R1−1,(3.8.56)
µ√01112
√λXj01−λ−1
0whereRj=µj√λ−1XjandS=√12λ1.
õjProof.First,usingequations(3.8.52),(3.7.27)and(3.7.28),wehave
Φ0=αδ00β00κκ0211κκ0212yν12y11,(3.8.57)
0101y11
y1112νy12
Φ1=α1β1yν11y11.(3.8.58)
δ11ν12y12
yieldsThisA=Φ0Φ1−1=α0β0κ1102κ1202α1β1.(3.8.59)
0101−1
δ00κ11κ12δ11
er,vmoreoSince,jXjXjj−1
δjj2λXj112µj01
αjβj=(i)µj−µj=√(i√)RjSλ0,(3.8.60)
41

(3.8.56)

(3.8.57)(3.8.58)(3.8.59)(3.8.60)

and,consequently,−1
αjβj=(−i)j√2√µjλ0S−1R−1,(3.8.61)
δjj01j
withupendew√−10101√01−101
A=−i√µ1R0Sλ0κ0211κ0212λ0S−1Rj−1=−i√µ1R0Sκ1102λ02κ12S−1R1−1.
µ001κ11κ1201µ0λκ11κ12
(3.8.62)

Moreover,wehavethefollowingresult:
Lemma3.39.ThematrixAdefinedin(3.8.56)satisfies
det(A)=1.(3.8.63)
articular,pInκ1101κ1202−κ1201κ1102=−µ0.(3.8.64)
µ1Proof.SinceΦ0andΦ1takev−1aluesinΛrSL(2,C)σ,theidentitydet(A)=1followsdirectlyfrom(3.8.55).
Moreover,asdet(Rj)=det(R1)=det(S)=det(S−1)=1,weobtaininviewof(3.8.56)
√01−101
1=det(A)=det(−i√µ1κ1102λ02κ12)=−µ1(κ1101κ1202−κ1201κ1102),(3.8.65)
µ0λκ11κ12µ0
whichalsoprovestherelation(3.8.64).
Takingintoaccount(3.8.55),wearenowabletoexplicitlycomputethemonodromymatricesofΦ0at
z0andz1,aswellasthemonodromymatricesofΦ1atz0andz1.SinceΦ0andΦ1arelinkedby(3.8.55),
itsufficestoconsiderthesolutionΦ=Φ0.
solutionThe3.40.TheoremΦ=Φ0=AΦ1(3.8.66)
tothedifferentialequation(3.8.9)satisfies
Φ(z,γj,λ)=Mj(λ)Φ(z,λ),(3.8.67)
ewherM0(λ)=−e2πiD0=−cos(2πµ0)10+isin(2πµ0)X00µ0,(3.8.68)
X0
100µ00X1
M1(λ)=−Ae2πiD1A−1=−cos(2πµ1)10+isin(2πµ1)AX1µ1A−1.(3.8.69)
100µ1Proof.By(3.8.37),weknowthatΦj(z,γj,λ)=Mj∗(λ)Φj(z,λ),whereMj∗(λ)=−e2πiDj.Consequently,
(3.8.55),usingΦ(z,γ0,λ)=Φ0(z,γ0,λ)=M0∗(λ)Φ0(z,λ)=M0∗(λ)Φ(z,λ),(3.8.70)
Φ(z,γ1,λ)=AΦ1(z,γ1,λ)=AM1∗(λ)A−1AΦ1(z,λ)=AM1∗(λ)A−1Φ(z,λ).(3.8.71)
esvproThis

M0(λ)=M0∗(λ)=−e2πiD0,
M1(λ)=AM1∗(λ)A−1=−Ae2πiD1A−1.
Finally,referringto(2.6.23)(notethatbyassumptionsj>tjforj=0,1),wehave
e2πiDj=RjSe−20πiµS−1Rj−1.
2πiµj
je042

(3.8.66)(3.8.67)

(3.8.72)(3.8.73)(3.8.74)

Usinge±2πiµj=cos(2πµj)±isin(2πµj),theaboveequationyields
Xj0e2πiDj=cos(2πµj)I+isin(2πµj)RjSσ3S−1Rj−1=cos(2πµj)I+isin(2πµj)Xjµ0j.(3.8.75)
µjAltogether,thisfinishestheproof.
theRemarkdifferen3.41.tialInequationviewof(3.8.9)remarkis(at3.26least)andwtheelldefineddefinitionsandofΦ0,holomorphΦ1andicAin,λourforλsolution∈C∗\Φ(=WΦ0∪=WA∪ΦW1to).
ThesameholdsforthemonodromymatricesM0andM1ofΦ.123
RecallingthestructureofthesetsW1,W2andW31(inparticularthefactthat∗noneofthesesets
pconossessestainsforanyatleastaccumsomeulationrp∈oin(0,ts1)onantheopenunitanncircleulusScon),wtaininegnotetherthat-circletheCset(r).C\(More(W1∪Wprecisely2∪,Wan3)y
r∈(0,1)“closeenough”to1willdo.)
3.9Simultaneousunitarizationofthemonodromymatrices
Intheprevioussection,wehavedeterminedasolutionΦto(3.8.9),givenexplicitlyby
Φ=Φ0=eln(z)D0P0V+−,01=δ0y01+ν0y02α0y01+β0y02,(3.9.1)
α0y01+β0y02
νδ0y01+0y02
wherethefundamentalsystemy01,y02isgivenby(3.7.23)and(3.7.24)andtheconnectioncoefficients
αD0,⊆β0,Mδ0,and0thareusdefinedinducesinby(3.8.53)remarkand3.8a(3.8.54).solutionΦΨis=Φ◦definedπto(inthez)ondifferenthetialdoubleequationcutcomplexplane
dΨ=Ψ˜η,(3.9.2)
whichisdefinedforz∈M˜.
excludedByfromsurroundingthezthe-domainofsingularitiesdefinitionz0andofΦ,z1ΦinpicMks,upandthethusmono“crossingdromythematricescuts”,M0(whicλ)handhaveM1b(λeen),
respectively,whichwehaveexplicitlycomputedintheorem3.40:
0X0
M0(λ)=−e2πiD0=−cos(2πµ0)0110+isin(2πµ0)X0µ00,(3.9.3)
µ00X1
M1(λ)=−Ae2πiD1A−1=−cos(2πµ1)0110+isin(2πµ1)AX1µ01A−1.(3.9.4)
µ1Asexplainedearlier,M0(λ)andM1(λ)arealsothemonodromymatricesofΨwithrespecttothe
coveringtransformationsγ˜0andγ˜1,respectively,whichcorrespondtotheloopsγ0andγ1inMenclosing
thesingularitiesz0andz1,respectively:
Ψ(γ˜j(z),λ)=Mj(λ)Ψ(z,λ)j=0,1.(3.9.5)
Byremark3.41,Φ(andthusΨ)iswelldefinedandholomorphicinλforλ∈C∗\(W1∪W2∪W3).
RecallingthedefinitionsofthesetsW1,W2andW3,respectively,(wr)eobservethatC∗\(W1∪W2∪W3)
conenoughtainsto(a1.sufficien(Since,tlyassmallmenoptionedeneneigharblier,orho1o∈dWof2),antherunit-circleCcircleSfor1issomenot0con<rtained<1,inwhicC∗h\is(W1close∪
W2neigh∪bW3orho).)odThisof)animpliesr-circlethatCΦ(r).andThΨus,arecarryinparingticoutularanr-Iwholomorphicasawa(indecompλ)onosition(ainsufficienthetlysecondsmallstepopofen
theloopgroupmethod,Ψproducesbyevaluatingtheassociatedextendedframe(whichisholomorphic
forλintheannulusA(r))at3λ=1aCMC-immersionψ:M˜→R3.Referringtotheorem2.11,ψyieldsa
2.11,or,CMC-immersionbyremarkφ:M3.19→Requivifalenandtly,onlyififandtheonlymonoifthedromymatri“generating”cesofΨmonomeetdromtheymatrconditiicesonsMof0andtheoremM1
offorΨλ∈meetS1,theMiscondiintionsgeneralofnottheoremunitary2.11.forHoλw∈evS1er,(cf.inappgeneral,endixthisD).isnotthecase:WhileM0isunitary
1Therefore,inthefollowing,wemodifyΨtoˆobtainanothersolutionΨˆtothedifferentialequation
(3.9.2)theorem2.with11.More(generating)preciselymono,wedrommoydifyΨmatricesbyanMj(jappropri=0,ate1),λ-depwhichendenactutallydressingmeetthematrixTconditions=T(λ),of

43

suchthatthemonodromymatricesMˆj=TMjT−1ofthedressedsolutionΨˆ=TΨsatisfytheconditions
oftheorem2.11.(NotethattheexistenceofsuchamatrixTisprovidedbytheorem3.14.)
totheNotedifferenthat,btialysectionequation2.2of(3.9.2),[17],whicanyhisCMC-immersionholomorphicinψλ:∈M˜C∗→R(cf.3cansectionbe2.2obtainedof[17])fromandathussolutioncomesΨ˜
∗alongassumewiththatmonothedrommonoydromymatrices,matriceswhichMˆjare=TMjholomorphicT−1(jin=λ0,∈1)Cofasthewell.dressedTherefore,solutionweΨˆ=TadditionallyΨare
holomorphicinλ∈C∗.
Altogether,ournextgoalwillbetocomputeexplicitlyadressingmatrixT=T(λ),suchthatthe
λmono∈C∗dromandysatisfymatricestheMˆj=conditionsTMjTof−t1he(orje=m0,2.11,1)ofi.e.thesuchdressedthatforjsolution=0,1Ψˆ=TΨareholomorphicin
Mˆj(λ)isholomorphicinλ∈C∗,(3.9.6)
Mˆj(λ)isunitaryforallλ∈S1,(3.9.7)
Mˆj(λ=1)=±Iand(3.9.8)
∂λMˆj(λ)|λ=1=0.(3.9.9)
difficultRemarky.This3.42.isAmongemphasizedthebyconditionsthename(3.9.6)ofthetocurren(3.9.9),tsection.satisfyingthecondition(3.9.7)posesthemain
Remark3.43.OurstartingsolutionΨtothedifferentialequation(3.9.2),aswellasthecorresponding
∗viamonodressingdromybyTmatricesfromMΨjaarenewholomorphicsolutionΨˆin=λTonΨthewithdomainmonoCdrom\y(W1matrices∪W2∪MˆjW=3).TInMjTorder−1,towhichobtainare
∗matricesholomorphicMjinofλΨ.∈C,ConseTquennecessarilytly,itisneedsnottoonlyp“cancel”ossiblethebutexistingevensinprobable,gularitiesthatfromTtheitselfmonopossessesdromy
singularitiesinC∗andthuswillnotbewelldefinedonthewhole(punctured)λ-planeC∗.
nowInignortheefollotheλwing,-domainwewillofdefinicomputetionoftheT,pdressingostponingmatrixthisTissuepurelytoremarkformally.3.57.Inparticular,wewillfor
givenLetbyΦb(3.9.3)easinand(3.9.1),(3.9.4).ΨW=eΦw◦anπttoandthedeterminea(common)dressingmonomatrixdromyT=Tmatrices(λ),Msucjh(jthat=0,the1)ofmonoΦanddromΨy
0matricesMˆj=TMjT−1(j=0,1)ofthedressedsolutionΨˆ=TΨsatisfytheconditions(3.9.6)Xtoj(3.9.9).
First,wecomputethegeneralformofMˆj:ObservingthatanyconjugateofthematrixXjµjbya
0prµj
λ-dependentmatrixwillbeoftheformqjj−jpjforsomeλ-dependentfunctionspj,qj,rj,thedressed
monodromymatricesMˆjareinviewof(3.8.68)and(3.8.69)ofthegeneralform
Mˆj=TMjT−1=−cos(2πµj)0110+isin(2πµj)qpjj−rpjj,(3.9.10)
where0X0
qp00−rp00=TX0µ00T−1,(3.9.11)
µ0X1
qp11−rp11=TAX01µ01A−1T−1.(3.9.12)
µ1Sincedet(Mj)=1andconjugatingamatrixdoesnotaffectitsdeterminant,wemoreoverinferthatthe
functionspj,qj,rjsatisfy
pj2+qjrj=1.(3.9.13)
Remark3.44.InordertoensurethatthemonodromymatricesMˆ,j=0,1oftheform(3.9.10)are
elementsofthetwistedloopgroupΛrSU(2)σwefurthermorerequirejforj=0,1that
pjisanevenfunctionofλ,
qjisanoddfunctionofλ,(3.9.14)
rjisanoddfunctionofλ.
44

rjoInccurringviewofinthe(3.9.10),dressedwereformmonoulatedromythematricesconditionsMˆj.T(3.9.6)hetocondition(3.9.9)in(3.9.6)termsoftranslatestheinfunctionstothepj,folloqjwingand
constraintsonthefunctionspj,qjandrj(j=0,1):
sin(2πµj)pjisholomorphicforλ∈C∗,
sin(2πµj)qjisholomorphicforλ∈C∗,(3.9.15)
sin(2πµj)rjisholomorphicforλ∈C∗.
(Writingcos(2πµj)initspowerseriesrepresentation,
k2∞cos(2πµj)=(−1)k(2π(2µkj)!),(3.9.16)
=0kwephicseethatfunctioncos(2onπCµ∗j,)cos(2onlyπinµvo)lvisesevenholomorphicpowersforofλµj.∈C∗Since,.)byremark3.13,µj2isdefinedasaholomor-
jInordertoexpress(3.9.7)inadifferentway,wenotethatamatrixUisinSU(2)ifandonlyifUis
oftheformU=−uv¯u¯vwithuu¯+vv¯=1.Applyingthisto(3.9.10),weseethatMˆj(λ)isinSU(2)
forallλ∈S1ifandonlyifthefunctionspj,qjandrj(j=0,1)satisfy(inadditiontoalreadyexisting
conditions)pj(λ)=pj(λ)forallλ∈S˜1,
(3.9.17)rj(λ)=qj(λ)forallλ∈S˜1,
111˜sin(2whereπµSj(λ))denotes=0.th(Ineunitparticular,spheretheSfunctionexcludingspj,theqjanddiscreterjaresubsetnecesofsarilvyalueswellofdeλfi∈nedSon,fS˜or1.whicNoteh
thatatλ∈S1\S˜1,thisisnotnecessarilythecase:here,pj,qjandrjmightpossesssimplepoles,which
cancelwiththe(simple)zerosofsin(2πµj(λ)).)
Remark3.45.Inviewof(3.9.15),thefunctionspj,qjandrj,andthusalsothefunctionspjandqj
ybdefinedpj(λ):=pj(λ1),(3.9.18)
qj(λ):=qj(λ1),(3.9.19)
respectively,areatleastholomorphiconthe(common)subdomainofC∗,where(forallj∈{0,1,∞})
µj(λ)isholomorphicandsin(2πµj(λ))=0,i.e.onC∗\(W1∪W4),whereW1isgivenin(3.7.5)and
W4={λ∈C∗;sin(2πµj(λ))=0}.(3.9.20)
∗C(As∗\(Wstated1∪W4earlier,).Moreothever,mappingsthesetsµj,Wj1=and0,1,W4∞,areare“symmetric”holomorphicwithonCresp\ectW1toandthethusunitincirlceparticularS1intheon
sensethat(fork=1,4)λ∈Wifandonlyif1∈W.Thisensuresthat,togetherwiththefunctions
pj,qjandrj,alsothefunctionskpjandqjarewλelldefinedk(andholomorphic)onC∗\(W1∪W4).)
Noteforlateruse1(cf.remark3.57)that,as∗W1before,W4doesnotpossessanyaccumulationpoints
onanntheulusunitconcircletainingS.ther-circleConsequenC(rtly),.the(MoresetC\precisely(W1,∪anWy4r)∈con(0,1)tainsfor“closeatleastenough”someto1r∈will(0,do.)1)anopen
Combiningtheresultabovewith(3.9.17),weconcludethatpjandpj(resp.rjandqj)defineholo-
morphicfunctionsonC∗\(W1∪W4),whichcoincideonS˜1:
1pj(λ)=pj(λ)=pj(λ)=pj(λ),(3.9.21)
rj(λ)=qj(λ)=qj(λ1)=qj(λ).(3.9.22)
Consequently,pjandpj(resp.rjandqj)coincideeverywhereonC∗\(W1∪W4),i.e.wecanactually
yb(3.9.17)replacepj=pj,
rj=qj.(3.9.23)
45

Finally,bytwosimplecalculationsbasedontherelationsµj(λ=1)=21and(∂λµj)(λ=1)=0
fromlemmaB.3,theconditions(3.9.8)and(3.9.9)translateintothefollowingfurtherconstraintsonthe
functionspj,qjandrj(j=0,1):
pj,qj,rjtakefinitevaluesinCforλ=1,(3.9.24)
pj,qj,rjareholomorphic(inλ)atλ=1.(3.9.25)
Inthefollowing,wefocuson(3.9.7),thefirstconditionoftheorem2.11.Thus,ournextgoalwillbe
tocomputeexplicitlyadressingmatrixT=T(λ),suchthatΨˆ=TΨhasunitarymonodromymatrices
Mˆ0andMˆ1forallλ∈S1.Inviewofourconsiderationsabove,weseethatMˆj(j=0,1)isunitaryand
ofdeterminant1onS1ifanonlyifitisoftheform
Mˆj=−cos(2πµj)10+isin(2πµj)pjqj(3.9.26)
01qj−pj
withλ-dependentfunctionspj,pj(definedin(3.9.18)),qjandqj(definedin(3.9.19))satisfying
pj2+qjqj=1andpj=pj(3.9.27)
and

0X0
qp0−qp0=TX0µ0T−1,(3.9.28)
000µ0p1q1=TAX0µ1A−1T−1.(3.9.29)
X1
q1−p1µ110
Remark3.46.Wecanassumethatq0,q1≡0onS1:If,byaccident,someTshouldgiveqj≡0forsome
j,itiseasytoshowthatwecanmodifyTbymultiplyingfromtheleftbyaunitarymatrixU=U(λ)
suchthatTˆ=UTwillyieldq0,q1≡0onS1,whileunitarityofthedressedmonodromymatricesonS1
willbemaintained.Thatis,ifsimultaneouslyunitarizingM0,M1ispossibleatall,itisalsopossibleina
waysuchthatq0,q1≡0.Notethatmo3difyingTbyUwillresult(only)inarotationand/ortranslation
ofthegeneratedCMC-surfaceM→R.
Remark3.47.Notethat(3.9.26)holdsalsoforj=∞,asthemonodromymatrixMˆ∞isaunitary
conjugateoftheDelaunaymonodromymatrix−e2πiD∞.Notethat,however,inthespecialcasethat
s∞=t∞=41,wehaveµj(λ)=41(λ+λ−1),i.e.µjdefines(incontrasttothegeneralcases∞=t∞)an
oddfunctionofλ.Thusinthecases∞=t∞=41,theconditionthatthemonodromymatrixMˆ∞isan
elementofthetwistedloopgroupΛrSU(2)σisequivalenttorequiring
pjisanoddfunctionofλ,(3.9.30)
qjisanevenfunctionofλ,
replacingtherelations(3.9.14),whichapplyforj=0,1andforj=∞inthecasethats∞=t∞.
Remark3.48.Asaconsequenceofequation(3.3.12),theunitarizedmonodromymatricesMˆjsatisfy
Mˆ0(λ)Mˆ1(λ)Mˆ∞(λ)=TM0(λ)T−1TM1(λ)T−1TM∞(λ)T−1=TM0(λ)M1(λ)M∞(λ)T−1=I,(3.9.31)
i.e.Mˆ0(λ)Mˆ1(λ)Mˆ∞(λ)=I.(3.9.32)
RewritingthisasMˆ∞=Mˆ1−1Mˆ0−1,Mˆ1=Mˆ0−1Mˆ∞−1orMˆ0=Mˆ∞−1Mˆ1−1,respectively,andapplying
(3.9.26),weobtainthefollowingthreepairsofscalarequations,whereeachpairisequivalentto(3.9.32).
Thefirstpairofequationsreads
cos(2πµ∞)+isin(2πµ∞)p∞=−cos(2πµ0)cos(2πµ1)+icos(2πµ0)sin(2πµ1)p1
(3.9.33)+isin(2πµ0)cos(2πµ1)p0+sin(2πµ0)sin(2πµ1)(p0p1+q0q1),
isin(2πµ∞)q∞=icos(2πµ0)sin(2πµ1)q1+isin(2πµ0)cos(2πµ1)q0
(3.9.34)+sin(2πµ0)sin(2πµ1)(p0q1−p1q0).
46

Thesecondpairofequationsreads
cos(2πµ1)+isin(2πµ1)p1=−cos(2πµ∞)cos(2πµ0)+icos(2πµ∞)sin(2πµ0)p0(3.9.35)
+isin(2πµ∞)cos(2πµ0)p∞+sin(2πµ∞)sin(2πµ0)(p∞p0+q∞q0),
isin(2πµ1)q1=icos(2πµ∞)sin(2πµ0)q0+isin(2πµ∞)cos(2πµ0)q∞
(3.9.36)+sin(2πµ∞)sin(2πµ0)(p∞q0−p0q∞).
Thethirdpairofequationsreads
cos(2πµ0)+isin(2πµ0)p0=−cos(2πµ1)cos(2πµ∞)+icos(2πµ1)sin(2πµ∞)p∞(3.9.37)
+isin(2πµ1)cos(2πµ∞)p1+sin(2πµ1)sin(2πµ∞)(p1p∞+q1q∞),
isin(2πµ0)q0=icos(2πµ1)sin(2πµ∞)q∞+isin(2πµ1)cos(2πµ∞)q1(3.9.38)
+sin(2πµ1)sin(2πµ∞)(p1q∞−p∞q1).
Equations(3.9.26)to(3.9.29)givenecessaryandsufficient1conditionsforTtounitarizebothM0
pand,q,Mp1.,qMoredependingpreciselyon,λTsucwillhthatrendertheM0andequationsM1(3.9.27),unitaryon(3.9.28)Sifandandonly(3.9.29)ifthhold.ereInexistthiscasefunctionsthe
1100unitarizedmonodromymatricesMˆjaregivenby(3.9.26).Inthefollowing,wewilldiscusstheunitarizing
conditions(3.9.28)and(3.9.29)inmoredetail.
Lemma3.49.Theunitarizingconditions(3.9.28)and(3.9.29)holdifandonlyif
(Δ0S)σ3(Δ0S)−1=(TR0S)σ3(TR0S)−1,(3.9.39)
(Δ1S)σ3(Δ1S)−1=(TAR1S)σ3(TAR1S)−1,(3.9.40)
ewher√√λXj0−1−1
j0Rj=µj√λ−1Xj,S=√12λ1−λ1,σ3=01−01,Δj=λ1−1q01λλ−1qpjj.
√µj(3.9.41)Remark3.50.Inorderto∗ensurethatthe(λ-dependent)matricesΔjarewelldefined,wehaveto
excludediscreteanysubsetvaluesofC∗of,λwhic∈hCdoesfromnotourpossessanconsiderations,yaccumforulationwhicphoinqjts(λ)on=the0.unitThesecirclevaluesS1.ofInλorderformtoa
ensurethatthematricesΔjareholomorphicinλ,weexcludeforeachλ∈C∗withqj(λ)=0theradial
cutfromλto0(if|λ|<1)orfromλto∞(if|λ|≥1),respectively,fromourconsiderations.
C∗\W5DenotingofC∗thebyunionconstructionofallthesestillconcutsbtainsyWfor5,wsomeenoter∈for(0,1)lateranuseopen(cf.annremarkuluscon3.57)tainingthatthether-circlesubset
C(r).(Moreprecisely,anyr∈(0,1)“closeenough”to1willdo.)
Proofoflemma3.49.First,recalling(2.6.21),wehave
Xj0Xjµ0j=(RjS)σ3(RjS)−1.(3.9.42)
µjMoreover,astraightforwardcomputationyields
qpjj−qpjj=(ΔjS)σ3(ΔjS)−1.(3.9.43)
Insertingthesetworelationsinto(3.9.28)and(3.9.29),theseequationsreadas
(Δ0S)σ3(Δ0S)−1=(TR0S)σ3(TR0S)−1,(3.9.44)
(Δ1S)σ3(Δ1S)−1=(TAR1S)σ3(TAR1S)−1,(3.9.45)
whichprovestheclaim.
Remark3.51.Thedecompositonσ1=01=Sσ3S−1isconvenientforcomputationalpurposes.
01howevOtherwiseer,theonewmatricesouldLjreaddefined(3.9.39)in(3.and9.48)(3.9.40)belowaswequouldtakationseabetmoreweencomplicatedconjugatesofform.σ1.Inthiscase,

47

(3.9.42)

(3.9.43)(3.9.44)(3.9.45)

Lemma3.52.LetA,B,C∈Gl(2,C).ThenBAB−1=CAC−1ifandonlyifthereexistssomeL∈
Gl(2,C)satisfyingC=BLandLA=AL.
Proof.First,letA,B,C∈Gl(2,C),suchthatBAB−1=CAC−1.−1SettingL−:=1B−1C∈Gl(2,C),we
oboneviouslydirectionhaveofCthe=BLclaim..Moreover,usingtheassumption,LA=BCA=ABC=AL,whichproves
1−CLF−1orALCthe−1other=CL−1direction,LAC−let1=A,CB,ACC,−L1.∈Gl(2,C),suchthatC=BLandLA=AL.Then,BAB=
Bytheabovelemmas3.49and3.52,equations(3.9.39)and(3.9.40)areequivalentto
TR0S=Δ0SL0,(3.9.46)
TAR1S=Δ1SL1(3.9.47)
forMoreosomever,sincematricestheLjwhicmatriceshTcomm,Rjute,S,withΔjσ3and.AThishaveimpliesindeterminantparticular1(fortThat,thethisfollomatriceswsLfromjarethedrelationiagonal.
Ψˆ=TΨ),equations(3.9.46)and(3.9.47)implydet(Lj)=1aswell.Henceweobtain
Lj=ω0jω0−1,(3.9.48)
jwhereWithωj=theseωj(λ)preparationsdenotessomemade,λw-depecanendenprotvefunction.thefollowingtheorem.
M0TheoremandM13.53.simultaneLetM0ouslyandonMS11beifgivenandbyonlyif(3.9.3)itisandofthe(3.9.4)form,respectively.Then,amatrixTunitarizes
√
T=√1λ−1X0(ω0+ω0−1)+p0(−ω10−ω0−1)λ−1λX0−1(√ω0−ω0−1)+p0(−ω10+ω0−1),
2µ0λ−1q0λ−1X0q0(ω0−ω0)λλX0q0(ω0+ω0)
(3.9.49)forfunctionsp0,q0,p1,q1,ω0andω1ofλ,whichsatisfy
pj2+qjqj=1andpj=pjforj=0,1,(3.9.50)
p0p1+q0q12+q0q1=cos(2πµ0sin(2)πcos(2µ)πµ1sin(2)+πµcos)(2πµ∞),(3.9.51)
10√ω0=δκ1202√−q0+q1−p0q1+p1q0,(3.9.52)
√κ1201q0+q1+p0q1−p1q0
κ12q0+q1−p0q1+p1q0
ω1=δ˜κ011101√−q0+q1−p0q1+p1q0,(3.9.53)
whereδ,δ˜∈{±1},suchthat
√√√δδ˜=√µ0q0+q1+p0q1−p1q0q0+q101−p0q102+p1q0.(3.9.54)
µ1−2iλλ−1q0λ−1q1λκ11κ12
Moreover,ifTisoftheform(3.9.49),theunitarizedmonodromymatricesMˆj=TMjT−1aregivenby
.(3.9.26)Proof.Asexplicatedearlier,amatrixTunitarizesM0andM1simultaneously(forλ∈S1)ifandonly
pifj2T+qjqjsatisfies=1theandpj=equationspj.Moreo(3.9.39)ver,andpresuming(3.9.40)Tforsimλ-depultaneouslyendentfunctionsunitarizesp0M,0q0,andp1M1and,theq1unitarizedsatisfying
monodromymatricesMˆj=TMjT−1arethenoftheform(3.9.26).Thus,itremainstoprove,thatT
satisfiesAsstated(3.9.39)before,and(3.9.39)(3.9.40)ifandand(3.9.40)onlyifareitisofequivthealentformtothe(3.9.49).equations(3.9.46)and(3.9.47).Wecan
furthertransformtheseequationsequivalentlyinto
T=Δ0SL0S−1R0−1,(3.9.55)
S−1R0−1AR1S=L0−1S−1Δ0−1Δ1SL1.(3.9.56)
48

Thus,Tsatisfies(3.9.39)and(3.9.40)ifandonlyifthereexistλ-dependentfunctionspj,qjandωj,such
that(3.9.56)holds.(Ofcourse,westillneedtoadditionallyensurethattheconditionsgivenin(3.9.27)
aremet.)Oncewehavefoundfunctionsωj,pj,qjsatisfying(3.9.56)and(3.9.27),weareabletocompute
Tfrom(3.9.55),thatis
1λ−1X0(ω0+ω0−1)+p0(ω0−ω0−1)λ−1√λX0(ω0−ω0−1)+p0(ω0+ω0−1)
T=2√µ0λ−1q0λ−1X0q0(ω0−ω0−1)λ−1√λX0q0(ω0+ω0−1),
(3.9.57)claimed.asTofinishtheproof,wenowfocusonequation(3.9.56),whichreadsmoreexplicitly(cf.(3.8.56)for
)Amatrixthe−i√√µ1κ1101λ−1κ1201
µ0λκ1102κ0212
1ω0−1ω1λ−1(q0+q1−p0q1+p1q0)ω0−1ω1−1λ−1(−q0+q1−p0q1+p1q0)
=2λ−1q0λ−1q1ω0ω1(−q0+q1+p0q1−p1q0)ω0ω1−1λ−1(q0+q1+p0q1−p1q0).
(3.9.58)Naturally,thismatrixequationgivesrisetofourscalarequations.Bytakingintoaccount(3.7.19)to
(3.7.22)computationsand(3.9.27),necessarythesetocarrymayboutethisequivtranalensftlyormationtransformedinvolvinetothetheusefolloofwinsomegidenthreetitiesequations.forGammaThe
functions,butarestraightforwardapartfromthatandgivenexplicitlyinappendixE.Weendupwith
√ω0=δκ1202√−q0+q1−p0q1+p1q0,(3.9.59)
κ1201√q0+q1+p0q1−p1q0
ω1=δ˜κ1101√−q0+q1−p0q1+p1q0,(3.9.60)
κ1201q0+q1−p0q1+p1q0
p0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞),(3.9.61)
2sin(2πµ0)sin(2πµ1)
whereδ,δ˜∈{±1},suchthat
√√√δδ˜=√µ0q0+q1+p0q1−1−p1q0−1q0+q101−p0q021+p1q0.(3.9.62)
µ1−2iλλq0λq1κ11κ12
equationsThus,abovequatione.(Again,(3.9.56)noteholdsthatifwandeonlyadditionallyiftheinvassumeolvedallfunctionsthetimepj,qthatjandtheωjconditionssatisfythegiventhreein
(3.9.27)aremet.)Altogether,theclaimisproved.
Corollary3.54.LetM0andM1begivenby(3.9.3)and(3.9.4),respectively.Then,tofindamatrixT,
whichsimultaneouslyunitarizesM0andM1onS1,onecanproceedasfollows:
1.Solve(3.9.51)forfunctionsp0,q0,p1,q1satisfying(3.9.50).
2.Computeω0from(3.9.52).
3.ComputeTfrom(3.9.49).
Remark13.55.Wewouldliketoremarkthatequation(3.9.51)issolvableforfunctionsp0,q0,p1,q1in
pλ∈otenStialηassatisfyingexplicated(3.9.50)inifsectionandonly3.5ifmeetthetheeigenunvaluesitarizabilitµjofytheconditionDelaunay(3.5.28)matricesforallDλj∈S1inducing.Thistheis
provedinappendixF.
Remark3.56.Givenfunctionsp0,q0,p1,q1satisfying(3.9.27)and(3.9.51),thecorrespondingfunctions
pandq(withp2+qq=1)occurringinthemonodromymatrixMˆcanbeexplicitlycom-
∞putedfrom∞equations∞(3.9.33)∞∞and(3.9.34)representingthematrixidentityM∞ˆ0Mˆ1Mˆ∞(λ)=I.Bya
straightforwardcomputation,whichisgiveninappendixG,wecanprovethefollowingstatement:For

49

j=0,1,∞,letpj,qjbethefunctionsoccurringintheunitarymonodromymatrixMˆjasin(3.9.26),
satisfying(3.9.27).Inviewoftheidentity(3.9.32),wehave
p0,q0,p1,q1solvep0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞)
2sin(2πµ0)sin(2πµ1)
andp∞,q∞aregivenby(3.9.33)and(3.9.34)
⇐⇒p0,q0,p∞,q∞solvep0p∞+q0q∞+q0q∞=cos(2πµ0)cos(2πµ∞)+cos(2πµ1)
2sin(2πµ0)sin(2πµ∞)(3.9.63)
andp1,q1aregivenby(3.9.35)and(3.9.36)
⇐⇒p1,q1,p∞,q∞solvep1p∞+q1q∞+q1q∞=cos(2πµ1)cos(2πµ∞)+cos(2πµ0)
2sin(2πµ1)sin(2πµ∞)
andp0,q0aregivenby(3.9.37)and(3.9.38).
Remarkpreceeding3.57.considerationsReturningaretovalid.remarkFirst,3.43,recallwenofromwremturnartokthe3.41,thatquestion,–forforthewhictimehbveingalues–ofweλhavoure
∗nowrestrictedwhatλwetohavtheedonedomaininCthis\(Wsection,1∪Ww2e∪Wobserv3),ewherethat,inevordererythingtokiseepallholomorphicexpressionsinλ.welldefinedRecapitulatingand
holomorphicinλ,weneedtoexcludefurtherpointsfromtheλ-domain.Namely,thisisnecessarywhen
dealingwiththefunctionspj,qjandqjoccurringintheunitarizedmonodromymatricesMˆj(excluding
thesubsetsW1andW4ofC∗from∗ourconsiderations,cf.remark3.45)andwhenintroducingthematrices
Δ(excludingthesubsetWofC,cf.remark3.50).
jConsequently,ourcomp5utationsarevalidforallλfromtheλ-domain
C∗\(W1∪W2∪W3∪W4∪W5).(3.9.64)
Inparticular,thematrixTiswelldefinedandholomorphicforλ∈C∗\(W1∪W2∪W3∪W4∪W5).
Forourstudy,itiscrucialtoobservethatthesetC∗\(W1∪W2∪W3∪W4∪W5)containsanr-circle
C(r)forsomer∈(0,1).Moreprecisely,C∗\(W1∪W2∪W3∪W4∪W5)containsforeachr∈(0,1)
“closeenough”to1anopenannuluscontainingther-circleC(r).(Cf.remarks3.41,3.45and3.50for
(this.)r)Altogether,thematrixTisinparticularwelldefined(andholomorphic)forλfromsomer-circle
Candthusactuallydefinesanr-dressingmatrix(ashighlydesired).
theRemarkfollowing,3.58.Bygeneralizedaslighretsulmot,whicdificationhdoofesthenotrequirecomputationsexplicitknocarriedwledgeoutinofapptheendixconnectionE,onecocanefficienprovtse
κ01,κ01,κ02andκ02:
11Let12M011andM112begivenby(3.9.3)and(3.9.4),respectively.Then,amatrixTunitarizesM0andM1
simultaneouslyonS1ifandonlyifitisoftheform
√
T=√1λ−1X0(ω0+ω0−1)+p0(ω0−ω0−1)λ−1λX0(√ω0−ω0−1)+p0(ω0+ω0−1),
2µ0λ−1q0λ−1X0q0(ω0−ω0−1)λ−1λX0q0(ω0+ω0−1)
(3.9.65)forfunctionsp0,q0,p1,q1,ω0andω1ofλ,whichsatisfy
pj2+qjqj=1andpj=pj,(3.9.66)
µ−4q0q1µ01κ1101κ1202=(q0+q1)2−(p0q1−p1q0)2,(3.9.67)
µ−4q0q1µ01κ1102κ1201=(q0−q1)2−(p0q1−p1q0)2,(3.9.68)
√κ1201q0+q1+p0q1−p1q0
ω0=δκ1202√−q0+q1−p0q1+p1q0,(3.9.69)
√01κ12q0+q1−p0q1+p1q0
ω1=δ˜κ0111√−q0+q1−p0q1+p1q0,(3.9.70)
whereδ,δ˜∈{±1},suchthat
√√√δδ˜=√µ0q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0.(3.9.71)
µ1−2iλλ−1q0λ−1q1κ1101κ1202
50

Moreover,ifTisoftheform(3.9.65),theunitarizedmonodromymatricesMˆj=TMjT−1aregivenby
(3.9.26).Insertingtheconnectioncoefficientsκ1101,κ1201,κ1102andκ1202from(3.7.19)to(3.7.22)restoresthestate-
mentoftheorem3.53.(Actually,thisisprovedinlemmaE.1.)
Wecannowstatethefollowingresult:
Theorem3.59.LetM=C\{0,1},M˜=Handπ:M˜→Mtheuniversalcovering∗mapasdefinedin
(3.2.2).Letηbeastandardizedtrinoidpotentialoftheform(3.6.16)onMandη˜=πηthecorresponding
˜givenstandarindized(3.9.1)trinoidandpΨ=otentialπ∗ΦontheMc.orrMorespeondover,ingletΦsolutionbethetothesolutiondiffertoentialtheedifferquatioentialndΨe=quationΨη˜.dΦThen,=Φtoη
findadressingmatrixT=T(λ),whichdr∗essesthesolutionΨintoanewsolutionΨˆ,whichp˜ossesses3
holomorphicmonodromymatricesinλ∈CandwillproduceadescendingCMC-immersionM→R
viatheloopgroupmethod,onecanproceedasfollows:
1.Findλ-dependentfunctionsp0,q0,p1,q1satisfying
(a)sin(2πµj)pjandsin(2πµj)qjareholomorphicforλ∈C∗(holomorphicitycondition),
(b)pjisanevenfunctionofλ,qjisanoddfunctionofλ(twistingcondition),
(c)pj2+qjqj=1(determinant1condition),
(d)pj=pj(unitaritycondition),
(e)p0p1+q0q12+q0q1=cos(2πµ0sin(2)πcos(2µ0)πµ1sin(2π)+cos(2µ1)πµ∞)(simultaneousunitarizabilitycondition),
(f)pjandqjtakefinitevaluesinCandareholomorphicatλ=1(closingcondition).
2.Computeω0from(3.9.52).
3.ComputeTfrom(3.9.49).
Remark3.60.Theorem3.59allowsfortheconstructionofafamilyofdressingmatricesT=T(λ),which
dressaspecialstartingsolutionΨtothedifferentialequationdΨ=Ψη˜intonew˜solu3tionsΨˆ=TΨ,
whichwill(viatheloopgroupmethod)produceadescendingCMC-immersionM→Randthusa
trinoidφ:M→R3inthesenseofdefinition3.1.Bytheorem3.14,thechoiceofanappropriatedressing
remarkmatrixT3.18,willallensuretrinoidsthatwiththepropcorresperlyemondingbeddedtrinoidannφularhasendspropcanerblyeembobtainededded(byanndiffereularntends.choicesInofviewTof).
Nevertheless,weconjecturethatthegivenfamilyofdressingmatricesiseven“larger”inthesensethat,
forappropriatechoicesofT,eventrinoidsM→R3with“non-properly3embedded”or“non-embedded”
sucendshcantrinoidsbeprowouldduced.notItisnotnecessarilyclearshohowwthethecorrespasymptoticondingbehasurfacesviourofinRDelaunawouldysulookrfaceslik.e.ForIntheparticular,restof
thisthesis,werestrictourconsiderationstotrinoidswithproperlyembeddedannularends.
Remark3.61.EachdressingmatrixT=T(λ)providedbytheorem3.59producesasolutionΨˆ=TΨ
tothedifferentialequationdΨ=Ψη˜(3.9.72)
ˆˆ∗withholomorphicmonodrom(inyλ)onmatricesC∗.MNevj,whicertheless,harebasedholomorphiconremarksinλ∈3.21C.andHo3.39wevofer,[18],Ψwitselfehaisvenotthefollonecessarilywing
result:AssumeTisasintheorem3.59.Denote∗byΨˆ=TΨthecorrespondingsolutionto(3.9.72),whichhas
holomorphic˜mono3dromymatrices˜inλ∈CandgeneratesviatheloopgroupmethodadescendingCMC-
solutionimmersionΨ˜=MT˜→ΨRto.(3.9.72),AssumeTwhicishhasanotherholomorphicdressingmatrixmonopdromroyvidedbmatricesytheoreminλ∈C3.59,∗,whicgenerateshproviaducesthea
loopgroupmethodadescendingCMC-immersionM˜→R3andisholomorphicinλ∈C∗.Inparticular,
bothTandT˜simultaneouslyunitarizethemonodromymatricesofthestartingsolutionΨ.Thus,by
remarks3.21˜and3.39of[18],TandT˜onlydifferbyanelementU=U(λ)ofΛrSU(2)σ(foranappropriate
r∈(0,1)):T=UT.
Altogether,anydressedsolutionΨˆ=TΨto(3.9.72)obtainedbytheorem3.59isrelatedtoanother
solutionΨ˜=T˜Ψto(3.9.72),whichisholomorphicinλ∈C∗bysomeU∈ΛrSU(2)σ:Ψ˜=UˆΨ.
(BothΨˆandΨ˜haveholomorphicmon˜odrom3ymatricesinλonC∗andproduceviatheloˆopgroup˜
methoddescendingCMC-immersionsM→R.)TheadditionaldressingbyUtransformingΨintoΨ
correspondsontheleveloftheassociatedCMC-immersionstoaλ-dependentrotationand/ortranslation
ofthemembersoftherelatedassociatedfamilies.

51

symmetriesrinoidT4Sofar,wehaveclassifiedtrinoidsφ:M→R3producedbythe(standardized)trinoidpotentialηinterms
offunctionsp0,p1,q0,q1solving(3.9.51).Inthefollowing,wearegoingtorefinethisclassificationby
functionstranslatingp0p,p1,ossibleq0,q1.Fsymmetryromnowpropon,ertiesweofrestricttheimageouroftheconsiderationstrinoidφto(M)trinoiintodswithfurtherpropconstrainerlyemtsboneddedthe
ends.ularannAsp0,p1,q0,q1app˜earin3themonodromymatricesoftheholomorphicframeΨassociatedwiththe
mappingψ:=φ◦π:M→R(andthus,byremark2.10,inthemonodromymatricesoftheextended
frameFassociatedwithψ),wetranslateanysymmetry˜propertiesofφintosymmetrypropertiesofψ
andtransferthesetotheleveloftheextendedframeF:M→ΛSU(2)σafterwards.
Remark4.1.Boththedefinitionsandtheresultsofthesections4.1and4.2arevalidandthusformulated
inthegeneralizedsettingofanarbitrary3Riemann˜surface3MwithuniversalcoverM˜anda˜pairof
ψ=conformalφ◦π.WereturnCMC-immersionstotheφ:trinoidM→Rsettingandwithψ:MM=→CR\{0link,1}edandviaMth˜e=univHinersalsectioncovering4.3.π:M→M,
Definitions4.1LetIso(R3)denotetheisometrygroupofR3,i.e.thegroupofalldistancepreservingaffineisomorphisms
onR3withrespecttotheEuclideanmetric
d:R3×R3→R,d(x,y)=(y1−x1)2+(y2−x2)2+(y2−x2)2.(4.1.1)
TheelementsofIso(R3),oftenreferredtoasEuclideanmotionsonR3,areoftheform
T:R3→R3,x→T(x):=ATx+tT,(4.1.2)
3anywhereelemenATtofdenotesIso(Ra3)(reisal)comporthogonalosedof3an×orthogonal3-matrix,andtTtransformationdenotesaandatranslationtranslatvion.ectorinMoreoRv.er,ThTus,∈
Iso(R3)oftheform(4.1.2)preservesorientationonR3ifandonlyiftheassociatedmatrixATsatisfies
3det(orienATtation)=on+1,R3i.e.ififandandonlyonlyififtheATasso∈ciatedSO(3).matrixAccordinglyAT,Tsatisfies∈Iso(det(RA)Tof)=the−1,formi.e.if(4.1.2)andrevonlyerseifs
AT∈O(3)\SO(3).
GivenaconformalCMC-immersionφ:M→R3ofaRiemannsurfaceMintoR3,wedefinethe
symmetrygroupofφ(M)by
Sym(φ(M)):={T∈Iso(R3)|T(φ(M))=φ(M)}.(4.1.3)
ForthecorrespondingconformalCMC-immersionψ:=φ◦π:M˜→R3,whereπdenotestheuniversal
coveringM˜→M,wedefineanalogously
Sym(ψ(M˜)):={T∈Iso(R3)|T(ψ(M˜))=ψ(M˜)}.(4.1.4)
Byconstruction,wehaveψ(M˜)=φ(M)andthereforeobviouslySym(φ(M))=Sym(ψ(M˜)).
Definition4.2.Letφ:M→R3beaconformalCMC-immersionofaRiemannsurfaceMintoR3.Let
˜π:withMres→pMectbtoethesomeuniverEuclideansalcovmotioneringofTM∈Iso(andRψ3):=ifφand◦π.onlyTheifT∈mappinSym(gφφ(M(or))ψ)=isSym(calledψ(M˜)),symmetrici.e.if
ifonlyandT(φ(M))=φ(M)or,equivalently,T(ψ(M˜))=ψ(M˜).(4.1.5)
Inthiscase,Tiscalledasymmetryofφ(orψ).
frameextendedThe4.233andψThroughout:=φ◦πthisthesection,correspletφondingbeaconformalconformalCMC-immersionCMC-immersionMM˜→→RR3,ofawhereMRiemann˜denotessurfacetheMunivintoersalR,
coverofMandπ:M˜→Mtheassociatedcoveringmap.
Inthefollowing,wereviewtheprocedureofconstructingthecorrespondingextendedframeFfrom
ψgiveasanpresenoutlinetedofinthe[10],basicslighprotlymocedure,difyingtheitreatadertheissamereferredtimetotothemakappeitendixfitofour[10]needs.forAsmorewedetails.willonly

52

˜3˜ds2=Sinceeu(dthex2+dy2)CMC-immersionforsomeψreal:Mvalued→Risfunctionuconformal,:M˜→theR.metricMoreonψexplicitly(M),weinducedhavebyψisgivenby
<ψx(x,y),ψx(x,y)>=<ψy(x,y),ψy(x,y)>=eu(x,y),(4.2.1)
<ψx(x,y),ψy(x,y)>=0,(4.2.2)
where<∙,∙>denotesthe˜standardinnerpro9ductofR3andweinterpretψasamappingoftworeal
variablesxandy,writingz∈Masz=x+iy.Defining
N(x,y)=|ψψxx((x,x,yy))××ψψyy((x,x,yy))|,(4.2.3)
weobtainanorthogonalmatrix(dependingonz=x+iy∈M˜)
U=(e−u(2x,y)ψx(x,y),e−u(2x,y)ψy(x,y),N(x,y))∈SO(3)(4.2.4)
andrepresenshiftingtingittheafterwnaturalards,weorthonormalcanassumeframeU(z∗)corresp=G(1)ondingandtoψ(zψ∗.)By=p1e3ossiblyforanyrotatingpreassignedthesurfacez∗∈ψ(M˜M˜=)
H2H,wheree3=(0,0,1)∈R3and
cos(t)sin(t)0
100G(λ)=G(λ)−1=sin(t)−cos(t)0∈SO(3)forallλ=eit∈S1.(4.2.5)
−reasonsRemarkfor4.3.ourThenormalizationnormalizationwillofbψecomegivenabapparenovetisbydifferenthetfollofromwing:theoneInsteadofof[10],tranwhereslatiUng(z∗the)=moI.vingThe
frameUofψdirectlyintoanextendedframeF:M˜→ΛSU(2)bytheproceduregivenin[10],wewill
considerthemovingframeU˜ofψ˜:=G(1)ψwhichsatisfiesU˜(zσ∗)=I.Byapplyingthemethodof[10],
wesomeobtainrotateF˜dwhicandhexactlytranslatedproversionducesoftheψasoriginalin[10]).immersionψviatheSym-Bobenkoformula(ratherthan
ForψasaboveandA∈O(3),consider˜the(alsoconformal)CMC-immersionψ˜:=Aψ:M˜→R3.
Thecorrespondingorthonormalframeofψisrepresentedby
U˜=(e−u˜(2x,y)ψ˜x(x,y),e−u˜(2x,y)ψ˜y(x,y),N˜(x,y))∈SO(3),(4.2.6)
whereψ˜x(x,y)×ψ˜y(x,y)
N˜(x,y)=|ψ˜x(x,y)×ψ˜y(x,y)|.(4.2.7)
TherelationbetweenU˜andUisstatedinthefollowinglemma.
Lemma4.4.Letψ:M˜→R3beaconformalCMC-immersion,A∈O(3)andψ˜:=Aψ.Thenthe
correspondingorthogonalmatricesUandU˜givenin(4.2.4)and(4.2.6),respectively,satisfy
U˜=AUifA∈SO(3),(4.2.8)
001U˜=AU010ifA∈O(3)\SO(3).(4.2.9)
100−Proof.Asindicatedabove,ψ˜isalsoaconformalCMC-immersion.Moreprecisely,wehave
ψ˜x=Aψx,ψ˜y=Aψy(4.2.10)
andthuseu˜=ψ˜x,ψ˜x=ψx,ψx=eu,whichimplies
(4.2.11)u.=u˜BecauseofA∈O(3),wehaveforanytwovectorsv,w∈R3therelation(Av)×(Aw)=det(A)A(v×w)
thereforeandN˜=|((AAψψxx))××((AAψψyy))|=|det(det(AA))AA((ψψxx××ψψyy))|=det(A)A|ψψxx××ψψyy|=det(A)AN.(4.2.12)
ws.folloclaimtheAltogether,9eitherItMis˜w=ellCˆorknoM˜wn=CthatortheM˜=univH.InersalthecovercaseM˜underofaRiemannconsideration,surfaceweMcanisin(upterprettoM˜asconformalasubsetofCˆtransformations).givenby
53

Fromnowon,letA:=G(1)∈SO(3),whereG(1)isdefinedby(4.2.5).Hence,weconsiderfromnow
CMC-immersionconformaltheonψ˜=G(1)ψ:M˜→R3.(4.2.13)
Forlaterusewenoteψ˜(z∗)=−21He3.ConsideringthecorrespondingorthonormalframeU˜asgivenin
(4.2.6),wederivebytheabovelemmathatU˜=G(1)UandthusU˜(z∗)=I.Fornow,weconsiderU˜
insteadofUandthereforefindourselvesbackinthesettingof[10].
InterpretingU˜asanautomorphismofR3,themappingJ◦U˜◦J−1,whereJ:R3→su(2)asin
(3.4.3),˜definesanautomorphismofsu(2).Furthermore(cf.equation(3.4.7)),thereexistsa(uptosign)
uniqueP∈SU(2)suchthat
(J◦U˜◦J−1)(X)=P˜XP˜−1forallX∈su(2).(4.2.14)
Additionally,asU˜(z∗)=I,wehaveP˜(z∗)=±I.W.l.o.g.,weassumeP˜(z∗)=+I,thusdefiningP˜
.uniquelyFollowingA.4of[10],P˜satisfies
P˜−1P˜z=−142uuzQe1−2,P˜−1P˜z¯=¯4u−z¯2u2e12H.(4.2.15)
1u11u
−2eH4uz−Qe−4uz¯
Introducingtheloopparameter1λ∈S1,wedefinethemappingP˜λ:M˜→ΛSU(2)intothe(untwisted)
loopgroupΛSU(2)={g:S→SU(2)smooth}by
1−2−u11u
P˜λ−1(P˜λ)z=−−1e42uuzHλ1uQe2,P˜λ−1(P˜λ)z¯=−λ24¯uQez¯−2u2−e12uH(4.2.16)
24z4z¯
andP˜λ(z∗)=I.AfterconjugatingP˜λbyG(λ)−1,where
G(λ)=01iλ−21,(4.2.17)
02iλobtainfinallyewF˜λ=G(λ)−1P˜λG(λ):M˜→ΛSU(2)σ(4.2.18)
withF˜λ(z∗)=I.
BysectionA.7of[10],theSym-Bobenkoformula(evaluatedatλ=1)allowstorecoverψ˜fromF˜λ
motion:Euclidianantoupi∂1[−2H(∂θF˜λ∙F˜λ−1+2F˜λσ3F˜λ−1)]|λ=1=G(1)−1J(ψ˜)G(1)+C(1),(4.2.19)
whereλ=eiθandC=C(λ)denotesaz-independenttranslationmatrixinsu(2).
KeepinginmindthatF˜λ(z∗)=Iforallλ∈S1,byevaluatingthelefthandsideof(4.2.19)atz=z∗
weobtain−21H2iσ3.Sinceψ˜(z∗)=−21He3asstatedearlierandthereforeJ(ψ˜(z∗))=21H2iσ3,theright
handsideof(4.2.19)forz=z∗readsasG(1)−1J(ψ˜(z∗))G(1)+C(1)=−21H2iσ3+C(1).Comparing
bframeothF˜proexpressionsducesforexactlythelefttheand(su(2)forvtheersionrighoft)handtheside,originalweimobtainmersionC(1)ψ=via0.theTherefore,Sym-Bobenktheoformextendedula
evaluatedatλ=1:
[−1(∂F˜λ∙F˜λ−1+iF˜λσ3F˜λ−1)]|λ=1=G(1)−1J(ψ˜)G(1)=J(ψ).(4.2.20)
2θ∂H2Tounderstandthelaststepintheequationabove,notethatwehave
(J◦G(λ)◦J−1)(X)=G(λ)XG(λ)−1forallX∈su(2)(4.2.21)
1˜−andFthromusJno(ψw)=on,(bJy◦Gabuse(1))(ofψ)the=G(1)notationJ(ψ)ofG(1)[10],.wedenotetheframeF˜asconstructedabovebyFto
λmatchournotationoftheprevioussections.Altogether,startingwitha(normalized)CMC-immersion
ψ,wehavereversedthelaststepoftheDPW-methodbyrecoveringtheextendedframeFwhichexactly
producesψbyevaluatingtheSym-Bobenkoformulaatλ=1.Thisresultissummarizedinthefollowing
theorem.

54

Theorem4.5.Letψ:M˜→R3beaconformalCMC-immersionandU,G(1)definedby(4.2.4),
(4.2.5),respectively.Moreover,letU(z∗)=G(1)andψ(z∗)=21He3atsomez∗∈M˜.Then,thenatural
orthonormalmovingframeofG(1)ψtranslatesbytheprocedurepresentedin[10]intoanextendedframe
F:M˜→ΛSU(2)σsatisfying
[−1(∂F∙F−1+iFσ3F−1)]|λ=1=J(ψ).(4.2.22)
2θ∂H24.3Trinoidswithproperlyembeddedannularends
Wereturntothetrinoidsetting.I.e.,letM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversal
coveringasdefinedin(3.2.2).Fromnowon,wewillrestrictourconsiderationstotrinoidsM→R3with
properlyembeddedannularends.
Hence,fromnowon,letφ:M→R3beatrinoidwithproperlyembeddedannularendsBj,j=0,1,∞,
atzj=j∈Cˆ,j=0,1,∞,respectively(cf.3.1).Moreover,assumeφisderivedviatheloopgroupmethod
fromastandardizedtrinoidpotentialη(onM)oftheform(3.6.16).Recallthatηisdeterminedbya
tripleofDelaunaymatricesD0,D1andD∞oftheform
X0jDj=Xj0,(4.3.1)


where

Xj=sjλ−1+tjλ,Xj=sjλ+tjλ−1,(4.3.2)
111sj∈[4,2),sj+tj=2.(4.3.3)
Asstatedearlier,foreachj∈{0,1,∞},theproperlyembeddedannularendBjofφasymptotically
showsthebehaviouroftheunduloidalDelaunaysurfaceproducedfromtheDelaunaypotentialz−1zjDjdz.
Wedefinethetrinoidaxisofφ(atzj)astheaxisofrevolutionoftheasymptoticDelaunaysurfaceofthe
properlyembeddedannularendBj.Thetrinoidaxisofφ(atzj)isdenotedbyAj=Cj+Rvj,involving
abasepointCj∈R3andaunitdirectionvectorvj∈R3,pointingtowardsthetrinoidendBj.Itiswell
knownthatthedirectionvectorsofthetrinoidaxesaresubjecttothebalancingformula10
w0v0+w1v1+w∞v∞=0,(4.3.4)
where,asbefore,wj=sjtj.
Wenotethefollowingresult:
Lemma4.6.LetM=C\{0,1}and,forj=0,1,∞,zj=j∈Cˆ.Moreover,letφ:M→R3beatrinoid
withproperlyembeddedannularendsBj=φ(Uj)atzjforj=0,1,∞.Then,forsomej∈{0,1,∞},
thereexistsanopensubsetUˆj⊆UjinM,suchthatBˆj:=φ(Uˆj)isaproperlyembeddedannularendof
φatzjwhichsatifies
Bˆj∩φ(M\Uˆj)=∅.(4.3.5)
Proof.WeassumewithoutlossofgeneralitythatthepuncturedneighborhoodsUjofzj,j=0,1,∞,are
openinMandpairwisedisjoint,i.e.
U0∩U1=U0∩U∞=U1∩U∞=∅.(4.3.6)
Inaddition,weassumewithoutlossofgeneralitythat,foreachj∈{0,1,∞},thesetUj∪{zj}issimply
ˆ.CinconnectedAsbefore,denoteforeachj∈{0,1,∞}byvjtheunitdirectionvectorofthetrinoidaxiscorresponding
tothetrinoidendBjatzj(andpointingtowardstheend).Thevectorsv0,v1andv∞aresubjecttothe
(4.3.4),ulaformbalancingw0v0+w1v1+w∞v∞=0,(4.3.7)
10Bychapter7of[29],thedirectionvectorsv0,v1andv∞ofthetrinoidaxessatisfytheequation
m0v0+m1v1+m∞v∞=0,
forsomerealconstantsmjcalledthetrinoidweights,whichareassociatedwiththecorrespondingasymptoticDelaunay
withsurfacestheoftheparameterstrinoidwjends=sjBtjj,ofresptheectivrespely.ectivByesectionDelaunay5.4ofsurfaces:[19],mjthese=wκweighj.tsareConsequenuptotlya,cothemmoformnulafactoraboκveidencantibcale
(4.3.4).asulatedreform

55

3thewhereotherwjt>wo,0.sayThisvkandimpliesvl.thatConsetherequentlyexists,aafterplaneinshrinkingR,thewhichendsifseparatesnecessaryone,ofwethehavvje’s,sayvr,from
Br∩Bk=∅=Br∩Bl.(4.3.8)
subsetcompacttheenGivMˆ=M\(U0∪U1∪U∞)(4.3.9)
ofM,weconsidertheopenset
Vr=φ−1(Br\φ(Mˆ))⊆M.(4.3.10)
WewillprovethatVr⊆Ur.(4.3.11)
Tothisend,letz∈Vr.Assumez∈/Ur.Inviewof(4.3.9),wehavez∈Mˆorz∈Ukorz∈Ul.Consider
atheconfirsttradiction.case,z∈WMˆe.turnThen,toφ(thez)∈φsecond(Mˆ)case,andzth∈usUφ(.z)∈/Then,φ(Vrφ),(z)whic∈hφ(Uimplies)=bByand(4.3.10)thusthbatyz∈/(4.3.8)Vr,
kkkφ(case,z)∈/alsoBr.theThisthirdyieldscase,againz∈φ(Uzl,)is∈/φled(Vrto)aandconthereforetradiction.z∈/Vr,aAltogether,conwetradiction.inferthatAnalogousnecessarilytothez∈secondUr,
whichproves(4.3.11).
SinceMˆiscompact(inM),φ(Mˆ)iscompactandinparticularbounded(inR3).Consequently,
sincegeneralitlimzy,→zwreφ(canz)=assume∞bythatUˆassumption,isopenVrinconMtainsandathatpthuncturedesetUˆneigh∪{bz}orhoisodUsimplyˆrofzr.connectedWithoutinCˆ.lossSetof
rrrBˆr=φ(Uˆr).(4.3.12)
AsUˆr⊆Urand,byassumption,φdefinesaproperembeddingofUr,alsoφ|Uˆrisaproperembedding.
WeinferthatBˆrdefinesaproperlyembeddedannularendofφatzr.
wshotoremainsItBˆr∩φ(M\Uˆr)=∅.(4.3.13)
Forastartweobserveby(4.3.6)and(4.3.9)that
M\Uˆr=Mˆ∪Uk∪Ul∪(Ur\Uˆr)(4.3.14)
usthandφ(M\Uˆr)=φ(Mˆ)∪Bk∪Bl∪φ(Ur\Uˆr).(4.3.15)
Asthus,Bˆrb=yφ(Uˆ(4.3.8),r)⊆Bˆφr(∩VrB)k==B∅r=\φBˆ(rMˆ∩),Blw.ehaFinallyveB,ˆr∩sinceφ(φMˆ|U)=is∅.anemMoreobver,edding,Bˆrw=eφha(Uvˆer)Bˆ⊆r∩φφ((UUrr)\=UˆrB)r=and∅.
robtainewAltogether,Bˆr∩φ(M\Uˆr)=(Bˆr∩φ(Mˆ))∪(Bˆr∩Bk)∪(Bˆr∩Bl)∪(Bˆr∩φ(Ur\Uˆr))=∅,(4.3.16)
claimed.as4.4Theextendedframesymmetrytransformations
bWeetheareinunivtheersaltrinoidcoveringsettingasdefinedpresentedinin(3.2.2).sectionMoreo4.3.vI.e.,er,letletMφ:=MC\→{0R,31}b,eM˜a=Htrinoidandπwith:M˜prop→erlyM
embeddedannularendsandψ=φ◦πthecorrespondingconformalCMC-immersionM˜→R3.Finally,
letFdenotetheextendedframecorrespondingtoψinthesenseoftheorem4.5.
WeareinterestedintranslatinganysymmetrypropertiesofφtotheleveloftheextendedframeF.
Thus,weassumethatφissymmetricwithrespecttoT∈Sym(φ(M)),i.e.
T(φ(M))=φ(M).(4.4.1)
Recallingfromsection4.1thatSym(φ(M))=Sym(ψ(M˜)),weinferthatTisalsoasymmetryofthe
i.e.,ψCMC-immersionconformalT(ψ(M))=ψ(M).(4.4.2)
theInordersymmetrytoproptranslateertyoftheψsymthroughmetrythepropproertycessofψtogeneratingthelevelFoffromtheψ,extendedwhichhasframebFeen,wepresenwilltedtracein
section4.2.Theorem4.9,basedonresultsof[12],associateswiththegivensymmetryTofψapair
56

ofbijections,γ:M→Mandγ˜:M˜→M˜,andiscrucialforourpurposes.Wemakethenecessary
preparationsforthistheoreminthefollowing.
ahLetolomUorandphicVbijectiondenoteopwithensubsetsholomorphicofC.invAersefunctionfunction.f:fUis→calledViscalledbi-antiholomorphicbiholomorphic,if,fifisfanis
antiholomorphicbijectionwithantiholomorphicinversefunction.Wedefinethefollowingsets:
Aut(M˜)={γ˜:M˜→M˜;γ˜biholomorphic},(4.4.3)
Aut∗(M˜)={γ˜:M˜→M˜;γ˜bi-antiholomorphic},(4.4.4)
Aut(M)={γ:M→M;γbiholomorphic},(4.4.5)
Aut∗(M)={γ:M→M;γbi-antiholomorphic}.(4.4.6)
(3.2.2)Lemma.Mor4.7.eover,LetletMφ=:CM\→{0,R13},beM˜a=trinoidHandwithπ:prM˜operly→Membbeeddethedannularuniversalcendsoveringandψas=φdefine◦πdthein
correspondingconformalCMC-immersionM˜→R3.Then,wehave
{γ˜∈Aut(M˜);π◦γ˜=π}={γ˜∈Aut(M˜);ψ◦γ˜=ψ}(4.4.7)
Proof.First,letγ˜∈Aut(M˜)withπ◦γ˜=π.Sinceψ=φ◦π,weinferthat
ψ◦γ˜=φ◦π◦γ˜=φ◦π=ψ,(4.4.8)
whichalreadyprovestherelation“⊆”.
Itremainstoprovetherelation“⊇”.Tothisend,letγ˜∈Aut(M˜)satisfyingψ◦γ˜=ψ.Since
φ:M→R3isatrinoidwithproperlyembeddedannularends,thereexistsbylemma4.6forsome
j∈{0,1,∞}apuncturedneighborhoodUjofzjinM,suchthatBj=φ(Uj)isaproperlyembedded
annularendofφatzjsatisfyingBj∩φ(M\Uj)=∅.Asstatedintheproofoflemma4.6,wecanassume
withoutlossofgeneralitythatUjisopeninMandthatUj∪{zj}issimplyconnectedinCˆ.
WeconsiderthetesselationofM˜inducedbythesheetF⊆M˜definedin(3.2.31)andthegroup
Aut(M˜/M)ofcoveringtransformationsonM˜givenin(3.3.7).Viatheuniversalcoveringπ,Ujcor-
respondstoanopensubsetU˜jofF.Inparticular,themappingπ|U˜j:tildeUj→Ujisbijective.
Consequently,wehaveψ(U˜j)∩ψ(F\U˜j)=∅.
Naturally,wehaveγ˜(U˜j)⊆M˜.However,byrestrictingγ˜toanappropriateopensubsetofU˜j,the
resultingrestrictedmaptakesvaluesinonlyonesheetF˜ofourtesselationofM˜.Therefore,weassume
withoutlossofgeneralitythatγ˜maps˜UjintosomesheetF˜,i.e.γ˜|U˜j:U˜j→F˜.
RecallingthatthegroupAut(M˜/M)actstransitivelyonthesetofthesheetsofourtesselationofM˜,
thereexistsδ˜˜∈Aut(M˜/M),suchthatδ˜(F˜)=F.Consideringthemappingδ˜◦γ˜:M˜→M˜,weobserve
byusingπ◦δ=πandψ◦γ˜=ψthat
ψ◦δ˜◦γ˜=φ◦π◦δ˜◦γ˜=φ◦π◦γ˜=ψ◦γ˜=ψ.(4.4.9)
Letnowz∈U˜j.Inthiscase,wehaveγ˜(z)∈F˜andthusδ˜(γ˜(z))∈F.By(4.4.9),wehave
ψ(z)=ψ(δ˜(γ˜(z))).(4.4.10)
Sinceψ(U˜j)∩ψ(F\U˜j)=∅,thisimpliesδ˜(γ˜(z))∈U˜j.However,sinceψ=φ◦πisinjectiveonU˜j,we
inferfrom(4.4.10)thatactuallyδ˜(γ˜(z))=z.
Altogether,wehaveprovedδ˜◦γ˜|U˜j=id.Sinceδ˜andγ˜areholomorphiconM˜,thisrelationcarries
overtoM˜:δ˜◦γ˜=id.Butthisimpliesγ˜=δ˜−1∈Aut(M˜/M)andinparticularπ◦γ˜=π,whichfinishes
of.protheBasedontheorem2.7of[12],wecanstatethefollowingresult:
33π:LemmaM˜→4.8.MbeLettheφ:Muniversal→RcbeoveringacofonformalMandψ:=CMC-immersionφ◦πacofaonformalRiemannsurfacCMC-immersioneMintoofRM˜.Lintoet
R3.Finally,letφ-or,equivalently,ψ-besymmetricwithrespecttoanorientationpreserving(resp.
orientationreversing)EuclideanmotionT∈Sym(φ(M))=Sym(ψ(M˜)).Then,ifMwiththemetric
inducedbyφiscomplete,thereexistsamappingγ˜∈Aut(M˜)(resp.γ˜∈Aut∗(M˜)),suchthat
T◦ψ=ψ◦γ˜.(4.4.11)
Themappingγ˜isuniqueuptocompositionfromtheleftwithanelementofAut(M˜/M).

57

Proof.InthecaseofanorientationpreservingEuclideanmotionTthisisprovedintheorem2.7of[12].
TheprooffororientationreversingTiscompletelyanalogous.
Withthesepreparationsmadewecannowturntotheannounced
Theorem4.9.LetM=C\˜{0,1}withuniversalcoverM˜=Handcoveringmapπ:M˜→Mas3
definedin(3.2.2).LetAut(M/M)denotetheautomorphismgroupofπ.Moreover,letφ:M→R
beaimmersiontrinoidM˜with→Rpr3.operlyFinally,embeletddeφd-or,annulareendsquivalently,andψψ-=beφ◦πsymmetricthecorrwithesprespondingecttocanonformalorientationCMC-
prexisteservingγ∈(rAut(esp.M)andorientationγ˜∈Aut(rM˜)eversing)(resp.γEuclide∈Autan∗(Mmotion)andTγ˜∈∈Sym(Aut∗φ((M˜M))))=satisfyingSym(ψ(M˜)).Thenthere
T◦φ=φ◦γ,(4.4.12)
T◦ψ=ψ◦˜γ.(4.4.13)
Whileγisunique,γ˜isuniqueuptocompositionfromtheleftwithanelementofAut(M˜/M).Further-
more,γandγ˜arerelatedby
π◦γ˜=γ◦π.(4.4.14)
Proof.First,recallthat(bylemma3.2)Mwiththemetricinducedbyφiscomplete.Thus,wecan
applylemma4.8to˜relatetotheorien∗tation˜preserving(resp.orientationreversing)EuclideanmotionT
amappingγ˜∈Aut(M)(resp.˜γ∈Aut(M))satisfying
T◦ψ=ψ◦γ˜.(4.4.15)
Moreover,γ˜isuniqueuptocompositionfromtheleftwithanelementofAut(M˜/M).
Next,weprovethesetidentity
γ˜Aut(M˜/M)γ˜−1=Aut(M˜/M).(4.4.16)
Letδ˜∈Aut(M˜/M).Bylemma4.7wehaveψ◦δ˜=ψ.Consequently,
ψ◦γ˜◦δ˜◦γ˜−1=T◦ψ◦δ˜◦γ˜−1=T◦ψ◦γ˜−1=ψ,(4.4.17)
which,notingthatγ˜◦δ˜◦γ˜−1isbiholomorphic(evenifγ˜isbi-antihlomorphic)andapplyingagainlemma
4.7,showsγ˜◦δ˜◦−1γ˜−1∈Aut(M˜/M)andthusproves−the1inclusionγ˜Aut(M˜/M)γ˜−1⊆Aut(M˜/M).
˜˜Aut(ReplacingM˜/M)γ˜⊆byγ˜γ˜Aut(inM˜/M)(4.4.17),γ˜−1.weAltogether,obtain(4.4.16)analogouslyfolloγ˜ws.Aut(M/M)γ˜⊆Aut(M/M),or,equivalently,
WeidentifyMwithM˜/Aut(M˜/M)bythemappingz→[w],wherew∈M˜withπ(w)=z.(Cf.
remarkA.15ofappendixA.1formoredetails.)Considerthemapping
γ:M→M,[w]→γ([w]):=[γ˜(w)].(4.4.18)
(Since[w]={δ˜(w);δ˜∈Aut(M˜/M)}and,by(4.4.16),[γ˜(δ˜(w))]=[σ˜(γ˜(w))]=[γ˜(w)]forallδ˜∈
˜Aut(M˜/M˜)andappropriateσ˜∈Aut(M˜/M),γiswelldefined.)WiththeidentificationofMand
M/Aut(M/M)givenabove,wecanwriteγas
γ:M→M,z→γ(z):=π(γ˜(w)),(4.4.19)
where˜w∈π−1(z)andthedefinitionofγisindependentofthechoiceofw.Bydefinitionofγ,wehave
Monγ◦π=π◦γ˜.(4.4.20)
Moreover,sinceπisconformal,γisbiholomorphic(resp.bi-antiholomorphic)ifγ˜isbiholomorphic(resp.
tiholomorphic).bi-anLetnowz∈Mandw∈π−1(z).Then,
φ◦γ(z)=φ◦π◦γ˜(w)=ψ◦γ˜(w)=T◦ψ(w)=T◦φ◦π(w)=T◦φ(z).(4.4.21)
Thisprovestherelation
T◦φ=φ◦γ,(4.4.22)
whichmeansthatwehaveconstructedγwiththeclaimedproperties.

58

Itremainstoprovetheuniquenessofγ.Thiscanbeseenasfollows:Assumeγ1,γ2∈Aut(M)(resp.
γ1,γ2∈Aut∗(M))bothsatisfytheclaimedrelations.Then,inparticularφ◦γ1=T◦φ=φ◦γ2,which
impliesφ◦γ1◦γ2−1=φ.(4.4.23)
Sinceφisatrinoidwithproperlyembeddedannularendsthereexistsbylemma4.6anopensubsetUof
M,suchthatφ|Uisanembeddingwith
φ(U)∩φ(M\U)=∅.(4.4.24)
Thisimpliesγ1◦γ2−1|U=id|U.Asγ1◦γ2−1isbiholomorphic(resp.bi-antiholomorphic)onM,thisimplies
thatactuallyγ1◦γ2−1|M=id|Mandthereforeγ1=γ2.
Bytheorem4.9,wecaninvestigatethebehaviourofψunderthesymmetryTbyconsideringthe
compositionψ◦γ˜.Inthefollowing,weexplainhowthisrelationcarriesovertothecorresponding
extendedframeFfromtheorem4.5.
˜matrixU˜According=G(1)toUthegivenpreviousin(4.2.6).section,Wewweouldconsiderliketoψ=computeG(1)ψtogetherwiththeassociatedorthogonal
U˜◦γ˜=G(1)(e−u2◦γ˜ψx◦γ˜,e−u2◦γ˜ψy◦γ˜,N◦γ˜).(4.4.25)
Forthis,weneedtocollectsometechnicalresults.
Lemma4.10.Interpretingγ˜:M˜→M˜asamapping(x,y)→(γ˜1(x+iy),γ˜2(x+iy))∈R2ofreal
variablesxandy,whereγ˜1:=(γ˜)andγ˜2:=(γ˜),wehave
e−u2◦γ˜=e−2u∂γ˜1+∂γ˜2.(4.4.26)
22
x∂x∂Proof.BecauseofT∈Iso(R3)andT◦ψ=ψ◦γ˜,γ˜formsanisometryofM˜.Thisimplies
eu(dx2+dy2)=eu◦γ˜(d(γ˜1)2+d(γ˜2)2)
=eu◦γ˜∂γ˜1dx2+2∂γ˜1∂γ˜1dxdy+∂γ˜1dy2+∂γ˜2dx2+2∂γ˜2∂γ˜2dxdy+∂γ˜2dy2
2222
∂x∂x∂y∂y∂x∂x∂y∂y
22
=eu◦γ˜∂γ˜1+∂γ˜2dx2+dy2.
x∂x∂Thelaststepfollowsfromthe(anti-)holomorphicityofγ˜:Weeitherhave∂∂γ˜x1=∂∂γ˜y2,∂∂γ˜y1=−∂∂γ˜x2or
∂∂γ˜x1=−∂∂γ˜y2,∂∂γ˜y1=∂∂γ˜x2.Altogether,theclaimfollows.
Lemma4.11.Decomp3osingthesymmetryT∈Sym(ψ(M˜))intoanorthogonalpartAT∈O(3)andan
translationalparttT∈R,
T:R3→R3,x→ATx+tT,(4.4.27)
thefollowingequationshold:
1∂˜γ2∂γ˜2
ψx◦γ˜=∂γ˜1∂γ˜2−∂γ˜2∂γ˜1AT∂yψx−∂xψy,(4.4.28)
∂x∂y∂x∂y
1∂γ˜1∂γ˜1
ψy◦γ˜=∂γ˜1∂γ˜2−∂γ˜2∂γ˜1AT−∂yψx+∂xψy,(4.4.29)
∂x∂y∂x∂y
N◦γ˜=ATN.(4.4.30)
Proof.Definingthedifferentialmatrix
∂∂γ˜x1(x+iy)∂∂γ˜y1(x+iy)
Dγ˜(x+iy):=∂∂γ˜x2(x+iy)∂∂γ˜y2(x+iy)(4.4.31)

59

(4.4.31)

andanalogouslythedifferentialmatricesDψandD(ψ◦γ˜),wederivefromtherelationT◦ψ=ψ◦γ˜
(providedbytheorem4.9)
(ψx◦γ˜,ψy◦γ˜)Dγ˜=((Dψ)◦γ˜)Dγ˜=D(ψ◦γ˜)=D(T◦ψ)=ATDψ,(4.4.32)
whence(ψx◦γ˜,ψy◦γ˜)=ATDψ(Dγ˜)−1.(4.4.33)
1Takingintoaccount∂γ˜2∂γ˜1
−1−(Dγ˜)=∂γ˜1∂γ˜2∂γ˜2∂γ˜1−∂∂yγ˜2∂γ˜∂1y(4.4.34)
∂x∂y−∂x∂y∂x∂x
weobtain(4.4.28)and(4.4.29).
RecallingAT∈O(3)andapplyingthesameargumentasin(4.2.12),weinfer
(ψx◦γ˜)×(ψy◦γ˜)∂∂γ˜y2ψx−∂∂γ˜x2ψy×−∂∂γ˜y1ψx+∂∂γ˜x1ψy
N◦γ˜==det(AT)AT.(4.4.35)
|(ψx◦γ˜)×(ψy◦γ˜)|∂∂γ˜y2ψx−∂∂γ˜x2ψy×−∂∂γ˜y1ψx+∂∂γ˜x1ψy
Continuingthecalculationbyusingthe(anti-)holomorphicityofγ˜weobtain
∂∂γ˜x1+∂∂γ˜x2(ψx×ψy)
22
N◦γ˜=det(AT)AT∂γ˜2∂γ˜2=det(AT)ATN,(4.4.36)
∂x1+∂x2(ψx×ψy)
where∈{±1}takesthevalue“+1”(resp.“−1”)inthecaseofholomorphic(resp.antiholomorphic)
γ˜.As(bytheorem4.9)γ˜isholomorphicfororientationpreservingT,i.e.inthecasedet(AT)=1,and
antiholomorphicfororientationreversingT,i.e.inthecasedet(AT)=−1,weobtain(mergingboth
cases)det(AT)=+1.ThereforeN◦γ˜=ATN,whichis(4.4.30).
Combiningtheresultsofthetwoproceedinglemmas,wecanwriteouthowU˜transformsunderthe
biholomorphic(resp.bi-antiholomorphic)mapping˜γ.Werecordthisinthefollowingtheorem.
Theorem4.12.Letψ:M˜→R3beaconformalCMC-immersion,whichissymmetricwithrespectto
T:x→ATx+tT.Assumeψcorrespondstoatrinoidφ:M→R3withproperlyembeddedannularends
viatheuniversalcoveringπ:M˜→M,ψ=φ◦π.Letγ˜beabiholomorphic(resp.bi-antiholomorphic)
mappingM˜→M˜associatedwithTbytheorem4.9.Furthermore,letψ˜=G(1)ψ,whereG(1)isgivenby
(4.2.5).ThentheorthogonalmatrixU˜correspondingtoψ˜asdefinedin(4.2.6)satisfies
U˜◦γ˜=G(1)AT(G(1))−1U˜KT,γ˜,(4.4.37)
ewher

0BA100KT,γ˜=−BA0ifTpreservesorientation,
0BA100KT,γ˜=B−A0ifTreversesorientation,
andA,B:M˜→Raredefinedby
∂γ˜1(x+iy)
A(x+iy)=∂x22,
∂∂γ˜x1(x+iy)+∂∂γ˜x2(x+iy)
∂γ˜2(x+iy)
B(x+iy)=∂x22.
∂∂γ˜x1(x+iy)+∂∂γ˜x2(x+iy)
60

(4.4.38)(4.4.39)

(4.4.40)(4.4.41)

Proof.Wedistinguishtwocases.IfTpreservesorientation,theassociatedmappingγ˜oftheorem4.9is
holomorphic.Otherwise,ifTreversesorientation,γ˜isantiholomorphic.Thus,setting
+1ifTpreservesorientation
:=(T):=−1ifTreversesorientation(4.4.42)
wehave∂∂γ˜x1=∂∂γ˜y2,∂∂˜γy1=−∂∂γ˜x2.Applyingthistoequations(4.4.28)and(4.4.29)oftheabovelemma,
obtainew1∂γ˜1∂γ˜2
+ψx◦γ˜=∂γ˜12∂γ˜22AT∂xψx−∂xψy,(4.4.43)
x∂x∂1∂γ˜2∂γ˜1
+ψy◦γ˜=∂γ˜12∂γ˜22AT∂xψx+∂xψy.(4.4.44)
x∂x∂Altogether,takingintoaccount(4.4.26),(4.4.30),(4.4.43)and(4.4.44),weobtainfrom(4.4.25):
U˜◦γ˜=G(1)(e−2uAT(Aψx−Bψy),e−2uAT(Bψx+Aψy),ATN),(4.4.45)
whereA,Bareasin(4.4.40),(4.4.41),respectively.Thistransformsfurtherinto
0BAU˜◦γ˜=G(1)AT(e−2uψx,e−2uψy,N)−B+A0=G(1)ATUKT,γ˜,(4.4.46)
100withKT,γ˜asin(4.4.38),(4.4.39),respectively.AsU˜=G(1)U,theclaimfollows.
Asaconsequence,theconjugationmatrixP˜correspondingtoU˜by(4.2.14)transformsasfollows:
Corollary4.13.Weretainthenotationandtheassumptionsoftheorem4.12.Theconjugationmatrix
P˜realizingtheorthogonalmatrixU˜inthesu(2)modeltransformsunderγ˜as
P˜◦γ˜=±G(1)ATG(1)−1P˜kˆT,γ˜,(4.4.47)
whereG(1),AT,kˆT,γ˜∈SU(2)arethecorrespondingconjugationmatricesrealizingG(1),AT,KT,γ˜∈O(3),
respectively,inthesu(2)model,andtheremainingfreedominthesigniscausedbythefactthatwework
insu(2)andnotinO(3).
Proof.Asafirststep,weinterprettheO(3)matricesappearingin(4.4.37)inthesu(2)model.Tothis
recallend,

001G(1)=0−10,
0BA00−1
100KT,γ˜=−BA0∈SO(3)ifTpreservesorientation,
0BA100KT,γ˜=B−A0∈O(3)\SO(3)ifTreversesorientation.
From(4.2.21)wealreadyknow
(J◦G(1)◦J−1)(X)=G(1)XG(1)−1forallX∈su(2),
whereG(1)isgivenby(4.2.17).ThecorrespondingequationsforKT,γ˜read
(J◦KT,γ˜◦J−1)(X)=kˆT,γ˜XkˆT−,1γ˜forallX∈su(2)ifTpreservesorientation,
(J◦KT,γ˜◦J−1)(X)=−kˆT,γ˜XkˆT−,1γ˜forallX∈su(2)ifTreversesorientation,

61

(4.4.48)

(4.4.49)50)(4.4.

(4.4.51)(4.4.52)(4.4.53)

where√kˆT,γ˜=A0+iB√A0+iBifTpreservesorientation,(4.4.54)
√kˆT,γ˜=√0−A+iBifTreversesorientation.(4.4.55)
0iB+A2iBThe)(Aiden+iBtities)=(A(4.4.52)+iBan)(dA−(4.4.53)iB)=are1vanderifiedthusas|Afollo+ws:iB|=Using1,twheefactconcludethat|thatA+|√iBA|+=iB|(2A=+
√A+iB√A+iB=1.Inviewofthis,equations(4.4.52)and(4.4.53)areobtainedbyadirectcompu-
11tation.ConcerningtheorthogonalpartATofthesymmetryT,weknowthatJ◦AT◦J−1definesan
automorphismofsu(2)which(recallingequations(3.4.7)and(3.4.8),respectively)isrealizedby
(J◦AT◦J−1)(X)=ATXAT−1forallX∈su(2)ifTpreservesorientation,(4.4.56)
(J◦AT◦J−1)(X)=−ATXAT−1forallX∈su(2)ifTreversesorientation,(4.4.57)
whereATdefinesaT-dependentelementofSU(2).Finally,werecallequation(4.2.14),whichreads
(J◦U˜◦J−1)(X)=P˜XP˜−1forallX∈su(2).(4.4.58)
Altogether,bytheorem4.12,weobtainforallX∈su(2)
(P˜◦γ˜)X(P˜◦γ˜)−1=(J◦(U˜◦γ˜)◦J−1)(X)=(J◦(G(1)AT(G(1))−1U˜KT,γ˜)◦J−1)(X)
=(J◦G(1)◦J−1)◦(J◦AT◦J−1)◦(J◦G(1))−1◦J−1)◦(J◦U˜◦J−1)◦(J◦KT,γ˜◦J−1)(X)
=G(1)ATG(1)−1P˜kˆT,γ˜XkˆT−,1γ˜P˜−1G(1)AT−1G(1)−1.(4.4.59)
NotethatthetwominussignsoccuringinthecaseofanorientationreversingsymmetryTcancel.
ofSosu(2).far,ButwehasincevesethisenisthattrueP˜◦forγ˜allandXG∈(1)Asu(2),TGw(1)e−1P˜kˆnecessarilyT,γ˜haconjugateveX∈su(2)intothesameelement
P˜◦γ˜=±G(1)ATG(1)−1P˜kˆT,γ˜,(4.4.60)
whichprovestheclaim.
Remark4.14.Wedefinethecomplexsquareroot√occurringinˆkT,γ˜(cf.(4.4.54)and(4.4.55))on
∗thez-planeCby√:C∗→C∗,z=reiθ→√z:=√rei2θ,(4.4.61)
√+iθ∗thewhereusualwe(real)writeszq∈uareCroinottheofr.formForz=rfutureewithrcalculations,∈Rweandθstate∈(the−π,folπ],loandwingridentitiesdefinesinvtheolvinvaluegtheof
complexsquarerootasdefinedabove.Notethattheseidentitiesare,asiswell-knownforcomplexsquare
rootsingeneral,onlydetermineduptosign.Forallz,z1,z2∈C∗wehave
√z1√z2=±√z1z2,(4.4.62)
√z=±√z,(4.4.63)
√z−1=±(√z)−1.(4.4.64)
InordertotranslatethetransformationpropertyofP˜underγ˜statedintheproceedingcorollary
intoacorrespondingrelationforP˜λasintroducedintheprevioussection,weneedtomakesomefurther
preparations.Westartbydefiningthedifferentialform
ζ:=P˜−1dP˜.(4.4.65)
for11anNoteythatcomplexthesquarestatedroidenotCtit∗y→√CA∗.+WiBe√A+explicitlyiB=1,definewhicthehsufficescomplextoprosquareverootequations√used(4.4.52)inthisandwork(4.4.53),inremarkisobtained4.14.
62

Inviewofthecorollaryabove,ζtransformsunderγ˜as
γ˜∗ζ=(P˜◦γ˜)−1d(P˜◦γ˜)=ˆkT−,1γ˜P˜−1G(1)AT−1G(1)−1d(G(1)ATG(1)−1P˜kˆT,γ˜)
=kˆT−,1γ˜P˜−1(dP˜kˆT,γ˜+P˜dkˆT,γ˜)=kˆT−,1γ˜ζkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜.(4.4.66)
Furthermore,takingintoaccountequations(4.2.15)wehave
uu
ζ=P˜−1P˜zdz+P˜−1P˜z¯dz¯=−−1e412uuzHQe1u−2dz+−¯41Qeu−z¯2u21−e12uHdz¯.(4.4.67)
24z4z¯
splittingtheerformingPζ=ζk+ζpdz+ζpdz¯,(4.4.68)
whereζk=−410uz1u0dz+410uz¯−10udz¯,(4.4.69)
2Qe04z−u4z¯
ζp=−21eu2H0,(4.4.70)
u1ζp=−¯Qe0−2u2e02H,(4.4.71)
implies(4.4.66)equationγ˜∗ζ=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜ζpkˆT,γ˜dz+kˆT−,1γ˜ζpkˆT,γ˜dz¯+kˆT−,1γ˜dkˆT,γ˜.(4.4.72)
Atthesametimewehave
γ˜∗ζ=γ˜∗ζk+γ˜∗(ζpdz)+γ˜∗(ζpdz¯)=γ˜∗ζk+(ζp◦γ˜)dγ˜(z)+(ζp◦γ˜)dγ˜(z).(4.4.73)
Comparingequations(4.4.72)and(4.4.73),wecanstatethefollowinglemma:
Lemma4.15.Letψ:M˜→R3beaconformalCMC-immersion,whichcorrespondstoatrinoidφ:
3MMore→over,Rletwithζpr=opP˜−erly1dP˜emb=eζdded+ζdannularz+ζdendsz¯asviaabtheove,wheruniversaleP˜ccorroveringespπonds,:M˜as→inM,(4.2.14)ψ=,φto◦theπ.
ppkoforthoψ,Tgonal:R3fr→ameR3U˜,Tasso(x)ciate=AdTxwith+tψT,withwhichAisT∈givenO(3)inand(4.2.6)tT.∈FR3,urthermorandlete,γ˜letTdenotedenoteaabiholomorphicsymmetry
(resp.bi-antiholomorphic)mappingM˜→M˜associatedwithTbytheorem4.9.
1.IfTpreservesorientation,thefollowingholds:
γ˜∗ζk=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜(4.4.74)
(ζp◦γ˜)∂zγ˜=kˆT−,1γ˜ζpkˆT,γ˜(4.4.75)
(ζp◦˜γ)∂z¯γ˜=kˆT−,1γ˜ζpkˆT,γ˜.(4.4.76)
2.IfTreversesorientation,thefollowingholds:
γ˜∗ζk=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜(4.4.77)
(ζp◦γ˜)∂z¯γ˜=kˆT−1,γ˜ζpkˆT,γ˜(4.4.78)
(ζp◦γ˜)∂zγ˜=kˆT−,1γ˜ζpkˆT,γ˜.(4.4.79)
WPreoof.startAswithstatedthebfirstefore,case,thei.e.claimsletTfollowpreservfromeoriencomparingtation.equationsTherefore,by(4.4.72)theoreandm4.9,(4.4.73).γ˜isholomorphic,
andequation(4.4.73)readsas
γ˜∗ζ=γ˜∗ζk+(ζp◦γ˜)∂zγ˜dz+(ζp◦γ˜)∂z¯γ˜dz¯.(4.4.80)
Weobservethatγ˜∗ζkisoftheform
0∗∗0dz+0∗∗0dz¯,(4.4.81)
63

while(ζp◦γ˜)∂z˜γdzisoftheform
∗00∗dz,(4.4.82)
and(ζp◦γ˜)∂z¯γ˜dz¯isoftheform
0∗dz¯.(4.4.83)
0∗AsinthepresentcasekˆT,γ˜isexplicitlygivenby(4.4.54),itisnoweasytoverifythatthesummandsof
γ˜∗ζoccuringinequation(4.4.72)areofthefollowingforms,respectively:kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜isofthe
form(4.4.81),kˆT−,1γ˜ζpkˆT,γ˜dzisoftheform(4.4.82),andkˆT−,1γ˜ζpkˆT,γ˜dz¯isoftheform(4.4.83).Therefore,
comparingequation(4.4.72)to(4.4.73)provestheclaiminthefirstcase.
Wthatenoturnw,tobythetheoremsecond4.9,case,γ˜isi.e.anlettiTrevholomorphicerseorienandthtation.usequWeationproceed(4.4.73)asinbtheecomesfirstcase.However,note
γ˜∗ζ=γ˜∗ζk+(ζp◦γ˜)∂z¯γ˜(z¯)dz¯+(ζp◦γ˜)∂zγ˜dz.(4.4.84)
Inthiscase,γ˜∗ζkisoftheform(4.4.81),(ζp◦γ˜)∂z¯γ˜dz¯isoftheform(4.4.83),and(ζp◦γ˜)∂zγ˜dzisof
theform(4.4.82).InviewofkˆT,γ˜,nowexplicitlygivenby(4.4.55),weinvestigatethesummandsofγ˜∗ζ
in(4.4.72)andobservethat,exactlyasinthefirstcase,kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜isoftheform(4.4.81),
kˆT−,1γ˜ζpkˆT,γ˜dzisoftheform(4.4.82),andkˆT−,1γ˜ζpkˆT,˜γdz¯isoftheform(4.4.83).Combining(4.4.72)and
(4.4.73)provestheclaiminthesecondcase.
ThetechnicalresultjustestablishedyieldsthefollowingtransformationpropertyofP˜λunderγ˜:
˜betheLemmamapping4.16.M˜We→retainΛSU(2)theasnotationsdefinedinandsethection4.2.assumptionsThen,ofthethefolprevilowingouslemma.statementsFhold:urthermoreletPλ
1.IfTisorientationpreserving,P˜λtransformsunderγ˜as:
P˜λ◦γ˜=M˜γ˜(λ)P˜λkˆT,γ˜,(4.4.85)
wherekˆT,γ˜isdefinedin(4.4.54)andM˜γ˜(λ)isindependentofz.Moreover,forλ=1,wehave
M˜γ˜(1)=±G(1)ATG(1)−1(4.4.86)
where,asbefore,G(1),AT,∈SU(2)arethecorrespondingconjugationmatricesrealizingG(1),AT∈
O(3),respectively,inthesu(2)-model.
2.IfTisorientationreversing,P˜λtransformsunderγ˜as:
P˜λ−1◦γ˜=M˜γ˜(λ)P˜λkˆT,γ˜,(4.4.87)
wherekˆT,γ˜isdefinedin(4.4.55)andM˜γ˜(λ)isindependentofz.Moreover,forλ=1,wehave
M˜γ˜(1)=±G(1)ATG(1)−1(4.4.88)
where,asbefore,G(1),AT,∈SU(2)arethecorrespondingconjugationmatricesrealizingG(1),AT∈
O(3),respectively,inthesu(2)-model.
Proof.Definethedifferentialformζλ:=P˜λ−1dP˜λ.(4.4.89)
Bytakingintoaccountequations(4.2.16),wecanwrite
uu
ζλ=P˜λ−1(P˜λ)zdz+P˜λ−1(P˜λ)z¯dz¯=−−1e412uuzHλ−21uQe−2dz+−λ241¯uQez¯−2u21−e12uHdz¯.(4.4.90)
24z4z¯
Recallingthesplittingζ=ζk+ζpdz+ζpdz¯ofζ=P˜−1dP˜,whereζk,ζpandζparegivenbyequations
(4.4.69)to(4.4.71),aneasycomputationallowstorelateζλtoζ:
ζλ=ζk+λ−1λ02λ021ζpλ02λ−021dz+λλ02λ021ζpλ02λ−021dz¯.(4.4.91)
−11−11
64

Applyingγ˜tothisequation,1weobtain111
γ˜∗ζλ=γ˜∗ζk+λ−1λ−201(ζp◦γ˜)λ2−01dγ˜(z)+λλ−201(ζp◦γ˜)λ2−01dγ˜(z).
0λ20λ20λ20λ2
(4.4.92)NowwedistinguishbetweenthetwocasesoforientationpreservingandorientationreversingT:IfT
preservesorientation,γ˜isholomorphic,andweobtain
γ˜∗ζλ=γ˜∗ζk+λ−1λ0λ021(ζp◦γ˜)λ0λ−021∂zγ˜dz+λλ0λ021(ζp◦γ˜)λ0λ−021∂z¯γ˜dz¯.
−2121−2121
(4.4.93)Asaconsequenceofthepreviouslemma,thisisequivalentto
γ˜∗ζλ=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜+λ−1λ−201kˆT−,1γ˜ζpkˆT,γ˜λ2−01dz
11
0λ20λ2
+λλ0λ012kˆT−,1γ˜ζpkˆT,γ˜λ0λ−021dz¯
−2121
=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜+λ−1kˆT−,1γ˜λ−201ζpλ2−01kˆT,γ˜dz
11
0λ20λ2
+λkˆT−,1γ˜λ0λ021ζpλ0λ−021kˆT,γ˜dz¯=kˆT−,1γ˜ζλkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜,(4.4.94)
−2121
wherewehaveusedthattheoccurringdiagonalmatricescommute.Butthisimplies
(P˜λkˆT,γ˜)−1d(P˜λkˆT,γ˜)=kˆT−,1γ˜P˜λ−1dP˜λkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜
=kˆT−,1γ˜ζλkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜=γ˜∗ζλ=(P˜λ◦γ˜)−1d(P˜λ◦γ˜),(4.4.95)
whichmeansthatP˜λ◦γ˜andP˜λkˆT,γ˜solvethesamedifferentialequationandthereforeonlydifferbya
matrixM˜γ˜(λ)independentofz:
P˜λ◦γ˜=M˜γ˜(λ)P˜λkˆT,γ˜,(4.4.96)
whichistheequationclaimedinthefirstcase.Forλ=1,wehaveP˜λ=1=P˜andthusobtain
P˜◦γ˜=M˜γ˜(1)P˜kˆT,γ˜.(4.4.97)
1˜−LetComparingnowTthisbetoorien(4.4.47),tationwerevinferersing.thatMThisγ˜(1)impli=±esG(1)thatATγ˜Gis(1)an.tiholomorphicandequation(4.4.92)
asreadsγ˜∗ζλ=γ˜∗ζk+λ−1λ−201(ζp◦γ˜)λ2−01∂z¯γ˜dz¯+λλ−201(ζp◦γ˜)λ2−01∂zγ˜dz.
1111
0λ20λ20λ20λ2
(4.4.98)Applyinglemma4.15again,usingthesecondpartthistime,weobtain
γ˜∗ζλ=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜+λ−1λ02λ021kˆT−,1γ˜ζpkˆT,γ˜λ02λ−021dz¯
−11
+λλ−201kˆT−1ζpkˆT,γ˜λ2−01dz.(4.4.99)
11
0λ20λ2
RecallingthatkˆT,γ˜is,inthepresentcase,givenby(4.4.55),weverifybyadirectcomputationthe
identity1−1
kˆT,γ˜λ02λ−021=λ02λ021kˆT,γ˜.(4.4.100)
Consequently,weobtain
γ˜∗ζλ=kˆT−,1γ˜ζkkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜+λ−1kˆT−,1γ˜λ2−021ζpλ2021kˆT,γ˜dz¯
1−1
λ0λ0λ210λ−210
+λkˆT−,1γ˜0λ−21ζp0λ21kˆT,γ˜dz=kˆT−,1γ˜ζλ−1kˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜,(4.4.101)

65

eforebassimilarlyand(P˜λkˆT,γ˜)−1d(P˜λkˆT,˜γ)=kˆT−,1γ˜P˜λ−1dP˜λkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜
=kˆT−,1γ˜ζλkˆT,γ˜+kˆT−,1γ˜dkˆT,γ˜=γ˜∗ζλ−1=(P˜λ−1◦γ˜)−1d(P˜λ−1◦γ˜).(4.4.102)
Sobyfollowingverbatimtheargumentofthefirstcase,wederive
P˜λ−1◦γ˜=M˜γ˜(λ)P˜λkˆT,γ˜,(4.4.103)
whereM˜γ˜(λ)isindependentofz.Moreover,forλ=1,wehaveP˜λ−1=P˜λ=P˜andthusobtain
P˜◦γ˜=M˜˜γ(1)P˜kˆT,γ˜.(4.4.104)
Comparingthisto(4.4.47),weinferthatM˜γ˜(1)=±G(1)ATG(1)−1.
Finally,westateintheorem4.17thetransformationbehaviouroftheextended˜frame˜F:M˜→
ΛSU(2)σwithrespecttothebiholomorphic(resp.bi-antiholomorphic)mappingγ˜:M→Massociated
wewithdefinethethesymmetrymatrixTkTof,γ˜a∈givSU(2),entrinoidwhichiswithindeppropendenerlytemofbλ,eddedbyannularends.Inpreparationofthis,
√kT,γ˜:=A0+iB√A0+iB,(4.4.105)
whereA,B:M˜→Rdependonγ˜andareexplicitlygivenbytheequations(4.4.40)and(4.4.41).
Moreover,theoccurringcomplexsquarerootsaredefinedasinremark4.14.
Theorem4.17.Letψ:M˜→R3beaconformalCMC-immersion,3whichissymmetricwithrespectto
Tvia:xthe→ATuniversalx+tcT.overingAssumeπ:ψM˜c→orrMesp,ψonds=φto◦aπ.Ltrinoidetγ˜φ:denoteM→aRwithbiholomorphicproperl(ryesp.embeddedannulbi-antiholomorphic)arends
mappingM˜→M˜associatedwithTbytheorem4.9.ThentheextendedframeF:M˜→ΛSU(2)σ
correspondingtoψbytheorem4.5transformsunderγ˜asfollows.
1.IfTpreservesorientation,then
F(γ˜(z),λ)=Mγ˜(λ)F(z,λ)kT,γ˜(z),(4.4.106)
ofwherz.eInkTp,γ˜articuisgivenlar,forinλ=(4.4.105)1,weandhaveMγ˜(λ)denotesanelementofΛSU(2)σ,whichisindependent
Mγ˜(1)=±AT,(4.4.107)
whereAT∈SU(2)denotestheconjugationmatrixrealizingAT∈O(3)inthesu(2)-model.
2.IfTreversesorientation,then
F(γ˜(z),λ−1)=Mγ˜(λ)F(z,λ)kT,γ˜(z),(4.4.108)
wherekT,γ˜isgivenin(4.4.105)andMγ˜(λ)denotesanelementofΛSU(2)σ,whichisindependent
ofz.Inparticular,forλ=1,wehave
10Mγ˜(1)=±AT−10,(4.4.109)
whereAT∈SU(2)denotestheconjugationmatrixrealizingAT∈O(3)inthesu(2)-model.
Proof.ByconstructionofF,wehaveF=G(λ)−1P˜λG(λ).Therearetwocasestoconsider.Inthecase
thatTpreservesorientation,weusetheabovelemmaandobtain
F(γ˜(z),λ)=G(λ)−1((P˜λ◦γ˜)(z))G(λ)=G(λ)−1M˜γ˜(λ)P˜λ(z)kˆT,γ˜(z)G(λ)
=G(λ)−1M˜γ˜(λ)G(λ)F(z,λ)G(λ)−1kˆT,γ˜(z)G(λ).(4.4.110)

66

AsTisorientationpreserving,wehaveby(4.2.17)and(4.4.54)
√G(λ)−1kˆT,γ˜G(λ)=A+iB√0=kT,γ˜,(4.4.111)
iB+A0andbydefiningMγ˜(λ):=G(λ)−1M˜γ˜(λ)G(λ),wealtogetherobtainF(γ˜(z),λ)=Mγ˜(λ)F(z,λ)kT,γ˜(z),
whichproves(4.4.106).Clearly,Mγ˜(λ)isindependentofz,andfurthermore,asF(γ˜(z),λ),F(z,λ)and
kT,γ˜(z)areelementsofΛSU(2)σ,soisMγ˜(λ).Finally,forλ=1,weinferfrom(4.4.86)that
Mγ˜(1)=G(1)−1M˜γ˜(1)G(1)=±G(1)−1G(1)ATG(1)−1G(1)=±AT.(4.4.112)
ThesecondcasetoconsideristhecaseoforientationreversingT.Usingtheabovelemma,weobtain
analogouslyF(γ˜(z),λ−1)=G(λ−1)−1((P˜λ−1◦γ˜)(z))G(λ−1)
=G(λ−1)−1M˜γ˜(λ)P˜λ(z)kˆT,γ˜(z)G(λ−1)=G(λ−1)−1M˜γ˜(λ)G(λ)F(z,λ)G(λ)−1kˆT,γ˜(z)G(λ−1)
=G(λ−1)−1M˜γ˜(λ)G(λ)010−1F(z,λ)010−1G(λ)−1kˆT,γ˜(z)G(λ−1).

−1010−1010
(4.4.113)Thistimeweconsiderequations(4.2.17)and(4.4.55)toobtain
√0−1G(λ)−1kˆT,γ˜G(λ−1)=A+iB√0=kT,γ˜.(4.4.114)
100A+iB
Moreover,wehave
0−1F(z,λ)01=F(z,λ),(4.4.115)
0101−andbydefiningMγ˜(λ):=G(λ−1)−1M˜γ˜(λ)G(λ)01wearriveatequation(4.4.108).Mγ˜(λ)is
01−independentofzand,asF(γ˜(z),λ−1),F(z,λ)andkT,γ˜areelementsofΛSU(2)σ,soisMγ˜(λ).Finally,
forλ=1,weinferfrom(4.4.88)that
Mγ˜(1)=G(1)−1M˜γ˜(1)G(1)01=±G(1)−1G(1)ATG(1)−1G(1)01=±AT01.

−10−10−10
(4.4.116)

AsweareinterestedinexplicitlycomputingthematrixkT,γ˜,westateitinamoreconvenientform,
whichinvolvesmoredirectlythemappingγ˜associatedwiththesymmetryTbytheorem4.9.
Lemma4.18.Letψ:M˜→R3beaconformalCMC-immersion,whichcorrespondstoatrinoidφ:M→
R3withproperlyembeddedannularendsviatheuniversalcoveringπ:M˜→M,ψ=φ◦π.Moreover,
letT∈bi-antiholomorphic)Sym(ψ(M˜))bemappinggivenMby˜T→:Mx˜→assoATxciate+dtT.withFTbyurthermortheore,emletγ˜4.9.denoteThen,athe(λbiholomorphic-indep(rendent)esp.
matrixkT,γ˜definedin(4.4.105)satisfies
∂zγ˜
kT,γ˜=|∂zγ˜|0ifTpreservesorientation,(4.4.117)
0|∂∂zzγ˜γ˜|
∂¯zγ˜0
kT,γ˜=|∂z¯γ˜|∂z¯γ˜ifTreversesorientation,(4.4.118)
0|∂z¯˜γ|
wherewewriteforeaseofnotation∂zγ˜(resp.∂z¯γ˜)for∂∂zγ˜(resp.∂∂z¯γ˜).

67

(4.4.117)(4.4.118)

Proof.Weknowfrom(4.4.105)that
√kT,γ˜=A0+iB√A0+iB,(4.4.119)
whereAandBaregivenin(4.4.40)and(4.4.41).Recallingγ˜=γ˜1+iγ˜2,weinfer
A+iB=∂∂γ˜x1(x+iy)+i∂∂γ˜x2(x+iy)=∂∂xγ˜.(4.4.120)
x∂x∂∂γ˜1(x+iy)2+∂γ˜2(x+iy)2|∂∂xγ˜|
As∂∂x=∂∂z+∂∂z¯and,moreover,∂∂z¯γ˜=0(resp.∂∂zγ˜=0)forholomorphic(resp.antiholomorphic)γ˜,i.e.in
thecaseofanorientationpreserving(resp.reversing)symmetryT,weconcludethat
γ˜∂zA+iB=|∂zγ˜|ifTpreservesorientation,(4.4.121)
γ˜∂z¯A+iB=|∂z¯γ˜|ifTreversesorientation.(4.4.122)
By(4.4.119),theclaimfollows.
frameAsFaspecialtransformscaseofunderatheoremcov4.17eringweformtransformationulatetheγ˜folloonM˜wing.Thiscorollaryresult,iswhicobhtainedstatesbhoywsettingtheTextended=I:
R3→R3,x→I(x)=xintheorem4.17andinterpretingγ˜asa(biholomorphic)mappinglinkedtoT
bytheorem4.9.(As,bydefinition,π◦γ˜=π,wehaveobviouslyT◦ψ=ψ=φ◦π=φ◦π◦γ˜=ψ◦γ˜.)
Corollary34.19.Letψ:M˜→R3beaconformalCMC-immersion,whichcorrespondstoatrinoid
˜γ˜φ:Mdenote→aRcwithoveringproptrerlyansformationembeddedonMannular˜,i.e.endsaviathebiholomorphicuniversalcmappingoveringM˜π→:M˜M→assoM,ciateψd=byφ◦theπ.orLemet
33F4.9:M˜with→theΛSU(2)identitycorrespmappingondingI:Rtoψ→byRtheor(interprem4.5etedtrasaansformssymmetryunderofγ˜ψas).folThenlows:theextendedframe
σF(γ˜(z),λ)=Mγ˜(λ)F(z,λ)kI,γ˜(z),(4.4.123)
wherekI,γ˜isgivenin(4.4.105)withT=IandMγ˜(λ)denotesanelementofΛSU(2)σ,whichis
.zofendentindep4.5Theextendedframemonodromyrelations
Weapplycorollary4.19tothecoveringtransformationsγ˜j,j=0,1,∞,onM˜fromsection3.3.Denoting
thecorrespondingmatriceskI,γ˜0,kI,γ˜1andkI,γ˜∞inequation(4.4.123)ofcorollary4.19byk0,k1and
k∞,respectively,weobtainforj=0,1,∞:
F(γ˜j(z),λ)=Mγ˜j(λ)F(z,λ)kj(z),(4.5.1)
whereMγ˜(λ)denotesanelementofΛSU(2)σ,whichisindependentofz.
Thejmatriceskj=kI,γ˜j,j=0,1,∞,aregivenin(4.4.105)withT=Iandγ˜=γ˜j.Wecomputek0,
k1andk∞explicitlyinthefollowing.First,recallfrom(3.3.4),(3.3.5)and(3.3.6)that
zγ˜0(z)=−2z+1,(4.5.2)
γ˜1(z)=z+2,(4.5.3)
γ˜∞(z)=−3z−2.(4.5.4)
1+z2Thus,wehave
11∂zγ˜0=(1−2z)2,∂zγ˜1=1,∂zγ˜∞=(1+2z)2,(4.5.5)
,ytlconsequenand,11|∂zγ˜0|=(1−2z)(1−2z¯),|∂zγ˜1|=1,|∂zγ˜∞|=(1+2z)(1+2z¯).(4.5.6)
68

Usinglemma4.18,wecaneasilycomputek0,k1andk∞fromequation(4.4.117):
01−2z¯
k0(z)=1−2z1−2z¯,(4.5.7)
0101−2z
k1(z)=01,(4.5.8)
1+2¯z0
0k∞(z)=1+2z1+2z¯.(4.5.9)
z1+2Equation(4.5.1)statesthetransformationbehaviourofFunderthecoveringtransformationγ˜jon
M˜.Recallthat,atthesametime,wehaveforourdressedsolutionˆΨ=TΨtoequation(2.4.1)the
relationydrommonoˆΨ(γ˜j(z),λ)=Mˆj(λ)ˆΨ(z,λ),(4.5.10)
whereMˆj(λ)denotesanelementofΛSU(2)σ,whichisindependentofz,andwhichisoftheform(3.9.26).
Asexplicatedinremark2.10,thisrelationcarriesovertotheextendedframeFcorrespondingtoˆΨ:
F(γ˜j(z),λ)=Mˆj(λ)F(z,λ)k(z,γ˜j),(4.5.11)
wherekdenotesadiagonalmatrixinSU(2),whichisindependentofλ.Likeequation(4.5.1)before,
equation(4.5.11)statesaswellthetransformationbehaviourofFunderthecoveringtransformationγ˜j
˜.MonThus,intheterminologyof[14],both(γ˜j,Mγ˜j)and(γ˜j,Mˆj)definesymmetriesoftheextendedframe
F.However,bytheorem2.1of[14],thisimpliesthatMγ˜jandMˆjdifferatmostbyasign:
Mγ˜j(λ)=αjMˆj(λ),(4.5.12)
whereαj∈{±1}.Insertingthisrelationintoequation(4.5.1),weobtainthefollowingresult:
Theorem4.20.LetM˜=Handψ:M˜→R3beaconformalCMC-immersion,whichcorrespondsto
atrinoidφ:M→R3withproperlyembeddedannularendsviatheuniversalcoveringπ:M˜→M,
ψ=φ◦π.Letγ˜j,j=0,1,∞,denotethecoveringtransformationsonM˜fromsection3.3.Then,the
extendedframeF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜jasfollows:
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),(4.5.13)
whereαj∈{±1},thematricesMˆj(λ)areoftheform(3.9.26)andthematriceskj(z)aregivenby
equations(4.5.7)to(4.5.9).

symmetriesrinoidT4.6Inthissection,weinvestigateindetailthepossiblesymmetriesofatrinoidwithproperlyembedded
ends.ularannInthefollowing,letM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannular
ends,whichissymmetricwithrespecttoanEuclideanmotionT∈Sym(φ(M)),i.e.
T(φ(M))=φ(M).(4.6.1)
Bytheorem4.9,thereexistsauniquebiholomorhic(resp.bi-antiholomorhic)mappingγ=γ(T):M→M
ForsatisfyingeachTT˜◦∈φ=Sym(φφ◦(γM.)),Inwhicfact,hwealsoobservsatisfiesethatT˜◦theφ=correspφ◦γondence(forthebetwsameeenγT),wandehaγvisenecessarilyone-to-one:
3˜˜T|φ(MThe)≡folloT|φ(wingM)leandmmathusTexplicitly≡Tlists(onRall).biholomorhic(resp.bi-antiholomorhic)mappingsγ:M→M.
Lemma4.21.LetM=C\{0,1}andAut(M)={γ:M→M;γbiholomorphic},Aut∗(M)={γ:
M→M;γbi-antiholomorphic}.Then,thefollowingholds:

69

1.Aut(M)={γ(),γ(1∞),γ(0∞),γ(01),γ(01∞),γ(0∞1)},(4.6.2)
ewherγ():M→M,γ()(z)=z,(4.6.3)
zγ(1∞):M→M,γ(1∞)(z)=z−1,(4.6.4)
γ(0∞):M→M,γ(0∞)(z)=1,(4.6.5)
zγ(01):M→M,γ(01)(z)=1−z,(4.6.6)
1γ(01∞):M→M,γ(01∞)(z)=1−z,(4.6.7)
γ(0∞1):M→M,γ(0∞1)(z)=zz−1.(4.6.8)
2.Aut∗(M)={γ(),γ(1∞),γ(0∞),γ(01),γ(01∞),γ(0∞1)},(4.6.9)
ewherγ(∗):M→M,γ(∗)(z)=z¯,(4.6.10)
z¯γ(1∗∞):M→M,γ(1∗∞)(z)=¯z−1,(4.6.11)
γ(0∗∞):M→M,γ(0∗∞)(z)=z¯1,(4.6.12)
γ(0∗1):M→M,γ(0∗1)(z)=1−z¯,(4.6.13)
1γ(0∗1∞):M→M,γ(0∗1∞)(z)=1−z¯,(4.6.14)
γ(0∗∞1):M→M,γ(0∗∞1)(z)=z¯z¯−1.(4.6.15)
Remark4.22.Thenotationintroducedinthelemmaaboveforthedifferentbiholomorhic(resp.bi-
antiholomorhic)mappingsγ:M→Mismotivatedbythewayeachγ(extendedtoabiholomorhicor
bi-antiholomorhicmappingCˆ→Cˆ)permutestheset{0,1,∞}.Thisisexplainedinmoredetailinthe
of.prowingfollo∗Prbi-anoofoflemmatiholomorphic)4.21.Letmappingγ∈MAut(→MM),it(resp.canγ∈beAut(uniquelyM)).Sinceextendedγtodefinesaabiholomorphicbiholomorphic(resp.(rebi-sp.
antiholomorphic)mappingγextd:Cˆ→Cˆsuchthat
γextd|M=γ.(4.6.16)
Sinceγextdisbijective,wenecessarilyhave
γextd({0,1,∞})={0,1,∞},(4.6.17)
i.e.γextdpermutestheset{0,1,∞}accordingtoanappropriatepermutationσof{0,1,∞}:
γextd(zj)=zσ(j)(4.6.18)
forItalliszja=wjell∈{kno0,1wn,∞}result.ofcomplexanalysisthatγextdisoftheform
γextd:z→czaz++db(4.6.19)
withcomplexparametersa,b,c,dsatisfyingad−bc=0inthecasethatγ(andthusγextd)isbiholomor-
formtheofandphic,γextd:z→ca¯zz¯++bd(4.6.20)
70

withcomplexparametersa,b,c,dsatisfyingad−bc=0inthecasethatγ(andthusγextd)isbi-
antiholomorphic.Evaluatingtherelation(4.6.18)forj=0,1,∞in(4.6.19)and(4.6.20)(ofcourse,with
respecttoσ),theparametersa,b,c,dcanbeexplicitlycomputed(uptoacommoncomplexscale,which
cancelsin(4.6.19)and(4.6.20),respectively).Inviewof(4.6.16),weobtainthepossibleexplicitformsof
γ=γσ(resp.γ=γσ∗)giveninequations(4.6.3)to(4.6.8)andequations(4.6.10)to(4.6.15),respectively.
Thisprovestherelations“⊆”inthetwoclaimedidentitiesabove.Therelations“⊇”aretrivial.
Correspondingtotheelementsγ∈Aut(M)(resp.γ∈Aut∗(M)),wedefinethefollowingauxiliary
functionsh:M→C\{0},whichareholomorphic(resp.antiholomorphic)inM:
4.23.Definition

h():M→C\{0},h()(z)=1,(4.6.21)
h(1∞):M→C\{0},h(1∞)(z)=−i(z−1),(4.6.22)
h(0∞):M→C\{0},h(0∞)(z)=−iz,(4.6.23)
h(01):M→C\{0},h(01)(z)=−i,(4.6.24)
h(01∞):M→C\{0},h(01∞)(z)=1−z,(4.6.25)
h(0∞1):M→C\{0},h(0∞1)(z)=z,(4.6.26)
h(∗):M→C\{0},h(∗)(z)=1,(4.6.27)
h∗(1∞):M→C\{0},h∗(1∞)(z)=−i(z¯−1),(4.6.28)
h∗(0∞):M→C\{0},h∗(0∞)(z)=−iz¯,(4.6.29)
h∗(01):M→C\{0},h∗(01)(z)=−i,(4.6.30)
h∗(01∞):M→C\{0},h∗(01∞)(z)=1−z¯,(4.6.31)
h∗(0∞1):M→C\{0},h∗(0∞1)(z)=z¯.(4.6.32)
Lemma4.24.Letσbeapermutationoftheset{0,1,∞}.Denotebyγσ(resp.γσ∗)theelementof
Aut(M)(resp.ofAut∗(M))correspondingtoσasinlemma4.21.Moreover,denotebyhσ(resp.hσ∗)the
auxiliaryfunctioncorrespondingtoσasdefinedindefinition4.23.Then,forallz∈M,thefollowing
holds:1.∂zzhs(z)=0,∂z¯z¯hs∗(z)=0.(4.6.33)
2.21∗21
(hs(z))=∂zγσ(z),(hs(z))=∂z¯γσ∗(z).(4.6.34)
Proof.Thisisprovedbydirectcomputation.
Lemma4.25.Letη=η(z,λ)beastandardizedtrinoidpotentialonM=C\{0,1,∞}associatedwith
threeoff-diagonalDelaunaymatricesD0,D1,D∞possessingtheeigenvalues±µj(λ),respectively,
0λ−1
η=−λQ(z,λ)0dz,(4.6.35)
ewherQ(z,λ)=b0(2λ)+b1(λ)2+c0(λ)+c1(λ),(4.6.36)
z(z−1)zz−1
andthefunctionsbj,cjsatisfyequations(3.6.4),(3.6.5)and(3.6.6).
1.Letσbeapermutationoftheset{0,1,∞}.Denotebyγ:=γσtheelementofAut(M)corresponding
toσasinlemma4.21.Moreover,denotebyh:=hσtheauxiliaryfunctioncorrespondingtoσas
givenindefinition4.23anddefineW+:M→Λ+SL(2,C)σby
h(z)0
W+(z,λ)=−λ∂zh(z)(h(z))−1.(4.6.37)

71

haveweThen,γ∗η=η#W+⇐⇒Q(γ(z),λ)=(h(z))4Q(z,λ),(4.6.38)
whereγ∗ηdenotesthetransformofηunderγandη#W+denotesthegaugedpotentialW+−1ηW++
W+−1dW+.
2.Letσbeapermutationoftheset{0,1,∞}.Denotebyγ:=γ∗σtheelementofAut∗(M)correspond-
ingtoσasinlemma4.21.Moreover,denotebyh+:=hσ∗theauxiliaryfunctioncorrespondingtoσ
asgivenindefinition4.23anddefineW+:M→ΛSL(2,C)σby
h(z)0
W+(z,λ)=−λ∂z¯h(z)(h(z))−1.(4.6.39)

haveweThen,γ∗η(z,λ)=η(z,λ−1)#W+⇐⇒Q(γ(z),λ)=(h(z))4Q(z,λ−1),(4.6.40)
whereγ∗ηdenotesthetransformofηunderγandη(z,λ−1)#W+denotesthegaugedpotential
W+−1η(z,λ−1)W++W+−1dW+.
Proof.Westartwiththeproofofthefirstcase.Using(4.6.34)fromlemma4.24,wehave
∗0λ−10λ−1h−2
γη=−λQ(γ(z),λ)0∂zγdz=−λh−2Q(γ(z),λ)0dz.(4.6.41)
Furthermore,using(4.6.33)fromlemma4.24,wecompute
η#W+=W+−1ηW++W+−1dW+
h−100λ−1h0h−10∂zh0
=λ∂zhh−λQ(z,λ)0−λ∂zhh−1dz+λ∂zhh0−h−2∂zhdz
−h−1∂zhλ−1h−2h−1∂zh00λ−1h−2
=−λh2Q−λ(∂zh)2h−1∂zhdz+λ(∂zh)2−h−1∂zhdz=−λh2Q0dz.(4.6.42)
Together,thisproves(4.6.38).
Wenowturntothesecondcase.Usingagain(4.6.34)fromlemma4.24,wehave
∗0λ−10λ−1h−2
γη=−λQ(γ(z),λ)0∂z¯γ(z)dz¯=−λh−2Q(γ(z),λ)0dz¯.(4.6.43)
Furthermore,usingagain(4.6.33)fromlemma4.24,wecompute(λ∈S1)
η(z,λ−1)#W+=W+−1η(z,λ−1)W++W+−1dW+
h−100λ−1h0h−10∂z¯h0
=λ∂z¯hh−λQ(z,λ−1)0−λ∂z¯hh−1dz¯+λ∂z¯hh0−h−2∂z¯hdz¯
−h−1∂z¯hλ−1h−2h−1∂z¯h0
=−λh2Q(z,λ−1)−λ(∂z¯h)2h−1∂z¯hdz¯+λ(∂z¯h)2−h−1∂z¯hdz¯
0λ−1h−2
=−λh2Q(z,λ−1)0dz.(4.6.44)

Together,thisproves(4.6.40).
Inviewoftheone-to-onecorrespondenceexplicatedearlier,betweenpossibleorientationpreserving
(resp.orientationreversing)symmetriesofagiventrinoidφ:M→R3withproperlyembeddedannular
endsontheonehandandbiholomorphic(resp.bi-antiholomorphic)mappingsγ:M→Monthe
otherhand,itisadirectconsequenceoflemma4.21thatagiventrinoidφ:M→R3withproperly
embeddedannularendsallowsforatmosttwelvesymmetries,namelysixorientationpreservingonesand
sixorientationreversingones.
Asseenbefore,abiholomorphic(resp.bi-antiholomorphic)mappingM→Misentirelycharacterized
bythewayit(or,moreprecisely,itsbiholomorhicorbi-antiholomorhicextensionCˆ→Cˆ)permutesthe

72

set{0,1,∞}.Recallthatwewriteγσ(resp.γσ∗)fortheuniquebiholomorphic(resp.bi-antiholomorphic)
mappingM→M,whichpermutestheset{0,1,∞}accordingtothepermutationσ.
Inviewofthediscussed3correspondence,weadoptthisnotationforthetwelvepossiblesymmetriesT
{of0,a1,giv∞}en,fortrinoidtheφorien:Mtat→ionRwithpreservingproponeserlyandembT=eddedTσ∗,annσpularermends,utationwritingof{0,T1,=∞}T,σ,forσptheermorienutationtationof
ones.ersingrevInfact,denotingthetwelvepossiblesymmetriesofatrinoidφ:M→R3withproperlyembedded
annularendsbyTσandTσ∗,whereσrepresentsthesixpossiblepermutationsoftheset{0,1,∞},canbe
motivatedmoredirectly,asshowninthefollowinglemma.
Lemma4.26.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannularends
Bj=φ(Uj),whichissymmetricwithrespecttotheEuclideanmotionT∈Sym(φ(M)).Moreover,let
γdenotethebiholomorphic(resp.bi-antiholomorphic)mappingM→MassociatedwithTbytheorem
pr4.9eservesandcharorientationacterizedbyandγthe=pγ∗,ermutationT=T∗σinofthethecsetase{0,that1,T∞}r,i.e.eversesγ=γσ,orientation.T=TσinThen,thethecasefolthatlowingT
σσholds:1.Foreachj∈{0,1,∞},thereexistsanopen,non-emptypuncturedneighborhoodUˆj⊆Ujofzjsuch
thatT(φ(Uˆj))⊆Bσ(j).(4.6.45)
2.Foreachj∈{0,1,∞},letAj⊆R3denotethetrinoidaxisofφatzj=j.Then,
T(Aj)=Aσ(j).(4.6.46)
Proof.Webeginwiththeproofoftheˆfirstˆclaim.Asbefore,denotebyγextdtheuniquebiholomorphic
(resp.bi-antiholomorphic)mappingC→Cwithγextd|M≡γ.Then,bydefinitionofγ=γσ(resp.
γ=γσ∗),γextdpermutestheset{0,1,∞}accordingtoσ,i.e.γextd(zj)=zσ(j)forallzj=j∈{0,1,∞}.
Consequently,sinceγiscontinuousonM,thereexistsforeachj∈{0,1,∞}anopen,non-empty
puncturedneighborhoodUˆjofzjinM,suchthat
γ(Uˆj)⊆Uσ(j).(4.6.47)
W.l.o.g.,wecanassumeUˆj⊆Uj.
Since,bytheorem4.9,γsatisfiesT◦φ=φ◦γ(onM),weconcludethat,foreachj∈{0,1,∞},there
existsanopen,non-emptypuncturedneighborhoodUˆj⊆Ujofzjsuchthat
T(φ(Uˆj))=φ(γ(Uˆj))⊆φ(Uσ(j))=Bσ(j),(4.6.48)
claimed.asThesecondclaimisadirectconsequenceofthefirstone:By(4.6.45),thereexistsforeachj∈{0,1,∞}
anopen,non-emptypuncturedneighborhoodUˆj⊆Ujofzj,suchthatTmapsBˆj:=φ(Uˆj)(whichformsa
“sub-end”ofBj)toBσ(j),i.e.T(Bˆj)formsa“sub-end”ofBσ(j).Recallingthat,foreachj∈{0,1,∞},the
properlyembeddedannularendBjofφasymptoticallyshowsthebehaviourofanunduloidalDelaunay
surfaceφj,weproceedasfollows:SinceT(Bˆj)⊆Bσ(j),weinferthatforeachj∈{0,1,∞},T(Bˆj)
T(andisconthustinalsouous,theit“supnecessarilyer-end”mapsT(Bjthe))(imagesasymptoticallyofthe)shocorrespwstheondingbehaviourDelaunaofyφσ(jsurfaces).Moreo(asver,subsetssinceof
R3)ontoeachother,i.e.T(im(φ))=im(φ).Consequently,alsotherelatedrevolutionaxesofthe
Delaunaysurfaces,i.e.thetrinoidjaxesAj,jσ(∈j){0,1,∞},ofφaremappedbyTontoeachother:
T(Aj)=Aσ(j),(4.6.49)
claimed.asRemark4.27.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannularends
Bj=φ(Uj),whichissymmetricwithrespecttotheEuclideanmotionT∈Sym(φ(M)).Bylemma4.26,
Tmapsforeachj∈{0,1,∞}atleastsome“outerpart”φ(Uˆj)ofthetrinoidendBj=φ(Uj)tothe
trinoidendBσ(j).Inthissense,Tpermutesthetrinoidendsaccordingtothepermutationσ.
Inadditiontotheresultabove,wehavethefollowing:

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Lemma4.28.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannularends
Bj=φ(Uj)andcorrespondingtrinoidaxesAj=Cj+Rvj⊆R3,involvingabasepointCj∈R3andaunit
directionvectorvj∈R3,pointingtowardsthetrinoidendBj.Moreover,letφbesymmetricwithrespect
totheEuclideanmotion∗T∈Sym(φ(M))associatedwiththepermutationσoftheset{0,1,∞},i.e.
T(r=esp.Tσby(rt∈esp.RT3)=theTσ)orthoforgonalorientationpart(rpresp.eservingthetr(resp.anslationalorientationpart)ofrT,eversing)i.e.TT(.x)=DenoteAx+bytA(on∈RO(3)3).
holds:lowingfoltheThen,1.Forallj∈{0,1,∞},wehave
Avj=vσ(j).(4.6.50)
2.Moreover,wecanassumewithoutlossofgeneralityforallj∈{0,1,∞}that
T(Cj)=Cσ(j).(4.6.51)
Proof.Thefirstclaimisprovedasfollows:Letj∈{0,1,∞}.Since,bylemma4.26,T(Aj)=T(Aσ(j)),
thereexistsforeachλ∈Rarealnumberµσ(j)=µσ(j)(λ)∈R,suchthat
T(Cj+λvj)=Cσ(j)+µσ(j)(λ)vσ(j).(4.6.52)
Inparticular,forλ=0,wehave
T(Cj)=Cσ(j)+µσ(j)(0)vσ(j).(4.6.53)
Notethat,asTdefinesacontinuousbijectionR3→R3,µdefinesacontinuousbijectonR→R.Moreover,
psinceermTuted,permcorresputestheondingtrinoidaxes:ends“outer”inthepointssenseonofthelemmaaxisAj4.26,(i.e.itppoinrestservonesAjtherelativorienelytation“close”ofthetoalsothe
endBj)aremappedbyTto“outer”pointsontheaxisAσ(j).Inotherwords,µσ(j):R→Rdefinesa
bijection.increasingstrictlyInviewofT(x)=Ax+t(onR3),weobtainbysubstracting(4.6.53)from(4.6.52)that
λAvj=(µσ(j)(λ)−µσ(j)(0))vσ(j).(4.6.54)
As(4.6.54)holdsforallλ∈R,andsincev=0=vandµisstrictlyincreasing,Avnecessarily
equalsapositivemultipleofvσ(j).Sincevjjandvσ(j)σ(jare)unitjvectorsinR3,andAisjorthogonal,we
thatconcludeAvj=vσ(j),(4.6.55)
whichfinishestheproofofthefirstclaim.
Wenowturntotheproofofthesecondclaim,i.e.weshowthatwecanalwayschoosethebasepoints
CjofthetrinoidaxesAj,suchthat(4.6.51)holdsforallj∈{0,1,∞}.
Forastart,weinferfrom(4.6.54)and(4.6.55)thatµσ(j)(λ)=µσ(j)(0)+λandthus,by(4.6.52),
T(Cj+λvj)=Cσ(j)+(µσ(j)(0)+λ)vσ(j).(4.6.56)
Denoteasbeforebyγthebiholomorphic(resp.bi-antiholomorphic)mappingM→Massociated
withTbytheorem4.9,T◦φ=φ◦γ.Inview6oflemma4.21,weobservethateitherγ2≡id(incase
thatWσe∈{(consider),(1∞the),(0first∞),case:(01)γ}2)≡oridatandleastσγ∈≡{(id),(in(1∞case),(0that∞σ),∈(0{(01)}1.∞)Com,(0∞bining1)})theholds.identities
T◦φ=φ◦γandγ2≡idyieldsT2|φ(M)≡idandthusT2|R3≡id.Consequently,wehaveforallx∈R
thatA2x+At+t=T2(x)=x,whichdirectlyimplies(settingx=0)At+t=0andthusA2=I.It
followsthatAhasonlyeigenvalues±1andthereforeinducesaneigenspacedecompositionofR3intoU+
andU−.NotethatAt+t=0impliesthatt∈U−.
intercThephangingermtheutationotherσtweithero.Inktheeepsfirstallthreecase,wpeoinctsho0,ose1forandeach∞j∈fixed{0,or1,k∞}eepsanoneparbitraryointCjfixed∈Awjhileas
basepointofAj.By(4.6.56),wehave
ACj+t=T(Cj)=Cj+µj(0)vj.(4.6.57)
Asσ(j)=j,wehaveAvj=vjandthusvj∈U+.Moreover,writingCj=Cj++Cj−withCj+∈U+,Cj−∈
U−andrecallingt∈U−,comparisonoftheU+-partsofbothsidesin(4.6.57)yieldsCj+=Cj++µj(0)vj

74

andthereforeµj(0)=0,whichby(4.6.57)impliesT(Cj)=Cj=Cσ(j).ThisfinishestheproofforT
associatedwithσ=().
Inthecasethatσkeepsonepointfixed(say,j),whileinterchangingtheothertwo(say,kandl),we
proceedasfollows:ChooseCj∈Aj,thenbytheargumentsgivenaboveT(Cj)=Cj=Cσ(j).Moreover,
Tc2ho=oseid:CkT∈(CAl)k=andT2(setCkC)l=:=CTk.(CkThis)∈Al.finishesIttheremainsprooftoforshoTwTasso(Cl)ciated=Ck.withButσ∈this{(1follo∞)ws,(0∞directly),(01)from}.
Itremainstoconsiderthesecondcase:γ6≡idandσ∈{(01∞),(0∞1)}.Combiningtheidentities
T◦φ=φ◦6γandγ56≡id4yields3T6|φ2(M)≡idandth6usT6|R3≡id.Consequently,wehaveforall
xA5∈t+RA4tthat+AA3tx++A2At+tA+t+Att=+0Aandt+thAustA6+=AtI.+Itt=folTlows(x)that=xA,3whichashonlydireeigenctlyvaluesimplies±1and(settingxtherefore=0)
inducesaneigenspacedecompositionofR3intoU+andU−.NotethatA5t+A4t+A3t+A2t+At+t=0
impliesthatA2t+At+t∈U−.
Weconsiderw.l.o.g.onlythepermutationσ=(01∞).(σ=(0∞1)istreatedcompletelyanal-
T(Cogously).)=CCho,i.e.oseTC3(0C∈)A=0C.andResetpCeated1:=T(applicationC0)∈ofA1,C(4.6.56)∞:=yieldsT(C1)∈A∞.Itremainstoshow
000∞A3Cj+A2t+At+t=T3(Cj)=Cj+(µj(0)+µσ(j)(0)+µσ2(j)(0))vj.(4.6.58)
As−σ3(j)=j,wehaveA3vj=vjandthusvj∈U+.Moreover,writingCj=Cj++Cj−withCj+∈U+,
Cj∈U−andrecallingA2t+At+t∈U−,comparisonoftheU+-partsofbothsidesin(4.6.58)yields
Cj+=Cj++(µj(0)+µσ(j)(0)+µσ2(j)(0))vjandtherefore(µj(0)+µσ(j)(0)+µσ2(j)(0))=0,whichby
(4.6.58)impliesT3(Cj)=Cj,inparticularT3(C0)=C0.ThisfinishestheproofforTassociatedwith
σ=(01∞)(andσ=(0∞1)).
Remark4.29.Notethat,by[28],inadditionto(4.3.4)thereholdsanother“balancingformula”involving
the3torquesofthetrinoidaxesAj,whichimplies(cf.[28])thatthethreetrinoidaxesarecoplanarin
Randeitherareallparallelormeetinacommonpoint.Infact,thisholdsmoregenerallyforall
surfaces.CMC-immersionsRecentcommwiththreeunicationannularwithR.endsKusnerwhichareandN.Scasymptotichmittto(nsuggestsotthatnecessarilyinthecaseunduloidal)oftrinoidsDelaunawithy
properlyembedded(i.e.asymptoticunduloidal)annularendsparallelaxescannotoccurandthusthe
threetrinoidaxesnecessarilymeetinonepoint.Inviewofthis,onecouldassumew.l.o.g.C0=C1=C∞
for4.28thearethrealmostebasetrivial.pointsMoreoforvtheer,trithenoidproofaxesofA0,theoremA1and4.31A∞.simplifiesInthissisettignificanng,tlythe.Hoclaimswever,ofmainllemmay
topreservetheadaptabilityoflemma4.28andtheorem4.31topossiblefuturework(e.g.,thestudy
ofCMC-immersionswiththreenotnecessarilyproperlyembeddedannularends),weretainthemore
general(andmorecomplicated)proofshere.
Thefollowingtheoremnowliststhetwelvepossibletrinoidsymmetriesexplicitly.Firstly,howeverwe
introducesomemorenotions.
Definition4.30.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannular
endsBj=φ(Uj)andcorrespondingtrinoidaxesAj=Cj+Rvj⊆R3,j=0,1,∞.Then:
1.ThepointC:=31(C0+C1+C∞)∈R3willbecalledthetrinoidcenter.
32.isLetCcalledadenotetrinoidthetrinoidplaneandcenter.willAnoftenybplaneeindenotedRconbyE.tainingMoreothever,affinedenotingsubspacetheC+(upRvto0+signRv1+Runique)v∞
unitnormalvectorofatrinoidplaneEbyn,thelineC+Rniscalledatrinoidnormalandwilloften
btheeplanedenotedCjby+ARnv.j+RFinallynis,cgivallenedaatrinoidtrinoidplanenormalEwithplanenormal(alongvtheectortrinoidn,foraxiseacAhj)j.∈{This0,1,plane∞}
willoftenbedenotedbyEj.
Theorem4.31.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannular
endsBj=φ(Uj)andcorrespondingtrinoidaxesAj=Cj+Rvj⊆R3,j=0,1,∞.DenotebyCthe
trinoidcenter31(C0+C1+C∞)∈R3.Moreover,letT∈Sym(φ(M))anddenotebyσthepermutationof
{0,1,∞}representingthetransformationbehaviourofthetrinoidendsBjunderT.Then,thefollowing
holds:1.IfTpreservesorientation,i.e.T=Tσ,wehave:

75

(a)Tσ=Iforσ=(),(4.6.59)
whereIdenotestheidentitymappingonR3.
(b)Tσ=R0forσ=(1∞),(4.6.60)
whereR0denotestherotationonR3bytheangleπaroundthetrinoidaxisA0.
(c)Tσ=R1forσ=(0∞),(4.6.61)
whereR1denotestherotationonR3bytheangleπaroundthetrinoidaxisA1.
(d)Tσ=R∞forσ=(01),(4.6.62)
whereR∞denotestherotationonR3bytheangleπaroundthetrinoidaxisA∞.
(e)Tσ=Rforσ=(01∞),(4.6.63)
whereRdenotestherotation2onR33bytheangle12±23πaroundthetrinoidnormalAn=
c{Case,+theλn;λve∈ctorsR},vj,wherj∈e{n0,∈1,S∞}⊆,Rspanasatisfiesplanenin⊥vRj3,forwhencallje∈n{is0,1,uniquel∞}.y(Notedeterminethat,dinupthisto
signandwecanspeakofthetrinoidnormalAnofφ.)
)(fTσ=R−1forσ=(0∞1),(4.6.64)
whereRisgivenabove.
2.IfTreversesorientation,i.e.T=Tσ∗,wehave:
(a)Tσ∗=Sforσ=(),(4.6.65)
whereSdenotesthereflectiononR3insometrinoidplaneE=C+(Rn)⊥,wheren∈S2⊆R3
satisfiesn⊥vjforallj∈{0,1,∞}.(Moreover,Eisuniquelydeterminedbytherelation
T◦φ=φ◦γ.)
(b)Tσ∗=S0forσ=(1∞),(4.6.66)
3wherwhereenS0∈S2denotes⊆R3thersatisfieseflenction⊥vjonforRallinj∈some{0,1tri,∞}noid.(Mornormaleover,planeE0Eis0=uniquelyC0+Rv0determine+Rnd,
bytherelationT◦φ=φ◦γ.)
(c)Tσ∗=S1forσ=(0∞),(4.6.67)
3wherwhereenS1∈S2denotes⊆R3thersatisfieseflenction⊥vjonforRallinj∈some{0,1,trinoid∞}.(Mornormaleover,planeE1Eis1=uniquelyC1+Rv1determine+Rnd,
bytherelationT◦φ=φ◦γ.)
(d)Tσ∗=S∞forσ=(01),(4.6.68)
whereS∞denotesthereflectiononR3insometrinoidnormalplaneE∞=C∞+Rv∞+Rn,
wheren∈S2⊆R3satisfiesn⊥vjforallj∈{0,1,∞}.(Moreover,E∞isuniquelydetermined
bytherelationT◦φ=φ◦γ.)
(e)Tσ∗=Sˆforσ=(01∞),(4.6.69)
whereSˆdenotestherotoreflectiononR3composedoftherotationbytheangle±23πaroundthe
trinoidnormalAn={C+λn;λ∈R},wheren∈S2⊆⊥R3satisfiesn⊥vjforallj∈{0,1,∞},
vjand,j∈the{r0,efle1,∞}ction,spinantheaplanetrinoidinRplane3,Ewhenc=eCn+is(Rn)uniquely.(Notedeterminethat,dinupthistocsignase,andthevewecctorsan
speakofthetrinoidnormalAnandthetrinoidplaneofφ,respectively.)
12discussedNoteinthatsecthetionsign5.oftherotationangleisdefinedwithrespecttotheorientationoftherotationaxis.Thisisfurther

76

)(fTσ∗=Sˆ−1forσ=(0∞1),(4.6.70)
whereSˆisgivenabove.
Proof.LetT∈Sym(φ(M)andγ:M→Mbethebi-(anti)holomorphicmappingassociatedwithTby
thetheoremset{04.9,1,and∞}explicitlyrepresentinggiventheinlemmatransformation4.21,bsatifyingehaviourT◦ofφ=theφ◦γtrinoid.LetσendsdenoteundertheT.pWermestartutationwithof
thecasethatTpreservesorientation,i.e.T=Tσ.
IfConsequenσ=(tly),,Twe|R3infer=Ib,ywherelemmaI4.21denotesthattheγ(z)iden=ztit.yThus,mappingT◦φon=R3φ.,whichimpliesthatT|φ(M)=id.
Ifσ=(1∞),weinferbylemma4.21thatγ(z)=z−z1andthusγ2=id.ThisimpliesT2◦φ=φ,i.e.
T2|=id,andconsequentlyT2|3=id.WritingTexplicitlyasT(x)=Ax+twithA∈SO(3)and
t∈φR(3M,)weobtainA2x+At+t=xRforallx∈R3andthereforeimmediately(forx=0)At+t=0,which
2yieldsaroundAsome=I.axisSinceinR3A∈conSO(3),taining0this∈R3.impliesThethatcaseAeither=IA=yieldsItor=0thatandAisthausT=rotationid,abycontheangletradictionπ
tothetheanfgleactπthat,aroundsomeaccordingaxistoinσ,RT3swconapstainingthe0∈trinoidR3.endsSince,B1byandlemmaB∞.4.28,Therefore,ApreservAisesatherotationdirectionby
vaxisectorRvv0.ofMoreothever,trinoidthesaxisameA0,lemmai.eAv0allo=wsv0,forAassumingactuallydefineswithoutthelossofrotationgeneralitbyythethatangleTkπeepsaroundthebasethe
0pointC0ofthetrinoidaxisA0fixed,i.e.T(C0)=C0.Consequently,wehavet=C0−AC0andthus
T(x)=Ax+C0−AC0=A(x−C0)+C0,(4.6.71)
i.e.whicharoundmeansthethattrinoidTdefinesaxisA0the=C0rotation+Rvb0y.theTheanglecasesπσ=around(0∞)theandσrotation=(0ax1)isofareAtreatedtranslatedbcompletelyyC0,
.analogouslyIfσ=(01∞),weinferbylemma4.21thatγ(z)=1−1zandthusγ3=id.ThisimpliesT3◦φ=φ,
i.e.T3|=id,andconsequentlyT3|3=id.WritingTexplicitlyasT(x)=Ax+twithA∈SO(3)
andt∈φ(RM3),weobtainA3x+A2t+AtR+t=xforallx∈R3andthereforeimmediately(forx=0)
32aAt+rotationAt+bty=the0,anglewhich±2πyieldsaroundA=I.someSinceaxisAin∈R3conSO(3),tainingthis0∈impliesR3.thatTheeithercaseAA==IIoryieldsthattA=0is
3andTherefore,thusTA=isid,aarotationconbytradictionthetoanglethe±2factπaroundthat,someaccordingaxistoinσ,R3Tcondoesn’ttainingpres0∈ervRe3.thetrinoidends.
3v=±Assumev.noApplyingwthatthistwotogeofththeerwithdirectionthevrelationectorsAv0v,v=1vandv∞fromofthelemmatrinoid4.28sevaxeseralaretimes,wcollinear,eobtaine.g.
01jσ(j)
v∞=Av1=±Av0=±v1=v0andthusv1=Av0=Av∞=v0,i.e.v0=v1=v∞,acontradictiontothe
thebalancingformula(4.3.4).(Similarly,theassumptionsv0=±v∞andv1=±v∞,respectively,yield
contradictions.)Consequently,notwoofthevj,j∈{0,1,∞},arecollin3ear.By(4.3.4),however,2these
threevectorsarecoplanarandthusnownecessarilyspanaplaneEinR.Thenormalvectorn∈Sof
Eisdetermined2πuptosign.AsApreservesEandthusalsothelineRn,Aactuallydefinestherotation
bofythegeneralitangley±the3basearoundpointstheCj,axisj∈Rn{.0,1,Moreo∞}vofer,thesincetrinoid(againaxesbyaclemcordmaing4.28)toσT,pi.e.ermT(Cutesj)=withoutCσ(j),loswse
thatconcludet=C1−AC0=C∞−AC1=C0−AC∞(4.6.72)
usthand1T(x)=Ax+3(C1−AC0+C∞−AC1+C0−AC∞)=A(x−C)+C,(4.6.73)
whichmeansthatTdefinestherotationbytheangle±23πaroundtherotationaxisofAtranslatedby
C,i.e.aroundthetrinoidnormalAn=−C1+Rn.−1
T−1If(φσ(M=))(0=∞φ(1),M).weThusobserv(asethatshownTabove),correspT−1onds=toR,σwhere=(0R1∞denotes)andtheisalsorotationaonsymmetryR3byoftheφ:
π2theangleplane±3EaroundspannedthebyaxistheAndirection=C+vRectorsn,vwhere0,v1nanddenotesv∞ofthethe(uptrinoidtosignaxes.unique)Consequennormaltly,vweectorhavofe
T=R−1.
WenowturntothecasethatTreversesorientation,i.e.T=T∗.
Ifσ=(),weinferbylemma4.21thatγ(z)=z¯andthusσγ2=id.ThisimpliesT2◦φ=φ,i.e.
T2|φ(M)=id,andconsequentlyT2|R3=id.WritingTexplicitlyasT(x)=Ax+twithA∈O(3)\SO(3)
77

323andwhicth∈Rimplies,weA2obtain=I.ASincex+AAt∈+tO(3)=x\forSO(3),allxw∈eRhaveandA˜:=therefore−A∈SO(3)immediatelywithA(for˜2x==I.0)ThisAt+timpl=ie0,s
thateitherA˜=IorthatA˜isarotationbytheangleπaroundsomeaxisinR3containing0∈R3.The
caseA˜=Iyieldst=0(setx=0inT(x)=−x+t)andthusT=−id,acontradictiontothefactthat,
accordingtoσ,Tpreservesthetrinoidends.Therefore,A˜isarotationbytheangleπaroundsomeaxis
inR3containing0∈R3.Since,bylemma4.28,Apreservesthedirectionvectorsvj,j∈{0,1,∞}ofthe
˜˜sometrinoidaxisaxes,Rn,i.e.Awherevjn=∈vj,R3wesatifiesinfernthat⊥vAjvjfor=all−jvj.∈{Th0,1us,,A∞}.defines(Sincethv0e,v1rotationandvb∞yarethecoplangleanarπbyaroundthe
reflectionbalancinginformtheulaplane(4.3.4),(Rn)⊥suchat0an∈nR3.exists.)FinalItly,folloasTwsbbyyalemmadirect4.21computationwithoutlossthatofA=generalit−A˜ydefinespreservthese
thebasepointsCj,j∈{0,1,∞},ofthetrinoidaxes,i.e.T(Cj)=Cj,weconcludethatt=Cj−ACj
obtainandjallforT(x)=Ax+31(C0−AC0+C1−AC1+C∞−AC∞)=A(x−C)+C.(4.6.74)
ThismeansthatT⊥definesthereflectioninthereflectionplaneofAtranslatedbyC,i.e.inthetrinoid
planeIfσE==(1C+∞(),Rwne).inferNotebythatlemmathe4.21symmethattrγy(z)=relationz¯Tand◦φth=usφγ◦2γ=id.determinesThisEimpliesTcompletely2◦φ.=φ,i.e.
1z¯−T2|φ(M)=3id,andconsequen2tlyT2|R3=id.WritingT3explicitlyasT(x)=Ax+twithA∈O(3)\SO(3)
andt∈R,weobtain2Ax+At+t=xforallx∈Randtherefore˜immediately(forx=0)˜2At+t=0,
whichinturnyields˜A=I.Since˜A∈O(3)\SO(3),wehaveA:=−A∈SO(3)withA3=I.This
0∈impliesR3.thatThecaseeitherA˜A==IIoryieldsthatt=A0is(setax=rotation0inbTy(x)the=−anglex+πt)andaroundthussomeT=axis−id,inaRconcontradictiontaining
tothefactthat,according3toσ,Tpreserves3thetrinoidendB0.Therefore,A˜isarotationbytheangle
πA˜v0around=−v0.someThaxisus,Ain˜Rdefinesconthetaining0rotation∈Rb.ythSince,eanglebyπlemmaaround4.28,sAomeaxissatisfiesRn˜A,v0where=v0n˜,∈weR3infersatifiesthat
n˜and⊥nv0.:=Itvfollo×n˜ws,bywherea“direct×”denotescomputationtheusualthatAcross=−proA˜ductdefinesontheR3.Byreflectiondefinition,inthen⊥planev.spannedActuallyb,yvw0e
00necessarilyhaven˜⊥vjforallj∈{0,1,∞},whichcanbeseenasfollows:Incasethatanytwoofv0,v1
andv∞arecollinear,allthreearecollinearbythebalancingformula(4.3.4),whencen⊥v0impliesthat
n˜⊥vjforallj∈{0,1,3∞}.Otherwise,i.e.inthecasethatnotwoofv0,v1andv∞arecollinear,these
threespanaplaneinR.(Recallthattheyarecoplanarbythebalancingformula.)Since,bylemma
4.28,Avj=vσ(j)forallj,thisplaneispreservedunderA,i.e.itisthereflectionplaneofAitselfor,
otherwise,orthogonaltothereflectionplaneofA.Inthefirstcase,weinferthatAx=xforallpointsx
vof1=thev∞re,flaeconctionplane,tradictionintotheparticularAassumptionv1=v1that,whnoichtwooftogetherthevjwith’sarethecollinear.relationAThv1us=wveσ(1)are=v∞necessarilyyields
inthe(spannedsecondbyv0case,andi.e.n).theSinceplanev0isspannedcontainedbyv0in,bv1othandvplanes,∞isweorthogonalconcludetothatthenriseflectionorthogonalplanetooftheA
planeAltogether,spannedbAyv0,definesv1theandv∞,reflectioni.e.nin⊥thevjforplaneallj∈spanned{0,1b,yv∞}.andn,wheren⊥vforallj∈{0,1,∞}.
j0AsTbylemma4.28preservesthebasepointC0ofthetrinoidaxisA0,i.e.T(C0)=C0,weconclude
thatt=C0−AC0foralljandobtain
T(x)=A(x−C0)+C0.(4.6.75)
ThismeansthatTdefinesthereflectioninthereflectionplaneofAtranslatedbyC0,i.e.inthetrinoid
T◦normalφ=φplane◦γ,EE00=andC0th+usRv0(up+toRnsign)alongalsothenaretrinoidaxisdeterminedA0.Notecompletelythat,.bTheythecasessσymmetry=(0∞)relationand
σ=(01)aretreatedanalogously.
Ifσ=(01∞),weinferbylemma4.21thatγ(z)=1−1z¯andthusγ6=id.ThisimpliesT6◦φ=φ,i.e.
T6|=id,andconsequentlyT6|3=id.WritingTexplicitlyasT(x)=Ax+twithA∈O(3)\SO(3)
andφt(M∈)R3,weobtainA6x+A5t+AR4t+A3t+A2t+At+t=xforallx∈R3andthereforeimmediately
(forx=0)A5t+A4t+A3t+A2t+At+t=0,whichinturnyieldsA6=I.SinceA∈O(3)\SO(3),
wehaveA˜:=−A∈SO(3)withA˜6=I.Therearefourpossiblecases:
1.A˜definestheidentitymappingonR3,or
2.A˜definesarotationbytheangleπaroundsomeaxiscontaining0∈R3,or
3.A˜definesarotationbytheangle23πaroundsomeaxiscontaining0∈R3,or
78

4.A˜definesarotationbytheangle3πaroundsomeaxiscontaining0∈R3.
Weleadeachofthefirstthreepossiblecasestoacontradiction,usingrepeatedlytherelationAvj=vσ(j)
herei.e.4.28,lemmafromAv0=v1Av1=v∞Av∞=v0.(4.6.76)
First,assumethatA˜=I,i.e.A=−I.Then,−v0=Av0=v1=−Av1=−v∞=Av∞=v0,which
yieldsv0=0,acontradiction.Second,assumethatA˜definesarotationbytheangleπaroundsomeaxis
containing0∈R3.Inparticular,A˜2=IandthusalsoA2=I.Then,v0=A2v0=v∞=A2v∞=v1,i.e.
v0=v1=v∞,acontradictiontothebalancingformula(4.3.4).Third,assumethatA˜definesarotation
bytheangle23πaroundsomeaxiscontaining0∈R3.Inparticular,wehaveA˜3=IandthusA3=−I.
Thisyields−v0=A3v0=v0,i.e.v0=0,anothercontradiction.
Aswehaveleadeachofthefirstthreecasestoacontradiction,wearenecessarilyinthefourthcase,
i.e.A˜definesarotationbytheangle3πaroundsomeaxiscontaining0∈R3.Byadirectcomputation,
thisimpliesthatA=−A˜definesarotoreflectiononR3composedofarotationbytheangle±23πaround
someaxisRncontaining0∈R3andthereflectionintheplane(Rn)⊥.
Assumenowthattwoofthedirectionvectorsv0,v1andv∞ofthetrinoidaxesarecollinear,e.g.
v0=±v1.ApplyingthistogetherwiththerelationAvj=vσ(j)fromlemma4.28severaltimes,weobtain
v∞=Av1=±Av0=±v1=v0andthusv1=Av0=Av∞=v0,i.e.v0=v1=v∞,acontradictionto
thethebalancingformula(4.3.4).(Similarly,theassumptionsv0=±v∞andv1=±v∞,respectively,
adictions.)trconyieldConsequently,notwoofthevj,j∈{0,1,∞}are3collinear,whichimpliesthatthesevectors(which
arecoplanarby(4.3.4))actuallyspanaplaneinR.NotethatthisplaneispreservedbyAandthus
necessarilycoincideswiththeplane(Rn)⊥introducedabove,i.e.(Rn)⊥=Rv0+Rv1+Rv∞.Therefore,
nsatifiesn⊥vjforallj∈{0,1,∞}andisdetermineduptosign.AsTbylemma4.28withoutlossof
generalitypermutesthebasepointsCj,j∈{0,1,∞},ofthetrinoidaxes,i.e.T(Cj)=Cσ(j),weconclude
thatt=Cσ(j)−ACjforalljandobtain
T(x)=Ax+1(C1−AC0+C∞−AC1+C0−AC∞)=A(x−C)+C.(4.6.77)
3ThismeansthatTdefinestherotoreflectiononR3composedoftherotationbytheangle±2πaround
thetrinoidnormalAn=C+RnandthereflectioninthetrinoidplaneE=C+(Rn)⊥.3
Ifσ=(0∞1),weobservethatT−1correspondstoσ−1=(01∞)andisalsoasymmetryofφ:
T−1(φ(M))=φ(M).Thus(asshownabove),T−1=SˆwithSˆasgivenabove.Consequently,wehave
T=Sˆ−1.
Theorem4.31explicitlyliststhetwelveEuclideanmotionsonR3,whichqualifyaspossiblesymmetries
ofthegiventrinoidφ:M→R3withproperlyembeddedannularends.Inthecasethatφisactually
symmetricwithrespecttooneofthese,sayT,lemma4.21providestheassociatedbiholomorphic(resp.
bi-antiholomorphic)mappingγ:M→M,whichallowsfortranslatingthesymmetrypropertytothe
levelofthetrinoiddomainM:T◦φ=φ◦γ.Thisenablesus,basedontheresultsofsection4.4,tostudy
indetailtheimpactofthepossiblesymmetriesofφonthemonodromymatricesoftheextendedframe
FassociatedwiththeconformalCMC-immersionψ:=φ◦π:M˜→R3,whereπ:M˜→Mdenotesthe
universalcoveringdefinedin(3.2.2).Thisisdoneinthesections5to9.

79

5Rotationalsymmetrywithrespecttothetrinoidnormal
Definition5.1Inthissectionwediscusstrinoidsφ:M→R3withproperlyembeddedannularendsonM=Cˆ2\π{0,1,∞}
whicharesymmetricinthesenseofdefinition4.2withrespecttotherotationRbytheangle±3around
thetrinoidnormalAn={C+λn;λ∈R},whereCdenotesthetrinoidcenter,andndenotesanormal3
vectorofthetrinoidplaneE.(Recallfromtheorem4.31that,inthecasethatatrinoidφ:M→R
withproperlyembeddedannularendsissymmetricwithrespecttothegivenEuclideanmotionR,there
existsauniquetrinoidplaneandauniquetrinoidnormalofφ,whichenablesustospeakofthetrinoid
planeandthetrinoidnormalofφ,respectively.)Risuniquelydeterminedbyadditionallyprescribing
thatRpermutesthetrinoidendsaccordingtothepermutationσ=(01∞)oftheset{0,1,∞}.Since
evhaewR(φ(M))=φ(M)⇐⇒R−1(φ(M))=φ(M),(5.1.1)
itisclearthatagiventrinoidφ:M→R3withproperlyembeddedannularendsissymmetricwithrespect
toR,ifandonlyifitissymmetricwithrespectR−1,definingtherotationaroundthetrinoidnormalAn
bytheinverseangle23π,permutingthetrinoidendsaccordingtothepermutationσ−1=(0∞1).
Remark5.1.Notethat,inordertodefine(thesignof)theangleofrotationforR,onefirstneeds
todetermineanorientationoftheaxisofrotationofRitself.Dependingonwhichchoicewemake
fortheorientationoftheaxisofrotation−1ofR,theangleof2πrotation2πofRwillbeeither+23πor−23π.
Accordingly,theanglsofrotationofRwillbeeither−or+.However,forourpurposesit
sufficestocharacterizeR(resp.R−1)bythepropertythatit3permutes3thetrinoidendsaccordingtothe
permutationσ=(01∞)(resp.σ−1=(0∞1)).
Definition5.2.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannularends.
LetM˜=Handψ=φ◦π:M˜→R3theconformalCMC-immersionassociatedwithφviatheuniversal
coveringπ:M˜→Mgivenin(3.2.2).LetAn={C+λn;λ∈R},whereCdenotesthetrinoidcenter
andnanormalvectorofthetrinoidplaneE,bethetrinoid2πnormal.Then,ifφ(or,equivalently,ψ)is
symmetricwithrespecttotherotationRbytheangle±3aroundAn,whichpermutesthetrinoidends
accordingtothepermutationσ=(01∞)oftheset{0,1,∞},
R(φ(M))=φ(M),R(ψ(M˜))=ψ(M˜),(5.1.2)
or,equivalently,ifφ(or,equivalently,ψ)issymmetricwithrespecttotheinverserotationR−1,
R−1(φ(M))=φ(M),R−1(ψ(M˜))=ψ(M˜),(5.1.3)
φ(orψ)iscalledrotationallysymmetricwithrespecttothetrinoidnormal.
Wearenowgoingtoapplytheresultsoftheprevioussectionsinordertotranslatetherotational
symmetryofφintofurtherconstraintsonthefunctionsp0,p1,q0,q1.Recallthatp0,p1,q0,q1occurin
themonodromymatricesoftheextendedframeFoftheconformalCMC-immersionψ=φ◦π:M˜→R3
associatedwithφviatheuniversalcoveringπ:M˜→M.Sothequestionwenowactuallyturnto
is:Whichmonodromymatricesarepossibleforrotationallysymmetrictrinoidswithproperlyembedded
ends?ularann

5.2Implicationsofrotationalsymmetrywithrespecttothetrinoidnormal
Thefollowingresultisanimmediateconsequenceofdefinition5.2:
3larLemmaendspro5.3.duceLdetfrMom=aCtrinoid\{0,p1}otentialandφη:asMin→theRorembea3.14.trinoidDenotewithbyprDop,erlDy,embDeddethedcorrannu-e-
10∞spondingDelaunaymatriceswitheigenvalues±µ0,±µ1,±µ∞,respectively,where,forj∈{0,1,∞},
µj=XjXj=41+wj(λ−λ−1)2andwj=sjtjasinsection3.5.Then,ifφisrotationallysymmet-
ricwithrespecttothetrinoidnormal,wehave
µ:=µ0=µ1=µ∞=1+w(λ−λ−1)2,(5.2.1)
4ewherw:=w0=w1=w∞.(5.2.2)

80

Proof.Bydefinition5.2,theendsofatrinoidwithproperlyembeddedannularends,whichisrotationally−1
intosymmetriceachotherwithrespaccordingecttotothetheptrinoidermutationnormal,σare=(0rotated1∞)by(resp.theσcorresp−1=(0ondin∞g1)).symmetryThisRmeans(resp.thatRthe)
DelaunaasymptoticysurfacesDelaunayonlysudifferrfacesbyassoarigidciatedmotionwiththeonRends3.Inareparticular,rotatedintothiseachimpliesotherthataswtheell.correspHence,ondingthese
DelaunaymatricesDj,j=0,1,∞,(seesection3.5formoredetails)allpossessthesameeigenvalues.
Thisyieldsµ0=µ1=µ∞andallowsfordefiningµ:=µ0=µ1=µ∞.UsinglemmaB.6,weinferthat
w0=w1=w∞,whencewgivenin(5.2.2)iswelldefined.Consequently,µ=41+w(λ−λ−1)2holds,
whichfinishestheproof.
3assoLetciatedMc=onC\formal{0,1},φ:CMC-immersionM→RonbeM˜a=Htrinoid,ψ=withφ◦πprop:M˜erly→emR3b,edwherededπannulardenotesendstheandunivψersalthe
tocovtheriengtrinoidM˜→Mnormal,givenandin(3.2.2).denotetheSupposecorrespφ(or,ondingequivalensymmetrytly,ψb)yisR.SincerotationallyRpreservsymmetricesorienwithtationrespecton
R3,weobtainbytheorem4.9apairofbiholomorphicmappings,γR:M→Mandγ˜R:M˜→M˜
satisfyingR◦φ=φ◦γR,(5.2.3)
R◦ψ=ψ◦γ˜R,(5.2.4)
π◦γ˜R=γR◦π.(5.2.5)
Analogously,weobtainforR−1apairofbiholomorphicmappings,γR−1:M→Mandγ˜R−1:M˜→M˜
satisfyingR−1◦φ=φ◦γR−1,(5.2.6)
R−1◦ψ=ψ◦γ˜R−1,(5.2.7)
π◦˜γR−1=γR−1◦π.(5.2.8)
ThemappingsγRandγR−1areuniquelydeterminedandexplicitlygivenbylemma4.21:
1γR(z)=1−z,(5.2.9)
γR−1(z)=zz−1.(5.2.10)
oftheThemappingsautomorphismγ˜Randgroupγ˜R−Aut(1Mare˜/M)uniquelyofπ.Thedeterminedfollowingupletommacompexplicitlyositionsfromtatestheapleftairofwithvalidancelemenhoicest
forγ˜Randγ˜R−1:
˜˜LetLemmaγR:M5.4.→LMetMand=γRC−\1{:0,M1},→MM=beHgivenandπby:M(5.2.9)→Mandbethe(5.2.10)u,niversalrespecctiveloveringy.asThen,giventheinfol(3.2.2)lowing.
holds:1.Themappingγ˜R:M˜→M˜,−z−1
γ˜R(z)=z,(5.2.11)
satisfiesandbiholomorphicisπ◦γ˜R=γR◦π,(5.2.12)
R◦ψ=ψ◦γ˜R.(5.2.13)
2.Themappingγ˜R−1:M˜→M˜,
1γ˜R−1(z)=−z−1,(5.2.14)
satisfiesandbiholomorphicisπ◦γ˜R−1=γR−1◦π,(5.2.15)
R−1◦ψ=ψ◦γ˜R−1.(5.2.16)
81

Proof.Westartwithpro˜vingthe˜firstclaim.Clearly,γ˜RdefinesaMoebiustransformationandthusa
biholomorphicmappingM˜→M.Moreover,byapplyingtherelations(3.2.10)and(3.2.11)oflemma
3.4,weobtainforallz∈M
−z−1111
π◦γ˜R(z)=πz=π−1−z=π−z1=1−π(z)=γR◦π(z),(5.2.17)
i.e.π◦γ˜R=γR◦π.Finally,
R◦ψ=R◦φ◦π=φ◦γR◦π=φ◦π◦γ˜R=ψ◦γ˜R,(5.2.18)
i.e.R◦ψ=ψ◦γ˜R.
morphicNowwemappingturntoM˜the→M˜second.Moreoclaim.ver,byClearly,γ˜applyingR−1thedefinesarelationsMoebius(3.2.11)andtransformation(3.2.10)andofthlemmausabi3.4,holo-we
obtainforallz∈M˜
π◦γ˜R−1(z)=π−z1−1=1−π(z+1)=1−π(1z)=π(πz()z)−1=γR−1◦π(z),(5.2.19)
i.e.π◦γ˜R−1=γR−1◦π.Finally,
R−1◦ψ=R−1◦φ◦π=φ◦γR−1◦π=φ◦π◦γ˜R−1=ψ◦γ˜R−1,(5.2.20)
i.e.R−1◦ψ=ψ◦γ˜R−1.
Remark5.5.Notethat,sinceγ˜R◦γ˜R−1=γ˜R−1◦γ˜R=idforthemappingsγ˜Randγ˜R−1definedin
(5.2.11)and(5.2.14),respectively,wehave
˜γR−1=γ˜R−1.(5.2.21)
trinoidBytheabsymmetriesoveRlemma,andwRe−1ha,verespectivexplicitlyely,inthedeterminedsenseofmappingstheoremγ˜R4.9.andThγ˜Rus,−1wecorrespcanapplyondingtotheoremthe
obtainto4.173andTheoremψthe5.6.assoLciateetdMc=onformalC\{0,1},φCMC-immersion:M→RofbM˜ea=H,trinoidψ=withφ◦prπop:M˜erly→embRe3,ddedwhereπannulardenotesends
theuniversalcoveringM˜→Masdefined−1in(3.2.2).Letφberotationallysymmetricwithrespectto
thetrinoidnormal.DenotebyRandRthe−c1orrespondingsymmetriespermutingthetrinoidends
byaccγ˜or−ding1thetothepbiholomorphicermutationsσmappings=(01M˜∞)→andM˜σasso=ciate(0d∞wi1)th,rRespeandctively.R−1,rMorespeeoverctively,,denoteasinbyγthe˜Rorandem
R4.9andexplicitlygiveninlemma5.4.Then,thefollowingholds:
1.TheextendedframeF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜Ras
F(γ˜R(z),λ)=MR(λ)F(z,λ)kR,γ˜R(z),(5.2.22)
ewherz¯kR,γ˜R(z)=z0z¯,(5.2.23)
0zandMRdenotesanelementofΛSU(2)σ,whichisindependentofz.
2.TheextendedframeF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜R−1
asF(γ˜R−1(z),λ)=MR−1(λ)F(z,λ)kR−1,γ˜R−1(z),(5.2.24)
0wherez¯+1
kR−1,γ˜R−1(z)=0z+1z¯+1(5.2.25)
+1zandMR−1denotesanelementofΛSU(2)σ,whichisindependentofz.
82

Proof.Westartwiththeproofofthefirstpart.Letγ˜(z)=γ˜R(z)=−zz−1forallz∈M˜3=H.(For
convenienceweomittheindexRthroughoutthisproof.)AsRpreservesorientationonR,weapply
thefirstpartoftheorem4.17toobtain
F(γ˜(z),λ)=Mγ˜(λ)F(z,λ)kR,γ˜(z),(5.2.26)
whereF:M˜→ΛSU(2)denotestheextendedframecorrespondingtoψbytheorem4.5andMdenotes
anelementofΛSU(2)σ,σwhichisindependentofz.kR,γ˜(z)isgivenbyequation(4.4.117)fromγ˜lemma
computingBy4.18.∂zγ˜(z)=z12(5.2.27)
thatinferew2|∂∂zγ˜γ˜((zz))|=|zz2|=zz¯(5.2.28)
zz¯0andthusobtainfrom(4.4.117)
kR,γ˜(z)=0zz¯.(5.2.29)
zAsγ˜=γ˜R,wedenoteMγ˜byMR.Thisfinishestheproofofequation(5.2.22).
Toprovethesecondpartofthetheorem,wedefineγ˜(z)=γ˜R−1(z)=−z1−1onM˜=H.Everything
isthendoneanalogously.Wehave
1∂zγ˜(z)=(z+1)2(5.2.30)
usthand2|∂∂zγ˜γ˜((zz))|=|(zz++11)|2=zz¯++11.(5.2.31)
zFormula(4.4.117)fromlemma4.18thenyields
0z¯+1
kR−1,γ˜(z)=0z+1z¯+1,(5.2.32)
+1zandbysettingMR−1(λ):=Mγ˜(λ),thefirstpartoftheorem4.17implies(5.2.24).
Remark5.7.ThemonodromymatricesMandM−1ofFunderthebiholomorphicmappingsγ˜and
γ˜R−1arelinkedasfollows:Asγ˜R−1=γ˜R−1,RwehaveRR
F(z,λ)=F((γ˜R◦γ˜R−1)(z),λ)=MR(λ)F(γ˜R−1(z),λ)kR,γ˜R(γ˜R−1(z))(5.2.33)
usthandMR−1(λ)F(z,λ)kR−1,γ˜R−1(z)=F(γ˜R−1(z),λ)=(MR(λ))−1F(z,λ)(kR,γ˜R(γ˜R−1(z)))−1.(5.2.34)
AdirectcomputationyieldskR−1,γ˜R−1(z)=±((kR,γ˜R(γ˜R−1(z)))−1,whichimplies
MR−1(λ)=±((MR(λ))−1.(5.2.35)
5.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,
whicharerotationallysymmetricwithrespecttothetrinoidnormal
Usingtheresultsoftheprevioussectionwearenowabletodescribethe(unitary)monodromymatrices
Mˆ0,Mˆ1,Mˆ∞associatedwithatrinoidwithproperlyembeddedannularends,whichisrotationallysym-
γ˜j,metricj=0with,1,∞resp,onectM˜tothegeneratingtrinoidthenormal.monoAsdromaystart,matricesrecallMˆfromj,j=section0,1,∞3.3:thecoveringtransformations
zγ˜0(z)=−2z+1(5.3.1)
γ˜1(z)=z+2(5.3.2)
γ˜∞(z)=−3z−2.(5.3.3)
1+z283

Lemma5.8.LetM˜=Handγ˜0,γ˜1,γ˜∞:M˜→M˜begivenasabove.
1.Forγ˜R:M˜→M˜,γ˜R(z)=−zz−1,thefollowingidentitieshold:
γ˜R◦γ˜0=γ˜1◦γ˜R,γ˜R◦γ˜1=γ˜∞◦γ˜R,γ˜R◦γ˜∞=˜γ0◦γ˜R.(5.3.4)
2.Forγ˜R−1:M˜→M˜,γ˜R−1(z)=−z1−1,thefollowingidentitieshold:
γ˜0◦γ˜R−1=γ˜R−1◦γ˜1,γ˜1◦γ˜R−1=γ˜R−1◦γ˜∞,γ˜∞◦γ˜R−1=˜γR−1◦γ˜0.(5.3.5)
Proof.Westartwiththefirstpart,i.e.γ˜R(z)=−zz−1.Theclaimisprovedbystraightforwardcomputa-
tion:Forz∈M˜wehave
γ˜R◦γ˜0(z)=γ˜Rz=z−1=γ˜1−z−1=γ˜1◦γ˜R(z)(5.3.6)
−2z+1zz
γ˜R◦γ˜1(z)=γ˜R(z+2)=−z+3=γ˜∞−z−1=γ˜1◦γ˜R(z)(5.3.7)
z+2z
γ˜R◦γ˜∞(z)=γ˜R−2z3z+−12=−3zz++12=γ˜0−zz−1=γ˜0◦γ˜R(z).(5.3.8)
1theNowinverseconsideringfunctiontheofγ˜secondRgivparteninoftthehefirstlemma,part.weSohavetheγ˜Riden−1(z)tities=−z(5.3.5)−1.folloObservwedirectlythatthisfromthemappingfirstis
partbyapplyingtheautomorphismγ˜R−1fromboththelefthandsideandtherighthandsidetothe
4).(5.3.titiesidenontheTheabmonoovedromlemmayismatricesneededoftotheproveextendedthefolloframewingFassotheorem,ciatedwhicwithhastatestrinoidfurtherwithnepropcessaryerlyembconditionsedded
annularends,whichisrotationallysymmetricwithrespecttothetrinoidnormal.
Theorem5.9.LetM=C\{0,1},φ:M→R3be˜atrinoidwithproperly˜embe3ddedannularends
andψtheassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotes
thetrinoiduniversalnormal.coveringDenoteM˜by→RMandasR−1definethedcinorresp(3.2.2).ondingLetφbesymmetriesrotationalplyermutingsymmetricthetrinoidwithrendsespeacctcortodingthe
˜1−tothethepextendeermudfrtatiameonsσass=o(0ciate1d∞)withandψσbythe=or(0em∞1)4.5.,respeDenotectively.byMFˆ0,Mˆ1urthermor,Mˆ∞e,∈letFΛSU(2:M,C)→σtheΛSU(2)unitaryσbe
monodromymatrices
10pjqj
Mˆj=−cos(2πµj)01+isin(2πµj)qj−pj(5.3.9)
associatedwithFasin(4.5.13)by
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),j=0,1,∞,(5.3.10)
whereαj∈{±1}andγ˜jdenotethec˜overing˜transformationsonM˜fr−1omsection3.3.Finally,letγ˜R,
γ˜R−1bethebiholomorphicmappingsM→MassociatedwithRandR,respectively,asintheorem4.9
ofandFasexplicitlygiveningiveneinquationslemma(5.2.22)5.4,andandletM(5.2.24)R(.λ),InMRview−1(oλf)rbeemarkthec5.7,orrespwesetondingmonodromymatrices
MR(λ)=±(MR−1(λ))−1=:−abRRabRR.(5.3.11)
Then,themonodromymatricessatisfy
Mˆ1(λ)=MR(λ)Mˆ0(λ)MR(λ)−1,(5.3.12)
Mˆ∞(λ)=MR(λ)Mˆ1(λ)MR(λ)−1,(5.3.13)
Mˆ0(λ)=MR(λ)Mˆ∞(λ)MR(λ)−1.(5.3.14)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(5.3.12)to(5.3.14)readas
p1=aRaRp0+aRbRq0+aRbRq0−bRbRp0,(5.3.15)
q1=−2aRbRp0+aR2q0−bR2q0,(5.3.16)
84

p∞=aRaRp1+aRbRq1+aRbRq1−bRbRp1,(5.3.17)
2q∞=−2aRbRp1+aR2q1−bRq1,(5.3.18)
p0=aRaRp∞+aRbRq∞+aRbRq∞−bRbRp∞,(5.3.19)
22q0=−2aRbRp∞+aRq∞−bRq∞.(5.3.20)
Proof.Considerthebiholomorphicmappingγ˜R:M˜→M˜givenin(5.2.11):γ˜R(z)=−z−1.Applying
z(thefirstpartof)theorem5.6weobtain
F(γ˜R(z),λ)=MR(λ)F(z,λ)kR,γ˜R(z),(5.3.21)
wherez¯0zz¯0kR,γ˜R(z)=.(5.3.22)
zCombiningthiswiththemonodromyequations(5.3.10),andapplyingtheidentities(5.3.4)fromthe
abovelemma,wecompute:
MR(λ)Mˆ0(λ)F(z,λ)k0(z)kR,γ˜R(γ˜0(z))=α0MR(λ)F(γ˜0(z),λ)kR,γ˜R(γ˜0(z))=α0F(γ˜R◦γ˜0(z),λ)
=α0F(γ˜1◦γ˜R(z),λ)=α1α0Mˆ1(λ)F(γ˜R(z),λ)k1(γ˜R(z))
=α1α0Mˆ1(λ)MR(λ)F(z,λ)kR,γ˜R(z)k1(γ˜R(z)).(5.3.23)
Analogously,weobtain
MR(λ)Mˆ1(λ)F(z,λ)k1(z)kR,γ˜R(γ˜1(z))=α∞α1Mˆ∞(λ)MR(λ)F(z,λ)kR,γ˜R(z)k∞(γ˜R(z)),(5.3.24)
MR,1(λ)Mˆ∞(λ)F(z,λ)k∞(z)kR,γ˜R(γ˜∞(z))=α0α∞Mˆ0(λ)MR(λ)F(z,λ)kR,γ˜R(z)k0(γ˜R(z)).(5.3.25)
Asz¯
1−2z¯0−2zz¯+10z¯0
k0(z)kR,γ˜R(γ˜0(z))=1−2z−2z+1=±z
1−2z0−2zz+1z
01−2z¯−2z¯z¯+10z¯
z¯z010
0=±z¯01=±kR,γ˜R(z)k1(γ˜R(z)),(5.3.26)
zwherechangesinsignmayoccurduetothepowerrulesforcomplexnumbers,equation(5.3.23)implies
MR(λ)Mˆ0(λ)=Mˆ1(λ)MR(λ)(5.3.27)
with∈{±α1α0},andtherefore
Mˆ1(λ)=MR(λ)Mˆ0(λ)MR(λ)−1.(5.3.28)
Takingintoaccountequation(5.3.9),wecomparetheupperleftandthelowerrightentriesofMˆ1(λ)
andMR(λ)Mˆ0(λ)MR(λ)−1.Thisyields
−cos(2πµ)−isin(2πµ)p1=
−cos(2πµ)−isin(2πµ)aRaRp0+aRbRq0+aRbRq0−bRbRp0,(5.3.29)
−cos(2πµ)+isin(2πµ)p1=
−cos(2πµ)+isin(2πµ)aRaRp0+aRbRq0+aRbRq0−bRbRp0.(5.3.30)
85

Addinguptheseequations,weobtain
−2cos(2πµ)=−2cos(2πµ),(5.3.31)
which,ascos(2πµ)doesn’tvanishidentically,implies=1andthusasclaimed
Mˆ1(λ)=MR(λ)Mˆ0(λ)MR(λ)−1.(5.3.32)
Moreover,thisequationtranslatesequivalentlyintothescalarequations(omittingredundantones)
p1=aRaRp0+aRbRq0+aRbRq0−bRbRp0,(5.3.33)
q1=−2aRbRp0+aR2q0−bR2q0.(5.3.34)
Similartotheargumentgivenabove,wehave
z¯+20z¯+20
k1(z)kR,γ˜R(γ˜1(z))=0110z+2z¯+2=z+2z¯+2
0z+20z+2
−z¯−1
z¯01+21+2−zz¯−10
0z01+2−zz¯z−1
=±zz¯z1+2−z¯−1=±kR,γ˜R(z)k∞(γ˜R(z))(5.3.35)
and−3z¯−2
1+2z¯0−23z¯z+1−20
01+2z0−23z¯z+1−2
k∞(z)kR,γ˜R(γ˜∞(z))=1+2z1+2z¯2z+1−3z¯−2
2z+1
3z¯+2z¯1−2−z¯z¯−10
3z+2001−2−z−1
=±3z¯+2=±zz¯z1−2−z¯−1=±kR,γ˜R(z)k0(γ˜R(z)),
03z+20z01−2−zzz¯−1
(5.3.36)whichleadsby(5.3.24)and(5.3.25)toMR(λ)Mˆ1(λ)=±α∞α1Mˆ∞(λ)MR(λ)andMR(λ)Mˆ∞(λ)=
±α0α∞Mˆ0(λ)MR(λ),respectively.Fromthisweobtain
Mˆ∞(λ)=±α∞α1MR(λ)Mˆ1(λ)MR(λ)−1,(5.3.37)
Mˆ0(λ)=±α0α∞MR(λ)Mˆ∞(λ)MR(λ)−1.(5.3.38)
Replacingintheargumentabove(Mˆ1,Mˆ0)by(Mˆ∞,Mˆ1)andby(Mˆ0,Mˆ∞),respectively,weobtain
±α∞α1=±α0α∞=1,whichyields
Mˆ∞(λ)=MR(λ)Mˆ1(λ)MR(λ)−1,(5.3.39)
Mˆ0(λ)=MR(λ)Mˆ∞(λ)MR(λ)−1,(5.3.40)
formscalarinand

finishesThisof.prothe

p∞=aRaRp1+aRbRq1+aRbRq1−bRbRp1,
2q∞=−2aRbRp1+aR2q1−bRq1,
p0=aRaRp∞+aRbRq∞+aRbRq∞−bRbRp∞,
2q0=−2aRbRp∞+aR2q∞−bRq∞.

86

(5.3.41)(5.3.42)(5.3.43)(5.3.44)

5.4Normalizedtrinoidswithproperlyembeddedannularends,whichare
rotationallysymmetricwithrespecttothetrinoidnormal
Letφ:M→R3beatrinoidwithproperlyembeddedannularends,whichisrotationallysymmetricwith
respectto−1thetrinoidnormal,andletψ=φ◦πbetheassociatedCMC-immersionM˜→R23π.Denoteby
RandRthecorrespondingsymmetriesofφ(andψ),i.e.therotationsbytheangles±3aroundthe
normal.trinoidWespecializetheresultsofsection5.3tothecasethattheextendedframeF:M˜→ΛSU(2)σ
associatedwithψasinsection4.2is“normalized”at
√z∗=−1+2i3∈M˜,(5.4.1)
i.e.,F(z∗,λ)=I,(5.4.2)
oforccuringallλ∈inS1the.Themonospdromecialychoicematricesofz∗ofF.resultsinmoreexplicitrequirementsonthefunctionsp0,p1,q0,q1
z∗∈M˜Recall,or,frommoresectionprecisely4.2,,thethattheunderlyingnormalizationnormalizationF(z∗,λof)=theIofthe(conformal)extendedframeCMC-immersionFatψsome,point
ψ(z∗)=21He3,U(z∗)=G(1),(5.4.3)
where(4.2.5),U∈correspSO(3)ondstorepresenrotatingtstheandnaturalshiftingtheorthonormal(imageofframethe)corresptrinoidondinginR3,tosucψ,handthatG(1)theisgivconditionsenin
(5.4.3)aremet.Itturnsout(cf.3corollary5.12),thatthechoiceofz∗asin(5.4.1)correspondsto3arranging
thethatthe(imageofrotationthe)axistrinoidofRin(andR,ofsucRh−1)thatisthetheztrinoid-axisinRplane3.ofφisparalleltothex-y-planeinR,and
3respAecttotrinoidtheφ:trinoidM→norRmalwithand,propinerlyaddition,embisedded“wellannpularositioned”ends,inwhicR3hinistherotationallysensethatsymmethetrassoicciatedwith
conformalCMC-immersionψ:M˜→Mmeetsthenormalizationconditions(5.4.3),iscalledanormalized
trinoidwithproperlyembeddedannularends,whichisrotationallysymmetricwithrespecttothetrinoid
normal.Wenowformulateamoreexplicitversionoftheorem5.6:
3andTheoremψtheasso5.10.ciateLetdcM=onformalC\{0,1},φCMC-immersion:M→RonbeM˜a=Htrinoid,ψ=withφ◦prπop:erlyM˜→embRe3,ddedwhereπannulardenotesends
theuniversalcoveringM˜→Masdefinedin(3.2.2).Assumethatψhasbeennormalizedatz∗givenin
corr(5.4.1)esp,ondingsuchtothatψψ(byz∗)the=or21Heme34.5.andMorF(ez∗,over,λ)=letI,φbwherereFotational:M˜ly→ΛSU(2)symmetricσwithdenotesresptheecttoextendethedtrifrnoidame
normal.DenotebyRandR−1thecorrespondingsymmetriespermutingthetrinoidendsaccordingto
1−thethepermutationsbiholomorphicσm=a(0ppin1gs∞)M˜and→σM˜asso=(0ciate∞d1),withrespReandctively.R−1,Morreespeover,ctively,denoteasbyin˜γtheRorandemby4.9γ˜Rand−1
explicitlygivenin(5.2.11)and(5.2.14),respectively:
11z−−˜γR(z)=z,γ˜R−1(z)=−z−1.(5.4.4)
holds:lowingfoltheThen,1.TheextendedframeFtransformsunderγ˜Ras
z¯00F(γ˜R(z),λ)=MR(λ)F(z,λ)zz¯,(5.4.5)
zwhereeπ3i0
MR(λ)=0e−π3i.(5.4.6)
Inparticular,MRisactuallyindependentofλ.
87

2.TheextendedframeFtransformsunder˜γR−1as
z¯+10
0F(γ˜R−1(z),λ)=MR−1(λ)F(z,λ)z+1z¯+1,(5.4.7)
+1zwheree−π3i0
MR−1(λ)=0eπ3i.(5.4.8)
Inparticular,MR−1isactuallyindependentofλ.
Proof.Inviewoftheorem5.6,weonlyhavetoproveequations(5.4.6)and(5.4.8).
Inthefirstcase,adirectcomputationshowsγ˜R(z∗)=−zz∗∗−1=z∗.Furthermore,byassumption,
F(z∗,λ)=I.Keepingthisinmind,weevaluateequation(5.4.5)atz=z∗toobtain
z¯∗0−πi
I=F(z∗,λ)=F(γ˜R(z∗),λ)=MR(λ)F(z∗,λ)z∗z¯∗=MR(λ)Ie03eπ03i,(5.4.9)
0z∗wherewehaveexplicitlycomputedtheoccurringcomplexsquarerootsaccordingtoremark4.14.This
(5.4.6).yieldsInthesecondcase,wehavebyadirectcomputationγ˜R−1(z∗)=−z∗1−1=z∗aswellas,byassumption,
F(z∗,λ)=I.Evaluatingequation(5.4.7)atz=z∗,weinferthat
z¯∗+1πi
∗I=F(z∗,λ)=F(γ˜R−1(z∗),λ)=MR−1(λ)F(z∗,λ)z∗+1z¯0+1=MR−1(λ)Ie03e−0π3i,
0+1z∗(5.4.10)(5.4.8).usthand3annRemarkularends,5.11.whicInhtheadmitspreviousthesections,rotationalwestartedsymmetrywithRaaroundtrinoidtheφ:Mtrinoid→Rnormal.withpropNoteerlythatembineddedthis
generalcase,theextendedframeF(z,λ)associatedwiththe˜conformal31CMC-immersionψ=φ◦π
prowhichducesarebinyvtheariantSym-BobundersenkomeoformEuclideanulaanassomotionciatedRλ,familyrespψectivλ:elyM,→Rinduced,λb∈ySthe,ofmonodromCMC-immersions,ymatrix
M(λ)ofFunderthebiholomorphicmappingγ˜associatedwithR=R.(Notethat,sincefor
λR=1theconditionsoftheorem2.11areingeneralRnotmet,ψλwillforλ=1λi=1ngeneralnotdescendto
aCMC-immersionφλ:M→R3.However,ψλ3willbesymmetricwithrespecttoRλ.)Notethatitis
notclear,apriori,whetherRλisarotation(inR).
Inthespecialcaseconsideredinthissection,i.e.inthecasethattheCMC-immersionψaswellas
theextendedframeFassociatedwithagiventrinoidφ:M→R3withproperlyembeddedannularends,
ofwhich(5.4.3)admitsandthe(5.4.2),rotationalwesobservymemetrythatRthearoundmonothedromytrinoidmatrixnormal,MR(λha)veofbFeennunderormalizethedinthebiholomorphicsense
mappingγ˜R1associatedwithRisactuallyindependentofλ.Thus,eachelementoftheassociatedfamily
ψλ,symmetryλ∈SR,ofinψR3.=ψλ(Ho=1wever,generatedstill,bψyλFwillviafortheλ=1Sym-Bobinenkgeneraloformnotula,descendadmitstotheaCMC-sameimrotationalmersion
φλ:M→R3.)
theCorollarysymmetries5.12.RWeandrRetain−1ofthethenotationnormalizeanddthetrinoidφassumptionsistheofz-axistheorinemR3.5.10.TheThetrinoidaxisofplanerotaoftioφnisof
paralleltothex-y-planeinR3.
Proof.Applying(thefirstpartof)theorem4.17,weknowthatthemonodromymatricesMR(λ)and
MR−1(λ)explicitlygivenintheorem5.10satisfyatλ=1
MR(1)=±AR,(5.4.11)
MR−1(1)=±AR−1,(5.4.12)

88

ARwhereofAtheR∈symmetrySU(2)R(resp.(resp.AR−1the∈SU(2))orthogonaldenotespartAtheR−1ofconjugationthesymmetrymatrixR−1realizing)inthethesu(2)-moorthogonaldel.partIn
viewofequations(5.4.6)and(5.4.8),thisyields
eπ3i0
AR=±0e−π3i,(5.4.13)
iπ−AR−1=±e03eπ03i.(5.4.14)
su(2)RecallingdefinedthatinAR(3.4.3)andAasRin(resp.(3.4.7),AR−i.e.1andAR−1)arelinkedviatheLieAlgebraisomorphismJ:R3→
(J◦AR◦J−1)(X)=ARXAR−1forallX∈su(2),(5.4.15)
(J◦AR−1◦J−1)(X)=AR−1XAR−1−1forallX∈su(2),(5.4.16)
weobtainbyadirectcomputationthat
√√
−√12230−√21−230
001001
AR=−23−210,AR−1=23−210.(5.4.17)
3π23Thquenus,tly,ARtheandsymmAR−1etriesdefineRandRrotations−1of(intheR)bynormalizedtheanglestrinoid±φ3arearoundrotationsthebzy-axistheinanRgles,R±e23π.aroundConse-
33anthisaxisaxisinofR,rotation,whichisisparallelparalleltotothethezx-y-axis.-planeIninRparticular,3.Asththeeptrinointoidψ(z∗plane)∈ofR3φ,withwhiczh∗isgivenorthogoninal(5.4.1)to
satisfiesR(ψ(z∗))=ψ(γ˜R(z∗))=ψ(z∗),(5.4.18)
R−1(ψ(z∗))=ψ(γ˜R−1(z∗))=ψ(z∗),(5.4.19)
11−itinferliesthatonththeeaxiscommonofraxisotationofofRrotationandofR−R1isandRactually.Sinthecezb-axisyinRassumption3.wehaveψ(z∗)=2He3,we
Applyingtheorems5.9and5.10,weobtainthefollowingresult:
Theorem5.13.LetM=C\{0,1},φ:M→R3b˜eatrinoidwithprop˜erlyembe3ddedannularends
andψtheassociatedc˜onformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotes
theuniversalcoveringM→Masdefinedin(3.2.2).Assumethatψhasbeennormalizedatz∗givenin
1corr(5.4.1)esp,ondingsuchthattoψψ(byz∗)the=or2Heme34.5.andLFet(zφ∗,bλe)r=I,otationalwherelyF:M˜symmetric→wiΛSU(2)thrσespectdenotestothethetrinoidextendedfrnormal.ame
Then,theunitarymonodromymatricesMˆ0,Mˆ1,Mˆ∞∈ΛSU(2,C)σassociatedwithFasin(5.3.10)are
formtheofMˆ0=−cos(2πµ)I−√2icos(πµ)cos(πµ)ζ0,(5.4.20)
3ζ0−2cos(πiπµ)
Mˆ1=−cos(2πµ)I−√2i3cos(πµ)e−cos(23ππiζµ0)−ecos(3ζπ0µ),(5.4.21)
iπ2Mˆ∞=−cos(2πµ)I−√2icos(πµ)cos(2πiπµ)e−3ζ0,(5.4.22)
3e3ζ0−cos(πµ)
whereζ0isanoddfunctioninλandasolutionto
ζ0ζ0=4sin2(πµ)−1.(5.4.23)
Proof.Asb˜efore,˜wedenotethesymmetriesof−1φbyRandR−1andbyγ˜R,γ˜R−1thebiholomorphic
mappingsM→MassociatedwithRandR,respectively,asintheorem4.9andexplicitlygivenin
inlemmaequations5.4.Moreo(5.2.22)ver,andletMR(5.2.24).(λ),MInR−1view(λ)ofbethetheoremcorresp5.10,ondingwehavemonodromymatricesofFasintroduced
MR(λ)=(MR−1(λ))−1=−abRRabRR(5.4.24)
89

whereaR=eπ3iandbR=0.(5.4.25)
MˆMoreoofvtheer,byextendedtheoremframe5.9,Fw:eobtainthefollowingrelationsbetweentheunitarymonodromymatrices
jMˆ1(λ)=MR(λ)Mˆ0(λ)MR(λ)−1,(5.4.26)
Mˆ∞(λ)=MR(λ)Mˆ1(λ)MR(λ)−1,(5.4.27)
Mˆ0(λ)=MR(λ)Mˆ∞(λ)MR(λ)−1,(5.4.28)
whichtranslateintothefollowingscalarequationsinvolvingthefunctionspjandqjoccurringinMˆj(cf.
(3.9.26)):p∞=p1=p0,e23πiq∞=q1=e−23πiq0.(5.4.29)
normal,Thus,weintheobtaincasetheofafollowingnormalizedequivalentrinoid,twhicreformhisulationsrotationallyof(3.9.50)symmetricandwith(3.9.51),respcecttoharacterizingthetrinoidthe
monodromymatricesMˆj:
p0=p0andp02+q0q0=1,(5.4.30)
p02−q0q0=cos2(2πµ2)+cos(2πµ).(5.4.31)
2sin(2πµ)
Here,thesecondequationfollowsinviewofq0q1+q0q1=2q0q0cos23π=−q0q0.
Wederivedirectlyfrom(5.4.30)thatq0q0=1−p02.(5.4.32)
Insertingthisintothesecondequation,weobtain
3p02−1=cos2(2πµ2)+cos(2πµ),(5.4.33)
22sin(2πµ)
,tlyalenequivor,21cos(2πµ)(cos(2πµ)+1)1cos2(πµ)−sin2(πµ)cos2(πµ)
p0=3+6sin2(πµ)cos2(πµ)=3+3sin2(πµ)=3sin2(πµ).(5.4.34)
impliesturninThiscos2(πµ)4sin2(πµ)−1
q0q0=1−3sin2(πµ)=3sin2(πµ).(5.4.35)
formasNext,(3.9.33)recallthatandthe(3.9.34).monoInsedromrytingthematricesMˆpreviousjsatisfyresults(3.9.32),togetheri.e.Mˆwith0Mˆthe1Mˆ∞iden=titI,yµwhic:=hµ0reads=iµn1=scalarµ∞
fromlemma5.3into(3.9.33),weobtain
√cos(2πµ)+isin(2πµ)p0=−cos2(2πµ)+2icos(2πµ)sin(2πµ)p0+sin2(2πµ)(p02−q0q0+i3q0q0),(5.4.36)
22whichinviewof(5.4.31)and(5.4.35)transformsinto
cos(2πµ)+isin(2πµ)p0=−cos2(2πµ)+2icos(2πµ)sin(2πµ)p0
+cos2(2πµ)+cos(2πµ)+√2i3cos2(πµ)(4sin2(πµ)−1),(5.4.37)
,tlyalenequivor,sin(2πµ)p0(1−2cos(2πµ))=√2cos2(πµ)(4sin2(πµ)−1).(5.4.38)
3Sincesin(2πµ)=2sin(πµ)cos(πµ)andcos(2πµ)=1−2sin2(πµ),thisimplies
p0=√2cos2(πµ)=√cos(πµ),(5.4.39)
3sin(2πµ)3sin(πµ)
90

determiningAltogether,p0wecompletelyconclude.Bythatathedirectfunctionscomputation,pjandweqjcoheckccurringthatin(3.9.34)theunitaryyieldsnomonofurtherdromyconditions.matrices
MˆjoftheextendedframeFofanormalizedtrinoid,whichisrotationallysymmetricwithrespecttothe
trinoidnormalsatisfy
)µπcos(p0=p1=p∞=√3sin(πµ),(5.4.40)
q0=e23πiq1=e−23πiq∞=√ζ0,(5.4.41)
)µπsin(3whereζ0isobtainedbysolvingζ0ζ0=4sin2(πµ)−1.(5.4.42)
Moreover,inviewofremark3.44,q0andthusalsoζ0arenecessarilyoddfunctionsinλ.ˆ
Applyingourresultsto(3.9.26),weobtaintheclaimedformsforthemonodromymatricesMj.
M˜→TheoremΛSU(2)5.13ofadescribtrinoidestheφ:(unM→itary)R3monowithdrompropyerlymatricesembeddedassoannciatedularwithends,thewhichextendedisframerotationallyF:
σψ(z∗)symmetric=1e3with,whererespectz∗∈toM˜theisgivtrinoideninnorma(5.4.1)l,andandψwhichdenoteshasbtheeenconformalnormalizedsuchCMC-immersionthatF(z∗)M˜=I→Rand3
corresp2Hondingtoφ.Itturnsoutthat,inthissetting,wecanalsoprovetheconverseresult:Atrinoid
φfromwith(5.4.1)properlyandembcorrespeddedondingannularmonoendsdromyandwimatricesthofextendedtheformframegivFeninsatisfyingtheoremF(z∗)5.13=isIatz∗necessarily∈M˜
rotationallysymmetricwithrespecttothetrinoidnormal.Thisresultisformulatedinthefollowing
theorem.matricTheoremesD,5.14.D,LetDηpbeaossessing(standarthedizesamed)trinoideigenvaluespotential±µ.assoDenoteciatedbywithφ:threMe→R3off-diagonalatrinoidDelaunaywith
properlyemb0edde1d∞annularendsonM=C\{0,1}generatedbyηviatheloopgroupmethod.Moreover,
FletF:urthermorM˜→e,letΛSU(2)F(zσ)b=eItheatzextende∈M˜dfrasamegivenassoinciate(5.4.1)d.withtheAssumemappingtheψunitary=φ◦monoπdrbyomytheoremmatric4.5.es
∗∗Mˆ0,Mˆ1,Mˆ∞∈ΛSU(2,C)σassociatedwithFaregivenby
Mˆ0=−cos(2πµ)I−√2icos(πµ)cos(ζ0πµ)−cos(ζ0πµ),(5.4.43)
32icos(πµ)e23πiζ
Mˆ1=−cos(2πµ)I−√3cos(πµ)e−23πiζ0−cos(π0µ),(5.4.44)
iπ2Mˆ∞=−cos(2πµ)I−√2icos(πµ)cos(23πiπµ)e−3ζ0,(5.4.45)
3eζ0−cos(πµ)
whereζ0isanoddfunctioninλandasolutionto
ζ0ζ0=4sin2(πµ)−1.(5.4.46)
Then,φisrotationallysymmetricwithrespecttothetrinoidnormal.
Prtheoof.sameSinceeigenthevaluesunderlying±µ,wecanDelaunawriteyηmatricesexplicitlyD0,Das1,D(cf.∞ofsectionthe3.6)standardizedtrinoidpotentialηpossess
1−η=−λQ0(z,λ)λ0dz,(5.4.47)
where111−1z2−z+1
Q(z,λ)=b(λ)z2+(z−1)2+z+z−1=b(λ)z2(z−1)2(5.4.48)
21γand(z)b(:=λ)1=4and−(µthe(λfu)).nctionh:ConsideringM→Cthe\{0},zbiholomorphic→h(z)=1−mappingz,weγ=γcomputeR:M→Mdefinedbyz→
z1−Q(γ(z),λ)=b(λ)(11−1z)2−11−1z+1=(z−1)2b(λ)1−1+z+1−2z+z2=(h(z))4Q(z,λ).(5.4.49)
(1−z)2(1−z−1)2z2
91

Recallingfromlemma4.21thatγRcorrespondstothepermutationσ=(01∞)oftheset{0,1,∞},
weapplylemma4.25toinferthatηtransformsunderγRas
γ∗η=η#W+,(5.4.50)
whereh(z)0
W+=W+(z,λ)=−λ∂zh(z)(h(z))−1.(5.4.51)
Applyingthepullbackconstructionwithrespecttothecoveringmappingπ:M˜→Mto(5.4.50),we
obtainπ∗(γ∗η)=π∗(η#W+)=η˜#W˜+,(5.4.52)
˜∗whereMoreov˜ηer,=πrecallηthatdenotesthethepullbacbiholomorphickpotentialmappingoftheγ˜=trinoidγ˜R:pM˜oten→tialM˜,ηz(cf.→−seczz−ti1onfrom2.3)andlemmaW+5.4=W+satisfies◦π.
γ◦π=π◦γ˜.Thus,thelefthandsideof(5.4.52)canbetransformedasfollows:
∗∗∗0λ−10λ−1
π(γη)=π−λQ(γ(z),λ)0dγ(z)=−λQ((γ◦π)(z),λ)0d(γ◦π)(z)
0λ−10λ−1
=−λQ((π◦γ˜)(z),λ)0d(π◦γ˜)(z)=γ˜∗−λQ(π(z),λ)0dπ(z)=γ˜∗(π∗η)=γ˜∗η˜.
(5.4.53)yields(5.4.52)Altogether,γ˜∗η˜=η˜#W˜+.(5.4.54)
thedifferenConsideringtialtheequationextendeddΨ=Ψframeη˜.FNoteassothatciatedΨpwithossessesthethetrinoidsameφ,(unweitary)obtainmonoasdoluromtionyΨ=matricesFB+astoF
ˆˆˆattheNaturallysingularities,theofmappingthepγ˜oten∗Ψtial=η,Ψ◦namelyγ˜Mdefines0,Ma1andsolutionM∞.tothedifferentialequationd(γ˜∗Ψ)=
(γ˜∗Ψ)(γ˜∗η˜),whichinviewof(5.4.54)readsas
d(γ˜∗Ψ)=(γ˜∗Ψ)(η˜#W˜+).(5.4.55)
SincethisdifferentialequationisalsosolvedbythemappingΨW˜+,i.e.
d(ΨW˜+)=(ΨW˜+)(η˜#W˜+),(5.4.56)
themappingsγ˜∗ΨandΨW˜+onlydifferbyaλ-dependentmatrixρ=ρ(λ):
γ˜∗Ψ=ρΨW˜+.(5.4.57)
˜Nowapplyingtherelation(5.3.4),γ˜◦γ˜0=˜γ1◦γ˜,involvingthecoveringtransformationsγ˜0andγ˜1
onMasgiveninsection3.3,wecompute
Mˆ1(λ)ρ(λ)Ψ(z,λ)W˜+(z,λ)=Mˆ1(λ)(γ˜∗Ψ(z,λ))=Mˆ1(λ)Ψ(γ˜(z),λ)=Ψ((γ˜1◦γ˜)(z),λ)
=Ψ((γ˜◦γ˜0)(z),λ)=γ˜∗Ψ(γ˜0(z),λ)=ρ(λ)Ψ(γ˜0(z),λ)W˜+(γ˜0(z),λ)=ρ(λ)Mˆ0(λ)Ψ(z,λ)W˜+(γ˜0(z),λ).
(5.4.58)˜˜AsWholomorphic+definesonMthe˜andpullbackthereforeofthedoesnotmappingpickWup+,anywhichmonoisdromyholomorphicunderγ˜on0,Mi.e.W˜(with+(γ˜0(respz),ectλ)=tozW˜),+(Wz,+λ).is
Thus,weconcludethat
Mˆ1(λ)ρ(λ)=ρ(λ)Mˆ0(λ).(5.4.59)
Analogously,applyingγ˜◦γ˜∞=γ˜0◦γ˜from(5.3.4),wehave
Mˆ0(λ)ρ(λ)Ψ(z,λ)W˜+(z,λ)=Mˆ0(λ)(γ˜∗Ψ(z,λ))=Mˆ0(λ)Ψ(γ˜(z),λ)=Ψ((γ˜0◦γ˜)(z),λ)
=Ψ((˜γ◦γ˜∞)(z),λ)=γ˜∗Ψ(γ˜∞(z),λ)=ρ(λ)Ψ(γ˜∞(z),λ)W˜+(γ˜∞(z),λ)
=ρ(λ)Mˆ∞(λ)Ψ(z,λ)W˜+(γ˜0(z),λ).(5.4.60)
92

UsingtheholomorphicityofW˜+onM˜,weknowthatW˜+(γ˜∞(z),λ)=W˜+(z,λ),whichyields
Mˆ0(λ)ρ(λ)=ρ(λ)Mˆ∞(λ).(5.4.61)
seteWρ(λ)=ca((λλ))db((λλ)),(5.4.62)
wherea,b,candddefinecomplexvaluedfunctionsofλsatisfyinga(λ)d(λ)−b(λ)c(λ)=1.Then,by
comparingthe2πiupperleftentriesofMˆ1(λ)ρ(λ)and2πρi(λ)Mˆ0(λ)(resp.ofMˆ0(λ)ρ(λ)andρ(λ)Mˆ∞(λ)),we
obtainc(λ)e3ζ0=b(λ)ζ0(resp.c(λ)ζ0=b(λ)e3ζ0).Combiningthesetwoequationsyieldsb(λ)≡
c(λ)≡ˆ0.Comparingnow2πitheupperrightentriesofMˆ1(−λ)2ρπi(λ)andρ(λ)Mˆ0(λ)(resp.ofMˆ0(λ)ρ(λ)and
ρ(λ)M∞(λ)),weobtaine3d(λ)=πai(λ)(resp.d(λ)=e3a(λ)).Togetherwitha(λ)d(λ)−0=1,we
concludethata(λ)=(d(λ))−1=±e3andthus
eπ3i0
ρ(λ)=±0e−π3i,(5.4.63)
inparticularρ(λ)∈ΛSU(2)σ.Thus,(ρFρ−1)(ρB+W˜+)definesanIwasawa-decompositionofρΨW˜+
(pointwiseforallz∈M˜)withρFρ−1∈ΛSU(2)σ,ρB+W˜+∈Λ+SL(2,C)σand(ρFρ−1)(z∗)=I.
writecanewTherefore,(F◦γ˜)(B+◦γ˜)=γ˜∗Ψ=ρΨW˜+=(ρFρ−1)(ρB+W˜+).(5.4.64)
∗ThisSymBob(impliesF◦γ)that,|λ=1usingandonthetheloopothergrouphandmethothed,γ˜rotatedΨprotrinoidducesρJon(ψ)theρ−1one=handSymBob(theρFtrinoidρ−1)J|λ(ψ=1.◦γ˜)Con-=
sequently,thesetwosurfacescoincide,i.e.
J(ψ◦γ)(M˜)=(ρJ(ψ)ρ−1)(M˜).(5.4.65)
UsingtheidentityρJ(ψ)ρ−1=J◦AR◦ψ,where
√−√21230
AR=−23−120,(5.4.66)
100fromtheproofofcorollary5.12,weswitchintotheR3modelandobtainψ◦γ˜=AR◦ψ.Asγ˜(M˜)=M˜,
yieldsthisψ(M˜)=AR(ψ(M˜)).(5.4.67)
Thismeansthatψ(andthusalsoφ)is2πsymmetricwithrespectto3theEuclideanmotionAR∈Iso(R3)
definingsymmetricthewithrotationrespectbytothetheangletrinoid±3normal.around(Intheviewz-axisofintheoremR.Th4.31,us,whicφhislistsneceallssarilypossiblerotationallytrinoid
π2Thus,symmetries,weinferonlythatthethezrotation-axisinbRy3thecoincidesangle±with3thearoundtrinoidthenormal,trinoidandnormalthatshoφiswstherotationallybehavioursymmeofAtrRic.
withrespecttothetrinoidnormal,coincidingwiththez-axisinR3.)
5.5Solvingζ0ζ0=4sin2(πµ)−1
Inordertodescribethesolutionsζ0to(5.4.42),weinvestigatetherighthandsideof(5.4.42).Tothis
1end,recallfromlemma5.3,that
µ(λ)=4+w(λ−λ−1)2,(5.5.1)
wherew=s0t0=s1t1=s∞t∞andsj,tjdenotetheparametersoccurringintheDelaunaymatricesDj
(3.5.7).indefined∗ForUnthetilfollnoow,wingwehaveconsiderations,treatedµwaseaextendholomorphicµtoC∗byfunctionofexplicitlyλinthedefiningdomaintheoC\ccurringW1(cf.complexremarksquare3.24).
rootonC∗.(Ofcourse,thisbreaks∗theholomorphicityofµ.However,byourdefinitionofthesquare
rootbelow,therestrictionofµtoC\W1willstillbeholomorphic.)

93

Wedefine∗thecomplexsquarerootontheλ-planeC∗inanalogytothecomplexsquarerootonthe
z-planeCgiveninremark4.14by
√√:C∗→C∗,λ=reiθ→λ:=√reiθ2,(5.5.2)
√usualwherewe(real)writesquareλ∈Cro∗otinofther.Informλ=particular,reiθ√withrmaps∈R+theandnegativθ∈(e−πreal,π],axisandontorthedefinespositivtheevalueimaginaryofthe
∗axis.TheWithfollothiswinglemmadefinition,wstateseintheterpretrootsµoffromthenowexpressiononasb4eingsin2(πdefinµ(eλd))for−1.λ∈C.
5.15.Lemma4sin2(πµ(λ))−1=0⇐⇒(µ(λ))2=(61+k)2fork∈Z⇐⇒λ∈Ik:={±λk,±λk−1}fork∈Z,(5.5.3)
where,fork∈Z,
λk:=21wdk+dk2−4w2(5.5.4)
anddk:=(61+k)2−41+2w.(5.5.5)
e,urthermorFλk,−λk−1∈iR+fork=0,(5.5.6)
λk,λk−1∈R+fork∈Z\{0}.(5.5.7)
Proof.Westarttheproofwiththefollowingobservation:
4sin2(πµ(λ))−1=0⇐⇒(2sin(πµ(λ))−1)(2sin(πµ(λ))+1)=0⇐⇒sin(πµ(λ))=±21
11⇐⇒µ(λ)=±(6+k)fork∈Z⇐⇒(µ(λ))2=(6+k)2fork∈Z.(5.5.8)
Thisalreadyprovesthefirstpartof(5.5.3).Furthermore,asµ(λ)=41+w(λ−λ−1)2,wehaveforall
k∈Zandallλ∈C∗
(µ(λ))2=(1+k)2⇐⇒1+w(λ−λ−1)2=(1+k)2
646⇐⇒wλ4+(−(61+k)2+41−2w)λ2+w=0
⇐⇒λ=±21w(61+k)2−41+2w±((61+k)2−41+2w)2−4w2.(5.5.9)
Definingdkasin(5.5.5)andforallk∈Z
λk,1:=−λk,3:=1dk+dk2−4w2(5.5.10)
w21λk,2:=−λk,4:=2wdk−dk2−4w2(5.5.11)
(5.5.12)obtainew1(µ(λ))2=(6+k)2⇐⇒λ∈{λk,1,λk,2,λk,3,λk,4}.(5.5.13)
Nextweshowforallk∈Zthat
λk,1,λk,2∈iR+andλk,3,λk,4∈iR−fork=0(5.5.14)
λk,1,λk,2∈R+andλk,3,λk,4∈R−fork=0(5.5.15)
94

Wefirstlookatthecasek=0.Wehaved02−4w2=(−92+2w)2−4w2=814−98w,whence,asw∈(0,181]
bylemmaB.6ofappendixB,weinferthat
d02−4w2≥0.(5.5.16)
Asaconsequence,d02−4w2liesonthepositiverealaxis.Moreover,wehaved02>d02−4w2≥0and
thus|d0|>d02−4w2.Becauseofw∈(0,181]wehaved0<0,whichimplies−d0=|d0|>d02−4w2.
Altogether,weobtain
d0±d02−4w2≤d0+d02−4w2<d0−d0=0(5.5.17)
thatconcludeand11[d0±d02−4w2]=i[−d0d02−4w2]∈iR+,(5.5.18)
w2w2whichproves(5.5.14).
Nowweturntothecasek=0.Here,(61+k)2−41>0,andhence,usingw>0,
11dk2−4w2=((+k)2−+2w)2−4w2>(2w)2−4w2=0.(5.5.19)
46Thisimpliesthatdk2−4w2liesonthepositiverealaxis.Furthermore,asaconsequenceof(61+k)2−41>
0andw>0,wehavedk>0.Hence,
dk+dk2−4w2>0.(5.5.20)
Moreover,asaconsequenceofdk2>dk2−4w2>0,weobtaindk=|dk|>dk2−4w2>0andthus
dk±dk2−4w2≥dk−dk2−4w2>dk−dk=0.(5.5.21)
1Weinferthat
[dk±dk2−4w2]∈R+,(5.5.22)
w2whichproves(5.5.15).
thatobserving,Finallyλ0,1λ0,4=−i2[d0+d02−4w2][d0−d02−4w2]=d02−(d02−4w2)=1,(5.5.23)
111
2w2w2w
obtainewλ0,2=−λ0−,11,λ0,3=−λ0,1,λ0,4=λ0−,11.(5.5.24)
Hence,asbydefinitionλ0=λ0,1,wehave
{λ0,1,λ0,2,λ0,3,λ0,4}={±λ0,±λ0−1},(5.5.25)
(5.5.24)ofviewinwhereλ0,−λ0−1∈iR+,(5.5.26)
whichproves(5.5.6).
Analogously,fork=0,wecompute
λk,1λk,2=[dk+dk2−4w2][dk−dk2−4w2]=dk2−(dk2−4w2)=1,(5.5.27)
111
2w2w2w
whichimplies(fork=0)
λk,2=λk−,11,λk,3=−λk,1,λk,4=−λk−,11.(5.5.28)
Inviewofλk=λk,1weinfer
1−{λk,1,λk,2,λk,3,λk,4}={±λk,±λk}fork=0,(5.5.29)
(5.5.28),usingwhere,λk,λ−1∈R+fork=0,(5.5.30)
kwhichproves(5.5.7).
Combiningtherelations(5.5.8)and(5.5.13)withtheequations(5.5.25)and(5.5.29),the(secondpart
ofthe)claimedrelation(5.5.3)follows.

95

(5.5.31)

(5.5.32)

Next,westatethefollowingtheorem.
Theorem5.16.Forallλ∈C∗thefollowingholds:
∞4sin2(πµ(λ))−1=4π2Ck(1−λ)(1+λ)(1−λ−1)(1+λ−1),(5.5.31)
k=−∞λkλkλkλk
ewher−λk2wfork=0
Ck:=λ2kw2fork=−1(5.5.32)
kk2λ(1+kw1)31fork∈Z\{−1,0}
andλkisgivenin(5.5.4).
Proof.TheproofofthistheoremisquitetechnicalandthereforegivenintheappendixH.
Wenowreturntoequation(5.4.42):
ζ0ζ0=4sin2(πµ)−1.(5.5.33)
Bytheorem5.16,wehave
∞4sin2(πµ(λ))−1=4π2Ck(1−λ)(1+λ)(1−λ−1)(1+λ−1),(5.5.34)
k=−∞λkλkλkλk
whereCkdefinesapositiverealnumberforallk∈Z.Thus,inordertosolve(5.5.33)forζ0,weneedto
splittheinfiniteproductgivenin(5.5.34),i.e.weneedtodistributethefactorsofthisproductamongζ0
andζ0.Inviewofthedefiningrelation
ζ0(λ)=ζ0(1)(5.5.35)
λdefiningζ0andtherelations(5.5.6)and(5.5.7)fromlemma5.15,weobservethatwenecessarilyhave
ζ0(±λ0)=0⇐⇒ζ0(±λ0−1)=ζ0(λ0−1)=0,(5.5.36)
ζ0(±λ0−1)=0⇐⇒ζ0(±λ0)=ζ0(λ0)=0(5.5.37)
and,forallk∈Z\{0},
ζ0(±λk)=0⇐⇒ζ0(±λk−1)=ζ0(±λk−1)=0,(5.5.38)
ζ0(±λk−1)=0⇐⇒ζ0(±λk)=ζ0(±λk)=0.(5.5.39)
Thus,whendistributingthefactorsoftheinfiniteproductgivenin(5.5.34)amongζ0andζ0,weneedto
respectthefollowingrelations:
1−(1±λλ)contributestoζ0⇐⇒(1λλ)contributestoζ0,(5.5.40)
λ−01λ0
(1±λ0)contributestoζ0⇐⇒(1λ0)contributestoζ0(5.5.41)
and,forallk∈Z\{0},
1−(1±λλ)contributestoζ0⇐⇒(1±λλ)contributestoζ0,(5.5.42)
λ−k1λk
(1±λk)contributestoζ0⇐⇒(1±λk)contributestoζ0.(5.5.43)

96

(5.5.42)(5.5.43)

Consequently,foreachk∈Z,ζ0necessarilycontainsexactlytwoofthefourfactors(1−λλk),(1+λλk),
(1−λλ−k1)and(1+λλ−k1)inoneofthefollowingfourpossiblecombinations:
λλζ0containspk(1)(λ):=(1−λk)(1+λk)(5.5.44)
(2)λ−1λ−1
orζ0containspk(λ):=(1−λk)(1+λk)(5.5.45)
(1−λ)(1−λ−1)fork=0
orζ0containspk(3)(λ):=(1−λλ0)(1+λλ−01)fork∈Z\{0}(5.5.46)
λλkk(4)(1+λ)(1+λ−1)fork=0
orζ0containspk(λ):=(1+λλλ0)(1−λλλ−01)fork∈Z\{0}(5.5.47)
kkSofar,wehaveinvestigatedthequestionhowtheλ-dependentfactorsoftheinfiniteproductgiven
in(5.5.34)canbedistributedamongζandζinordertosolve(5.5.33).Alsotheconstantfactors
4π2andCk,k∈Z,occurringinthe0represen0tation(5.5.34)oftheexpression4sin2(πµ(λ))−1asan
infiniteproduct,needtobedistributedamongζ0andζ0.Since4π2andCk(forallk∈Z)arepositive
real“equally”numbers,amongtheseζ0andfactorsζ0inarethe(insenseorderthattobtheeinresplineectivwithethsquareerodefinitionots,i.e.of2πζ0)and(fornecessarilyallk∈Zdistrib)√Cutedk,
contributeAltogether,tobwotheζset0andtheζ0follo.wing“basic”formofζ0:
∞ζ0(λ)=2πCkpk(νk)(λ),(5.5.48)
=k−∞where,forallk∈Z,νk∈{1,2,3,4}.
gg=Remark1.In5.17.particular,Noteg(that,λ)=inλisgeneral,avalidζ0ccanhoice,stillprobemoducingdifiedbyanyλ-dependentfunctiongsatisfying
∞ζ0(λ)=2πλCkpk(νk)(λ).(5.5.49)
=k−∞Sinceweareinterestedinafunctionζ0solving(5.5.33),whichisoddinλ,wekeepthispossibilityin
mindforlateruse:Ifζ0oftheform(5.5.48)solves(5.5.33)andiseveninλ,thefunctionλζ0isoddinλ
whileitaswellsolves(5.5.33).
Thefollowingobservationiscrucialforourfurtherconsiderations:Theinfiniteproduct
∞4π2Ck(1−λ)(1+λ)(1−λ−1)(1+λ−1),(5.5.50)
k=−∞λkλkλkλk
2∗o(Wccurringritinginsin(πµ((5.5.34)λ))iinswitsellpowdefinederforseriesallλ∈represenC,tation,sincethisweistheobservecasethatforthsine2(expπµre(λ))ssionin4volvsines(πµonly(λ))ev−en1.
22∗pthouswersalsoof4µ(sinλ2).(πµ(Since,λ))−by1areremark3.13,holomorphicµisandwdefinedellasdefinedaforholomorphicλ∈C∗.)functionWhenonsplittingC,sin((5.5.50)πµ(λin))twando
infiniteproductsrepresentingζandζ,westillneedtoensurethatthesetwoproductsarewelldefined
onC∗,i.e.thattheytakefinite0values0forallλ∈C∗.Thisleadstothenotionof(normal)convergence
ofaninfiniteproduct(cf.,e.g.[33],chapter1).
Definition5.18.1.Letν∈N0and(an)n≥νbeasequenceofcomplexnumbers.Then,thesequence
∞m
n=νan:=n=νanm≥ν(5.5.51)
∞ismanycalledn≥anν,i.e.infinitea=pro0ductfor.allnn=≥νann,isandsaidiftothebeclimitonvergentifandonlyifan=0foronlyfinitely
0nlllim→∞n=nan(5.5.52)
097

exists.Inthiscase,thevalueofn∞=νanisdefinedtobe
la:=aν∙aν+1∙∙∙an0−1∙llim→∞an.(5.5.53)
n=n0Afiniteproduct,whichdoesnotconverge,iscalleddivergent.
2.LetsequenceXbeofaconlotinuouscally-compactfunctionsmetricX→spaceC.X,Moreoe.g.vaer,subsetdefineofforC.allnLet≥νν∈theN0conandtin(fuousn)n≥νfunctionbea
gn:X→Cbygn=fn−1.Then,thesequenceoffunctions
∞m
n=νfn:=n=νfnm≥ν(5.5.54)
iscalledaninfiniteproduct(offunctions).∞fiscallednormallyconvergent(onX),ifthe
series∞gconvergesnormallyonX,i.e.,if,n=νfornanycompactsubsetKofX,∞|g|<∞,
where|gnn|=Kν:=nsupz∈K|gn(z)|.Afiniteproduct,whichisnotnormallyconvergentnon=νX,nisKcalled
.XongentdiverRemark5.19.Thenotionsofconvergenceofaninfiniteproductandofnormalconvergenceofan∞infinite
productoffunctionsgivenindefinition5.18aretransferredtoinfiniteproductsoftheforma
andinfiniteproductsoffunctionsoftheformn∞=−∞fn,respectively,asfollows:Forcomplexnn=um−∞bersn
an,n∈Z,theinfiniteproduct∞
n=−∞an(5.5.55)
∞−1∞
isconvsaidergeto.beconOtherwise,vergenitt,isifcalledanddivonlyergenift.theForinfinitecontinprouousductsfunctionsn=0fann:andX→nC=,n−∞∈anZ,:=onanlo=1acally-−n
compactmetricspaceX,theinfiniteproductoffunctions
∞n=−∞fn(5.5.56)
1−∞is∞calledf−narenormallynormallyconvconergenvtergen(ontX(on),ifX).andonlyOtherwise,iftheitisinficalledniteprodivductsergent(onn=0Xf).nandn=−∞fn:=
=1nWenotethefollowingusefulresult:
∞Lemma(an5.−20.1)cLetonverνges∈toN0someand(alimitn)na≥ν∈bRe.aseThen,quenctheeinofrfiniteealpronumbductersa∞n>an0.converAssumeges.theseries
n=νn=ν
Proof.Sincean=0foralln≥ν,itisenoughtoshowthatthelimitliml→∞ln=νanexists.Tothis
end,considerforalll≥νtheestimate
ll0≤an≤ean−1=enl=ν(an−1),(5.5.57)
n=νn=ν
adirectconsequenceoftherelationx≤ex−1forallx∈R.(5.5.57)implies
l0≤liman≤limenl=ν(an−1)=ea∈R,(5.5.58)
l→∞n=νl→∞
whichprovestheclaim.
Letnowζ0beoftheform(5.5.48).Inthefollowing,westudythequestion,if,or,moreprecisely,for
which“configurations”ofthefactorspk(νk)(λ),ζ0iswelldefined,i.e.normallyconvergent,onC∗.Asa
start,wehavethefollowingresult:
Lemma5.21.Forallk∈Z,letλkandCkbegivenby(5.5.4)and(5.5.32),respectively.Moreover,let
theλ-dependentfunctionspk(ν)(λ),ν∈{1,2,3,4}bedefinedby(5.5.44),(5.5.45),(5.5.46)and(5.5.47),
respectively.Then,wehave:

98

(5.5.58)

1.Theinfiniteproductk∞=−∞√Ckconverges.
2.Theinfiniteproductk∞=−∞pk(1)isnormallyconvergentonC∗.
3.Theinfiniteproductk∞=−∞pk(2)isnormallyconvergentonC∗.
4.Theinfiniteproductk∞=−∞pk(3)isdivergentonC∗.
5.Theinfiniteproductk∞=−∞pk(4)isdivergentonC∗.
Proof.ThisisprovedinappendixI.
Wereturntoζ0oftheform(5.5.48),i.e.
∞ζ0(λ)=2πCkpk(νk)(λ),(5.5.59)
=k−∞where,forallk∗∈Z,νk∈{1,2,3,4}.Wewanttoensurethatζ0iswelldefinedandthusnormally
convergentonC.Moreover,ζ0ismeanttobeanoddfunctionofλoranevenfunctionofλ(cf.remark
5.17).Applyingbasicresultsconcerninginfiniteproducts(cf.,e.g.,[33],chapter1),theabovelemma5.21
immediatelyallowsforthefollowingconclusions:
∗1.ζζ0isisdwivellergendefinedt(onandC)thifusandnormallyonlyifνkcon∈v{3ergen,4}tonforC∗,infinitelyweinfermanythatk∈νZ∈.{3Con,4s}forequenattly,mostpresumingfinitely
k0manyk∈Zandtherefore(naturally)νk∈{1,2}forinfinitelymanyk∈Z.
2.Byequations(5.5.44)to(5.5.47),weobservethat,forallk∈Z,pk(1)andpk(2)defineevenfunctions
ofλ,whilepk(3)andpk(4)definefunctionsofλ,whichareneitherevennorodd(inλ).Thus,any
infiniteproductinvolvingonlyfactorsoftheformpk(1)and/orpk(2)willbeeveninλ,whileanyfinite
productoffactorsoftheformpk(3)and/orpk(4)willbeneitherevennoroddinλ.Consequently,for
ζ0oftheform(5.5.48)withνk∈{1,2}forinfinitelymanyk∈Zandνk∈{3,4}foratmostfinitely
manyk∈Z,weinferthatζ0isevenoroddinλifandonlyifallfactorspk(νk)occurringinζ0are
oftheformpk(1)oroftheformpk(2).(Actually,thecasethatζ0isoddinλdoesnotoccur.)
Wesummarizeourconsiderationsaboveinthefollowinglemma:
Lemma5.22.Letζ0beoftheform
∞ζ0(λ)=2πCkpk(νk)(λ),(5.5.60)
k=−∞where,forallk∈Z,νk∈{1,2,3,4},Ckisgivenin(5.5.32),andthefunctionspk(νk)(λ)aredefinedby
(dependingonνk)(5.5.44),(5.5.45)∗,(5.5.46)or(5.5.47),respectively.Then,thefollowingholds:
k∈ζ0Z.isAwelctuallly,defineifdνkfor∈{al1l,2λ}∈forCallandk∈anZ,evenζ0isorwelanlodddefinedfunctionforaloflλλ,∈ifC∗andandonlyanifevenνk∈{1function,2}forofλal.l
equationInviewof(5.5.33),lemmawhich5.22iswandelldefinedremarkfor5.17,allλwe∈Ccan∗andgiveantheoddfollofuncwingtioninbasicλ:formofasolutionζ0to
∞ζ0(λ)=2πλCkp(kνk)(λ),(5.5.61)
=k−∞where,forallk∈Z,νk∈{1,2},Ckisgivenin(5.5.32),andthefunctionspk(νk)(λ)aredefinedby
(dependingonνk)(5.5.44)or(5.5.45),respectively.
Byremark5.17,ζ0canstillbemodifiedbyanyλ-dependentfunctiong,which,inordertopreserve
thepropertiesofζ0,necessarilysatisfiesg(−λ)=g(λ)andgg=1.Sincegmightpossesssingularitiesin

99

C∗,aftermodifyingζ0oftheform(5.5.61)byg,thenewsolutiongζ0mightnotbewelldefinedonC∗.
However,thisisnotnecessary:returningtotheorem3.59,weonlywanttoachieve
sin(2πµ)q0isholomorphicforλ∈C∗,(5.5.62)
q0takesafinitevalueinCatλ=1andisholomorphicatλ=1,(5.5.63)
whichinviewof(cf.section5.4)
ζ0q0=√3sin(πµ)(5.5.64)
tointranslatescos(πµ)ζ0isholomorphicforλ∈C∗,(5.5.65)
ζ0takesafinitevalueinCatλ=1andisholomorphicatλ=1.(5.5.66)
Altogether,thegeneralformofasolutionζ0toequation(5.5.33)isgivenby:
∞ζ0(λ)=2πλg(λ)Ckp(kνk)(λ),(5.5.67)
=k−∞suchthatζ0satisfies(5.5.65)and(5.5.66),thefunctiongsatisfies
g(−λ)=g(λ),(5.5.68)
gg=1,(5.5.69)
andwhere,forallk∈Z,νk∈{1,2},Ckisgivenin(5.5.32),andthefunctionsp(kνk)(λ)aredefinedby
(dependingonν)(5.5.44)or(5.5.45),respectively.
Infollowing,kwetranslatetheconditions(5.5.65)and(5.5.66)intofurtherconstraintsonthefunction
g,(5.5.67)whichwillimpliesleadthattogamaygeneralonlyformhaveofg.Wsingularitieseobservatevthealuesfolloofλwing:∈CIn∗,orderwheretocos(πsatisfyµ(λ))=(5.5.65),0.Moreoequationver,
µto(1)fulfill=1and(5.5.66),thusthecos(vπalueµ(1))λ==10).isSiexcludedncebyfrom(5.5.68)this,g(i.e.−1)g=needsg(1),togbisewalsoelldefinednecessarilyatλw=ell1defined(althoughat
λ=−1.2Thezerosoftheexpressioncos(πµ(λ))aregiveninthefollowinglemma.
5.23.Lemma1cos(πµ(λ))=0⇐⇒(µ(λ))2=(2+j)2forj∈Z⇐⇒λ∈Jj:={±λj,±λj−1}forj∈Z,(5.5.70)
where,forj∈Z,
λj:=21wdj+dj2−4w2(5.5.71)
anddj:=(1+j)2−1+2w.(5.5.72)
42Furthermore,wehaveforallj∈Zλ,λ−1∈R+.(5.5.73)
jjProof.Westarttheproofwiththefollowingobservation:
cos(πµ(λ))=0⇐⇒µ(λ)=1+jforj∈Z⇐⇒(µ(λ))2=(1+j)2forj∈Z.(5.5.74)
22Thisalreadyprovesthefirstpartof(5.5.70).Furthermore,asµ(λ)=41+w(λ−λ−1)2,wehavefor
allj∈Zandallλ∈C∗
(µ(λ))2=(21+j)2⇐⇒41+w(λ−λ−1)2=(21+k)2
⇐⇒wλ4+(−(1+j)2+1−2w)λ2+w=0
24
2w2424
⇐⇒λ=±1(1+j)2−1+2w±((1+j)2−1+2w)2−4w2.(5.5.75)
100

Definingdjasin(5.5.72)andforallj∈Z
λj,1:=−λj,3:=1dj+dj2−4w2(5.5.76)
12w
λj,2:=−λj,4:=2wdj−dj2−4w2(5.5.77)
(5.5.78)obtainew1(µ(λ))2=(2+j)2⇐⇒λ∈{λj,1,λj,2,λj,3,λj,4}.(5.5.79)
Nextweshowforallj∈Zthat
λj,1,λj,2∈R+andλj,3,λj,4∈R−.(5.5.80)
Since,forallj∈Z,(21+j)2−41≥0,andhence,usingw>0,
dj2−4w2=((1+j)2−1+2w)2−4w2≥(2w)2−4w2=0.(5.5.81)
42Thisimpliesthatdj2−4w2liesonthenon-negativerealaxis.Furthermore,asaconsequenceof(21+
j)2−41≥0andw>0,wehavedj>0.Hence,
dj+dj2−4w2>0.(5.5.82)
Moreover,asaconsequenceofdj2>dj2−4w2≥0,weobtaindj=|dj|>dj2−4w2≥0andthus
dj±dj2−4w2≥dj−dj2−4w2>dj−dj=0.(5.5.83)
Weinferthat
1[dj±dk2−4w2]∈R+,(5.5.84)
w2whichproves(5.5.80).
Finally,observingthatforallj∈Z
111
λj,1λj,2=[dj+dj2−4w2][dj−dj2−4w2]=dj2−(dj2−4w2)=1,(5.5.85)
2w2w2w
obtainewλj,2=λj−,11,λj,3=−λj,1,λj,4=−λj−,11.(5.5.86)
Inviewofλj=λj,1weinferthatforallj∈Z
{λj,1,λj,2,λj,3,λj,4}={±λj,±λj−1},(5.5.87)
where,using(5.5.86),forallj∈Z
λj,λj−1∈R+,(5.5.88)
whichproves(5.5.73).
Combiningtherelations(5.5.74)and(5.5.79)withtheequation(5.5.87),the(secondpartofthe)
ws.follo(5.5.70)relationclaimedBylemma5.23,thesetofzerosoftheexpressioncos(πµ(λ))(inC∗)isgivenby
Jj={±λj,±λj−1}.(5.5.89)
j∈Zj∈Z
Observingthat,forallj∈Z,λj=λ−1−jandthus
Jj=J−1−j,(5.5.90)
101

andthat,furthermore,Jjcontains±1ifandonlyifj=0(actually,J0={±1}),weconcludethat
{λ∈C∗\{±1};cos(πµ(λ))=0}=J:=Jj={±λj,±λj−1}.(5.5.91)
j∈Nj∈N
∗∗theAsfunctionindicatedgmabybefore,eJsingular.={λ∈MoreC\{±precisely1};,cos(sinceπµ(λeac))h=λ0∈}Jisdefinesasimplethesetrootofλof-vcos(aluesπµ)in(prCo,vedwhereby
directcomputation),gmayatmosthaveasimplepoleatλ∈J.
Foreachj∈N,therearefourpointsinC∗,atwhichgmayhavesimplepoles:±λjand±λj−1,where
λjisgivenby(5.5.71).Inviewof(5.5.68),weimmediatelyinferthatallpolesofgcomeinpairs,i.e.
λjisa(simple)poleofg⇐⇒−λjisa(simple)poleofg,(5.5.92)
λj−1isa(simple)poleofg⇐⇒−λj−1isa(simple)poleofg.(5.5.93)
Assumenowthat,forsomej∈N,gpλossesses−1apairλof−1simplepolesat±λj.Thus,writteninproduct
representation,gcontainsthefactor(1−λj)(1+λj).By(5.5.69),gconsequentlycontainsthe
inversefactor(1−λλj)(1+λλj).Inparticular,g(±λj)=0.Since,bydefinition,g(λ)=g(λ−1),this
impliesdirectly(usingλj∈R),thatg(±λj−1)=g(±λj−1)=0,whichmeansthatgalsocontainsthe
factor(1−λλ−j1)(1+λλ−j1).Altogetherweconcludethat,if,forsomej∈N,gpossessesapairofsimple
polesat±λj,thengcontainsinitsproductrepresentationthefactor
(1−λλ−j1)(1+λλ−j1)
gj,1(λ):=(1−λλj)(1+λλj).(5.5.94)
Analogously,weinferthat,if,forsomej∈N,gpossessesapairofsimplepolesat±λj−1,thengcontains
initsproductrepresentationthefactor
(1−λλ)(1+λλ)
gj,2(λ):=(1−λ−1j)(1+λj−1).(5.5.95)
λλjj(Obviously,foreachj∈N,gcanonlypossessapairofsimplepolesat±λjorat±λj−1,sincethe
respectiveotherpairwillautomaticallybecomeapairofzerosofg.)Wenotethatthefactorsgj,1and
gj,2areeveninλandsatisfytherelation(5.5.69).Thus,theyarevalidcomponentsofg.
Sofar,theoretically,gmaycontainaninfiniteproductoffactorsoftheform(5.5.94)and/oraninfinite
productoffactorsoftheform(5.5.95).Toprovethatthisisactuallypossible,weshowthatthefollowing
holds:Lemma5.24.Theinfiniteproducts(offunctions)
gj,1(λ)andgj,2(λ),(5.5.96)
j∈Nj∈N
C∗wher\e,for{±allλj}∈andN,Cgj∗,1\isgiven{±λin−1},r(5.5.94)espeandctively.gj,2isgivenin(5.5.95),arenormallyconvergenton
j∈Njj∈Nj
Proof.ThisisprovedinappendixI.
forallFinallyλ∈,Cw∗eandobserveveenthatinλg.canBybedefinitioncompletedofgbyandanequationadditional(5.5.69),factor,whthisichisfactorwellcandefinedbeandwritteninnon-zerothe
formeih(λ),(5.5.97)
wherehdenotesarealvaluedevenfunctionofλ.
Altogether,weconcludethatthefunctiongcanbewritteninthegeneralform
g(λ)=eih(λ)gj,1(λ)gj,2(λ)(5.5.98)
j∈N1j∈N2
wherehdenotesarealvaluedevenfunctionofλ,thefunctionsgj,1andgj,2areforallj∈Ngiven
in(5.5.94)and(5.5.95),respectively,andN1,N2denotesubsetsofthenaturalnumbersNsatisfying
N1∩N2=∅.InthecaseN1=∅orN2=∅,thecorrespondingproductoverjissettobe1.
Wesummarizeourresults:

102

Theorem5.25.Thegeneralsolutionζ0totheequation
ζ0ζ0=4sin2(πµ)−1(5.5.99)
satisfyingζ0isanoddfunctionofλ,(5.5.100)
cos(πµ)ζ0isholomorphicforλ∈C∗,(5.5.101)
ζ0takesafinitevalueinCatλ=1andisholomorphicatλ=1(5.5.102)
formtheofis∞
ζ0(λ)=2πλeih(λ)gj,1(λ)gj,2(λ)Ckpk(νk)(λ),(5.5.103)
j∈N1j∈N2k=−∞
wherehdenotesarealvaluedevenfunctionofλ∈C∗,thefunctionsgj,1andgj,2areforallj∈Ngiven
by(1−λλ−j1)(1+λλ−j1)
gj,1(λ)=(1−λλj)(1+λλj),(5.5.104)
(1−λλ)(1+λλ)
gj,2(λ)=(1−λ−1j)(1+λj−1)(5.5.105)
λλjjwithλjgivenin(5.5.71),N1,N2denotesubsetsofthenaturalnumbersNsatisfyingN1∩N2=∅,and
(wherνk)e,moreover,νk∈{1,2}forallk∈Z,Ckisforallk∈Zgivenin(5.5.32),andthefunctions
pk(λ)aredefinedby(dependingonνk)
pk(1)(λ)=(1−λλ)(1+λλ),(5.5.106)
−k1k−1
pk(2)(λ)=(1−λλ)(1+λλ)(5.5.107)
kkwithλkgivenin(5.5.4).(InthecaseN1=∅orN2=∅,thecorrespondingproductoverjissettobe1.)
assoRemarkciated5.26.withhIt(λis)≡susp0,N1ected=N2(based=∅onandνkcomputer=1forexpk<erimen0,νkts)=2thatforthek≥sp0,eciali.e.solutionζ0to(5.5.99)
−1∞
ζ0(λ)=2πλCkpk(1)(λ)Ckpk(2)(λ)
k=−∞k=0
=2πλ−1Ck(1−λ22)∞Ck(1−λ−22),(5.5.108)
k=−∞λkk=0λk
correspondstothetripleofunitarizedmonodromymatricesMˆj,j=0,1,∞,associatedwitha(com-
˜3inpletely)(5.5.108)proponlyerly“onembeddedfinitelymantrinoidypM→ositions”R.inOtherthesensesolutionsthatζthe0tocorresp(5.5.99),ondingwhichsubsetsdifferN˜f1,romN˜2ζof0Ngivareen
finiteand˜νk=νkonlyforfinitelymanyk∈Z,seemtoinducetrinoidswithfinitelymany“bubbles”(in
thesenseof[27])butstillwithproperlyembeddedannularends.Thisseemstobeperfectlyconsistent
awithsolutionthefollΨˆo=wingTΨtoobservtheation:differenWhiletialζ0equationgivenindΨ=(5.5.108)Ψη˜(cf.inducessectiona3.9),dressingζ˜inducesmatrixaTdrewhicssinghdetermatrixmineT˜s
0˜˜whicmatriceshM˜jdetermines,j=0,1,another∞,whicsolutionhdifferΨ=fromTΨthetotheunitarysamemonodifferendromtialymatricesequationMˆwithj,jun=0,itary1,∞monoofdΨˆromonlyy
byconjugationwithamatrixS,whichisgivenasafiniteproductofconjugationmatriceswhich“bring
in”thefactors(1)gj,1(λ)for(2)j∈N˜1andgj,2(λ)forj∈N˜2,respectively,andreplacefork∈Zwithν˜k=νk
thefactorsp(λ)byp(λ)(andviceversa).ThesingleconjugationmatricescontributingtoSseem
toplaythekroleofsimplkefactordressingmatricesinthesenseof[26].Consequently,thesolutionsΨˆ
andΨ˜=SΨˆwouldberelatedbyafiniteproductofsimplefactors,which,by[26],wouldimplythatΨ˜
producesatrinoidwithproperlyembeddedannularends.
infinitelyAccordingmanytoptheseositions”,wouldconsiderations,producesolutionstrinoidstowhich(5.5.99),donotwhicphossessdifferpropfromerlyζ0emgivbeneddedinann(5.5.108)ularends“on
(butarestillrotationallysymmetricwithrespecttothetrinoidnormal).
103

6Rotationalsymmetrywithrespecttoatrinoidaxis
Definition6.1Thesecondpossibletrinoidsymmetrytypewearegoingtostudyencompassesthesymmetrieswith
resprespectectivtoely.theRecallrotationsthat,bRy0,Rtheorem1and4.31,R∞Rb0y,Rthe1andangleR∞πpreservaroundeoritheentrinoidtationonaxesR3A0and,Ap1ermanduteAthe∞,
trinoidendsaccordingtothepermutations(1∞),(0∞)and(01)oftheset{0,1,∞},respectively.
Definition˜6.1.LetM=C\˜{0,1}3andφ:M→R3beatrinoidwithproperlyembeddedannularends.
LetM=Handψ=φ◦π:M→RtheconformalCMC-immersionassociatedwithφviatheuniversal
coverotationringπRl:bM˜yth→eManglegivπeninaround(3.2.2).thetrThen,inoidifaxisφ(or,Al,equivalently,ψ)issymmetricwithrespecttothe
Rl(φ(M))=φ(M),Rl(ψ(M˜))=ψ(M˜),(6.1.1)
φ(orψ)iscalledrotationallysymmetricwithrespecttothetrinoidaxisAl.
onIntheanalogymonotodromythepmatricesreviousassosection,ciatedwewithtranslatethetheextendedfrsymmetryameFpropofψert.y(6.1.1)intofurtherconstraints
6.2Implicationsofrotationalsymmetrywithrespecttoatrinoidaxis
Asadirectconsequenceofdefinition6.1,westatethefollowinglemma:
Lemma6.2.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannu-
splarenondingdsproDelaunayducedfrommatricaestrinoidwithpoteigenvaluesentialη±asµi0n,±theµ1or,em±µ∞,3.14.respeDenotectively,bywherD0,e,D1for,jD∞∈{the0,1c,orr∞}e-,
µj=XjXj=41+wj(λ−λ−1)2andwj=sjtjasinsection3.5.Moreover,denotebyB0,B1and
B∞thetrinoidendsandbyA0,A1andA∞thetrinoidaxes.Then,thefollowingholds:
1.IfφisrotationallysymmetricwithrespecttothetrinoidaxisA0,wehave
µ1=µ∞.(6.2.1)
2.IfφisrotationallysymmetricwithrespecttothetrinoidaxisA1,wehave
µ0=µ∞.(6.2.2)
3.IfφisrotationallysymmetricwithrespecttothetrinoidaxisA∞,wehave
µ0=µ1.(6.2.3)
toProtheof.WetrinoidcarryaxisoutA.theByprooftheoremforthe4.31,firstthecase,correspi.e.ondingsupposeφsymmetryisRrotationallypreservessymmetricthetrinoidwithendrespBect,
000whilesurfacesitassorotatesciatedthewithtrinoidtheendsendsBat1B1andandB∞B∞intoareeachrotatedother.intoThiseachmeansotherasthatwell.theHence,asymptotictheseDelaunaDelaunayy
surfacesonlydifferbyarigidmotiononR3.Inparticular,thisimpliesthatthecorrespondingDelaunay
matricesD1andD∞possessthesameeigenvalues,i.e.µ1=µ∞.
Theothertwocasesareprovedanalogously.
assoLetciatedMc=onC\formal{0,1},φ:CMC-immersionM→R3onbeM˜a=Htrinoid,ψ=withφ◦πprop:M˜erly→emR3b,edwherededπannulardenotesendstheandunivψersalthe
coveringM˜→Mgivenin(3.2.2).Supposeφ(or,equivalently,ψ)isrotationallysymmetricwithrespect
toonRthe3,wetrinoidobtainaxisbyAl,theoremanddenote4.9athepairofcorrespondingbiholomorphicsymmetrymappings,byRγlR.l:SinceM→RlMandpreservγ˜esRl:orienM˜→tationM˜
satisfyingRl◦φ=φ◦γRl,(6.2.4)
Rl◦ψ=ψ◦γ˜Rl,(6.2.5)
π◦γ˜Rl=γRl◦π.(6.2.6)
104

ThemappingsγRl,l=0,1,∞,areuniquelydeterminedandexplicitlygivenbylemma4.21:
zγR0(z)=,(6.2.7)
1z−1γR1(z)=,(6.2.8)
zγR∞(z)=1−z.(6.2.9)
Themappingsγ˜Rl,l=0,1,∞,areuniquelydetermineduptocompositionfromtheleftwithelements
oftheautomorphismgroupAut(M˜/M)ofπ.Thefollowinglemmaexplicitlystatesvalidchoicesforγ˜Rl,
l=0,1,∞:
Lemma6.3.LetM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversalcoveringasgivenin(3.2.2).
LetγRl:M→M,l=0,1,∞,begivenby(6.2.7),(6.2.8)and(6.2.9),respectively.Then,thefollowing
holds:1.Themappingγ˜R0:M˜→M˜,
2z−−γ˜R0(z)=,(6.2.10)
1+zsatisfiesandbiholomorphicisπ◦γ˜R0=γR0◦π,(6.2.11)
R0◦ψ=ψ◦γ˜R0.(6.2.12)
2.Themappingγ˜R1:M˜→M˜,
1z−−γ˜R1(z)=2z+1,(6.2.13)
satisfiesandbiholomorphicisπ◦γ˜R1=γR1◦π,(6.2.14)
R1◦ψ=ψ◦γ˜R1.(6.2.15)
3.Themappingγ˜R∞:M˜→M˜,
1γ˜R∞(z)=−,(6.2.16)
zsatisfiesandbiholomorphicisπ◦γ˜R∞=γR∞◦π,(6.2.17)
R∞◦ψ=ψ◦γ˜R∞.(6.2.18)
Proof.Directcomputationsshowthatγ˜Rl,l=0,1,∞definebiholomorphicmappingsM˜→M˜.More-
over,byapplyingtherelations(3.2.10)and(3.2.11)oflemma3.4,weobtainforallz∈M˜
−z−211
z+1z+1π−1
π◦γ˜R0(z)=π=π−1−=
+1z11π(z)
==1==γR0◦π(z),(6.2.19)
1−π(z+1)1−π(z)π(z)−1
−z−1z1
2z+12z+1πz
π◦γ˜R1(z)=π=π−1+=
+1z21111zzz=1−π−2z+1=1−π−2−1=1−π−1=π(z)=γR1◦π(z),(6.2.20)
1π◦γ˜R∞(z)=π−=1−π(z)=γR∞◦π(z),(6.2.21)
zi.e.π◦γ˜Rl=γRl◦πforl=0,1,∞.Consequently,
Rl◦ψ=Rl◦φ◦π=φ◦γRl◦π=φ◦π◦γ˜Rl=ψ◦γ˜Rl,(6.2.22)
i.e.Rl◦ψ=ψ◦γ˜Rlforl=0,1,∞.
105

trinoidBytheabsymmetriesovelemmRl,a,l=we0,1ha,v∞e,expresplicitlyectively,indeterminedthesensemappingsoftheoremγ˜Rl,l4.9.=0,Th1,us,∞,wecorrespcanapplyondingtotheoremthe
obtainto4.17Theorem6.4.LetM=C\{0,1},φ:M→R3beatrinoidwithproperlyembeddedannularends
theandψtheuniversalassocciateoveringdcM˜onformal→MasdefineCMC-immersiondin(3.2.2)on.M˜Let=φH,beψr=φotational◦π:lyM˜→symmetricR3,wherwitherπespectdenotesto
M˜the→Mtrinoid˜assoaxisciateAld.withDenoteRlastheincorrtheesporemonding4.9andsymmetryexplicitlybyRldefineanddbyinγ˜Rllemmathe6.3.biholomorphicThen,theextendemappingd
frameF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜Rlas
F(γ˜Rl(z),λ)=MRl(λ)F(z,λ)kRl,γR˜l(z),(6.2.23)
whereMRl(λ)denotesanelementofΛSU(2)σ,whichisindependentofz,and
0z¯+1
kR0,γR˜0(z)=0z+1z¯+1inthecasel=0,(6.2.24)
+1z2z¯+1
kR1,γR˜1(z)=2z+120z¯+1inthecasel=1,(6.2.25)
0z¯2z+1
0kR∞,γR˜∞(z)=0zz¯inthecasel=∞.(6.2.26)
zProof.AsRlpreservesorientation,weapplythefirstpartoftheorem4.17toobtain
F(γ˜Rl(z),λ)=MRl(λ)F(z,λ)kRl,γ˜Rl(z),(6.2.27)
˜Mwhereγ˜RF:denotesM→anΛSU(2)elementσofdenotesΛSU(2)σthe,whichextendedisfindeprameendentcorrespofzon.dingMoreotovψer,bkyRl,γR˜theoremisgiv4.5enbandyMRequationl:=
ll1(4.4.117)fromlemma4.18.Recallingfromlemma6.3thatγ˜R0(z)=−zz+1−2,γ˜R1(z)=−2zz+1−1andγ˜R∞(z)=
−z,wecompute
∂zγ˜R0(z)=(z+11)2,(6.2.28)
∂zγ˜R1(z)=(2z+11)2,(6.2.29)
∂zγ˜R∞(z)=z12,(6.2.30)
sieimplhwhic2|∂∂zγ˜γ˜R0((zz))|=|(zz++11)|2=zz¯++11,(6.2.31)
z0R∂zγ˜R1(z)=|2z+1|22=2z¯+1,(6.2.32)
|∂zγ˜R1(z)|(2z2+1)2z+1
∂zγ˜R∞(z)=|z2|=z¯.(6.2.33)
|∂zγ˜R∞(z)|zz
Hence,weobtainfrom(4.4.117)theclaimedexplicitformsforkRl,γR˜l,l=0,1,∞.Thisfinishesthe
of.pro6.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,
whicharerotationallysymmetricwithrespecttoatrinoidaxis
Wenowstudythe(unitary)monodromymatricesMˆ0,Mˆ1,Mˆ∞associatedwithatrinoidwithproperly
embeddedannularendswithrotationalsymmetrywithrespecttothetrinoidaxisAl.Ourconsiderations
106

arebasedontherelationsbetweenthebiholomorphicmappingsγ˜RlassociatedwiththesymmetriesRl
andthecoveringtransformationsγ˜jonM˜generatingthemonodromymatricesMˆj.Recallthelatter
3.3:sectionfromonesz−3z−2
γ˜0(z)=,γ˜1(z)=z+2,γ˜∞(z)=.(6.3.1)
−2z+12z+1
Thecorrespondinginversefunctionsaregivenby
2+zzγ˜0−1(z)=,γ˜1−1(z)=z−2,γ˜∞−1(z)=.(6.3.2)
2z+1−2z−3
Therelationsmentionedabovearestatedinthefollowinglemma.
Lemma6.5.LetM˜=Handγ˜0,γ˜1,γ˜∞:M˜→M˜begivenasabove.
1.Forγ˜R0:M˜→M˜,γ˜R0(z)=−zz+1−2,thefollowingidentitieshold:
11−−γ˜R0◦γ˜0=γ˜1◦˜γ∞◦γ˜R0,γ˜R0◦γ˜1=γ˜∞◦γ˜R0,γ˜R0◦γ˜∞=γ˜1◦γ˜R0.(6.3.3)
2.Forγ˜R1:M˜→M˜,γ˜R1(z)=2−zz+1−1,thefollowingidentitieshold:
1−γ˜R1◦γ˜0=γ˜∞◦γ˜R1,γ˜R1◦γ˜1=γ˜∞−1◦γ˜0◦γ˜R1,γ˜R1◦γ˜∞=γ˜0◦γ˜R1.(6.3.4)
3.Forγ˜R∞:M˜→M˜,γ˜R∞(z)=−z1,thefollowingidentitieshold:
11−−γ˜R∞◦γ˜0=γ˜1◦γ˜R∞,γ˜R∞◦γ˜1=γ˜0◦γ˜R∞,γ˜R∞◦γ˜∞=γ˜0◦γ˜1◦γ˜R∞.(6.3.5)
Proof.Forz∈M˜wehavebydirectcomputation
−(z+2)−2−z−4−3−zz+1−2−2
γ˜R0◦γ˜1(z)=(z+2)+1=z+3=−z−2=γ˜∞◦γ˜R0(z),(6.3.6)
1+2+1z−−3z−2−2z−z−2
+1z2γ˜R0◦γ˜∞(z)=−3z−2=z+1=z+1+2=γ˜1◦γ˜R0(z),(6.3.7)
1++1z2−−2zz+1−1−32−zz−+11−2
γ˜R1◦γ˜0(z)=z=z−1=−z−1=γ˜∞◦γ˜R1(z),(6.3.8)
2−2z+1+122z+1+1
2z+12z+1
−−3z−2−1−z−1−z−1
γ˜R1◦γ˜∞(z)=−3z−2=4z+3=−z−1=γ˜0◦γ˜R1(z),(6.3.9)
22z+1+1−22z+1+1
12z−11
γ˜R∞◦γ˜0(z)=−z==−+2=γ˜1◦γ˜R∞(z),(6.3.10)
−2z+1zz
11−γ˜R∞◦γ˜1(z)=−=1z=γ˜0◦γ˜R∞(z).(6.3.11)
z+22z+1
Theremainingidentitiesfollowfromtheonesabovebyuseofγ˜0◦γ˜1◦γ˜∞=idonM˜:
γ˜R0◦γ˜0(z)=γ˜R0◦γ˜∞−1◦γ˜1−1(z)=γ˜1−1◦γ˜R0◦γ˜1−1(z)=γ˜1−1◦γ˜∞−1◦γ˜R0(z),(6.3.12)
γ˜R1◦γ˜1(z)=γ˜R1◦γ˜0−1◦γ˜∞−1(z)=γ˜∞−1◦γ˜R1◦γ˜∞−1(z)=γ˜∞−1◦γ˜0−1◦γ˜R1(z),(6.3.13)
andγ˜R∞◦γ˜∞(z)=γ˜R∞◦γ˜1−1◦γ˜0−1(z)=γ˜0−1◦γ˜R∞◦γ˜0−1(z)=γ˜0−1◦γ˜1−1◦γ˜R∞(z).(6.3.14)

Inviewofthis,weareabletoprovethefollowingtheorem:

107

(6.3.3)(6.3.4)(6.3.5)(6.3.6)(6.3.7)(6.3.8)(6.3.9)(6.3.10)(6.3.11)(6.3.12)(6.3.13)

(6.3.14)

Theorem6.6.LetM=C\{0,1},φ:M→R3beatrinoidwithproperlyembeddedannularendsandψ
theassociatedconformalCMC-immersiononM˜=H,ψ=φ◦π:M˜→R3,whereπdenotestheuniversal
˜cAl.overingDenoteM→theMcorrasespdefineondingdin(3.2.2)symmetry.LbyetφRlb.erFotationalurthermorlye,letsymmetricF:M˜with→respΛSU(2)ectσtobethethetrinoidextendeaxisd
frameassociatedwithψbytheorem4.5.DenotebyMˆ0,Mˆ1,Mˆ∞∈ΛSU(2,C)σtheunitarymonodromy
matrices10pq
Mˆj=−cos(2πµj)01+isin(2πµj)qjj−jpj(6.3.15)
associatedwithFasin(4.5.13)by
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),j=0,1,∞,(6.3.16)
whereαj∈{±1}andγ˜jdenotethecoveringtransformationsonM˜fromsection3.3.Finally,letγ˜Rl,be
thebiholomorphicmappingM˜→M˜associatedwithRlasintheorem4.9andexplicitlygiveninlemma
and6.3,MRl(λ):=−abRRlabRRl(6.3.17)
llthecorrespondingmonodromymatrixofFsatisfying(6.2.23).
1.Inthecasel=0,themonodromymatricessatisfy
MR0(λ)Mˆ1(λ)=Mˆ∞(λ)MR0(λ),(6.3.18)
MR0(λ)Mˆ∞(λ)=Mˆ1(λ)MR0(λ),(6.3.19)
MR0(λ)Mˆ0(λ)=(Mˆ1(λ))−1(Mˆ∞(λ))−1MR0(λ).(6.3.20)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(6.3.18)to(6.3.20)areequivalent
to

aR0p1+bR0q1=aR0p∞−bR0q∞,(6.3.21)
−bR0p1+aR0q1=bR0p∞+aR0q∞,(6.3.22)
aR0p∞+bR0q∞=aR0p1−bR0q1,(6.3.23)
−bR0p∞+aR0q∞=bR0p1+aR0q1.(6.3.24)
2.Inthecasel=1,themonodromymatricessatisfy
MR1(λ)Mˆ∞(λ)=Mˆ0(λ)MR1(λ),(6.3.25)
MR1(λ)Mˆ0(λ)=Mˆ∞(λ)MR1(λ),(6.3.26)
MR1(λ)Mˆ1(λ)=(Mˆ∞(λ))−1(Mˆ0(λ))−1MR1(λ).(6.3.27)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(6.3.25)to(6.3.27)areequivalent
to

aR1p∞+bR1q∞=aR1p0−bR1q0,
−bR1p∞+aR1q∞=bR1p0+aR1q0,
aR1p0+bR1q0=aR1p∞−bR1q∞,
−bR1p0+aR1q0=bR1p∞+aR1q∞.
3.Inthecasel=∞,themonodromymatricessatisfy
MR∞(λ)Mˆ0(λ)=Mˆ1(λ)MR∞(λ),
MR∞(λ)Mˆ1(λ)=Mˆ0(λ)MR∞(λ),
MR∞(λ)Mˆ∞(λ)=(Mˆ0(λ))−1(Mˆ1(λ))−1MR∞(λ).

108

(6.3.28)(6.3.29)(6.3.30)(6.3.31)(6.3.32)(6.3.33)(6.3.34)

IntermsofthefunctionspjandqjoccurringinMˆj,equations(6.3.32)to(6.3.34)areequivalent
toaR∞p0+bR∞q0=aR∞p1−bR∞q1,(6.3.35)
−bR∞p0+aR∞q0=bR∞p1+aR∞q1,(6.3.36)
aR∞p1+bR∞q1=aR∞p0−bR∞q0,(6.3.37)
−bR∞p1+aR∞q1=bR∞p0+aR∞q0.(6.3.38)
Proof.Westartwiththecasel=0,i.e.withatrinoidwithproperlyembeddedannularends,thatis
symmetricwithrespecttotherotationR0bytheangleπaroundthetrinoidaxisA0.Combining(6.2.23)
fromtheorem6.4,equation(6.3.16)andtheidentities(6.3.3)fromtheabovelemma,weobtain
MR0(λ)α∞Mˆ∞(λ)F(z,λ)k∞(z)kR0,γ˜R0(γ˜∞(z))=MR0(λ)F(γ˜∞(z),λ)kR0,γ˜R0(γ˜∞(z))
=F(γ˜R0◦γ˜∞(z),λ)=F(γ˜1◦γ˜R0(z),λ)
=α1Mˆ1(λ)F(γ˜R0(z),λ)k1(γ˜R0(z))=α1Mˆ1(λ)MR0(λ)F(z,λ)kR0,γ˜R0(z)k1(γ˜R0(z))(6.3.39)
andMR0(λ)α1Mˆ1(λ)F(z,λ)k1(z)kR0,γ˜R0(γ˜1(z))=MR0(λ)F(γ˜1(z),λ)kR0,γ˜R0(γ˜1(z))
=F(γ˜R0◦γ˜1(z),λ)=F(γ˜∞◦γ˜R0(z),λ)
=α∞Mˆ∞(λ)F(γ˜R0(z),λ)k∞(γ˜R0(z))=α∞Mˆ∞(λ)MR0(λ)F(z,λ)kR0,γ˜R0(z)k∞(γ˜R0(z)).(6.3.40)
Computing(duetotheoccurringcomplexrootsuptosign)
zz¯+3+30
0k1(z)kR0,γ˜R0(γ˜1(z))=z¯+3
+3z1+2−z¯−2
+1z+1zz¯+101+2−z¯z+1−20
0z+101+2−z¯z+1−2
=±z¯+1−z¯−2=±kR0,γ˜R0(z)k∞(γ˜R0(z))(6.3.41)
1+2+1zand−3z¯−2+1
1+2¯z0−23z¯z+1−2+10
z¯1+2+101+2z0−23z¯z+1−2
k∞(z)kR0,γ˜R0(γ˜∞(z))=1+2z2z+1−3z¯−2
+1+1z20z¯+1
0¯zz+1+1
=±z+1=±kR0,γ˜R0(z)k1(γ˜R0(z)),(6.3.42)
thatconcludeewMR0(λ)Mˆ1(λ)=±α1α∞Mˆ∞(λ)MR0(λ),(6.3.43)
MR0(λ)Mˆ∞(λ)=±α∞α1Mˆ1(λ)MR0(λ).(6.3.44)
Thiscanbereformulatedas
Mˆ1(λ)=±α1α∞MR0(λ)−1Mˆ∞(λ)MR0(λ),(6.3.45)
Mˆ∞(λ)=±α∞α1MR0(λ)−1Mˆ1(λ)MR0(λ).(6.3.46)
Comparingtheupperleftentriesaswellasthelowerrightentriesofbothsidesineachoftheseequations,
obtainew−cos(2πµ1)−isin(2πµ1)p1
=±α1α∞−cos(2πµ∞)−isin(2πµ∞)(aR0aR0p∞−aR0bR0q∞−aR0bR0q∞−bR0bR0p∞),(6.3.47)
109

−cos(2πµ1)+isin(2πµ1)p1
=±α1α∞−cos(2πµ∞)+isin(2πµ∞)(aR0aR0p∞−aR0bR0q∞−aR0bR0q∞−bR0bR0p∞)(6.3.48)
and−cos(2πµ∞)−isin(2πµ∞)p∞
=±α∞α1−cos(2πµ1)−isin(2πµ1)(aR0aR0p1−aR0bR0q1−aR0bR0q1−bR0bR0p1),(6.3.49)
−cos(2πµ∞)+isin(2πµ∞)p∞
=±α∞α1−cos(2πµ1)+isin(2πµ1)(aR0aR0p1−aR0bR0q1−aR0bR0q1−bR0bR0p1),(6.3.50)
respectively.Bysummingupthefirsttwoequations(andrecallingthatµ∞=µ1),weconcludethatthe
factor±α1α∞necessarilyequals+1.Analogously,bysumminguptheothertwoequations,wededuce
aswell±α∞α1=+1.Therefore,
MR0(λ)Mˆ1(λ)=Mˆ∞(λ)MR0(λ),(6.3.51)
MR0(λ)Mˆ∞(λ)=Mˆ1(λ)MR0(λ).(6.3.52)
claimed.asWhatremainstoproveis(6.3.20).Butthisequationfollowsinviewof(3.9.32)directlyfromequations
(6.3.18):and(6.3.19)MR0(λ)Mˆ0(λ)=MR0(λ)(Mˆ∞(λ))−1(Mˆ1(λ))−1
=(Mˆ1(λ))−1MR0(λ)(Mˆ1(λ))−1=(Mˆ1(λ))−1(Mˆ∞(λ))−1MR0(λ).(6.3.53)
Asequation(6.3.20)isimpliedbyequations(6.3.18)and(6.3.19),thesethreeequationsareequivalent
tothescalarreformulationsoftheequations(6.3.18)and(6.3.19),whichread
−cos(2πµ1)aR0−isin(2πµ1)(aR0p1+bR0q1)=−cos(2πµ∞)aR0−isin(2πµ∞)(aR0p∞−bR0q∞),
(6.3.54)−cos(2πµ1)bR0−isin(2πµ1)(aR0q1−bR0p1)=−cos(2πµ∞)bR0−isin(2πµ∞)(bR0p∞+aR0q∞),
(6.3.55)and−cos(2πµ∞)aR0−isin(2πµ∞)(aR0p∞+bR0q∞)=−cos(2πµ1)aR0−isin(2πµ1)(aR0p1−bR0q1),
(6.3.56)−cos(2πµ∞)bR0−isin(2πµ∞)(aR0q∞−bR0p∞)=−cos(2πµ1)bR0−isin(2πµ1)(bR0p1+aR0q1),
(6.3.57)respectively.Astraightforwardsimplificationoftheseequationsyieldstheclaimedonesandfinishesthe
proofofthefirstcase,l=0.
Thecasesl=1andl=∞areprovedcompletelyanalogouslybysimplyshiftinglabels.Theonly
remainingidentities,whichstillneedtobechecked,are
k∞(z)kR1,γ˜R1(γ˜∞(z))=±kR1,γ˜R1(z)k0(γ˜R1(z)),(6.3.58)
k0(z)kR1,γ˜R1(γ˜0(z))=±kR1,γ˜R1(z)k∞(γ˜R1(z)),(6.3.59)
k0(z)kR∞,γ˜R∞(γ˜0(z))=±kR∞,γ˜R∞(z)k1(γ˜R∞(z)),(6.3.60)
k1(z)kR∞,γ˜R∞(γ˜1(z))=±kR∞,γ˜R∞(z)k0(γ˜R∞(z)).(6.3.61)
Thisisdonebydirectcomputations:
2−2z3¯z¯+1−2+1
1+21+2zz¯02−2z3z+1−2+1044zz¯+3+30
01+2z0−23z¯z+1−204z+3
k∞(z)kR1,γ˜R1(γ˜∞(z))=1+2z¯2−3z¯−2+1=±4z¯+3
+12+1z21−2z¯
2z¯+101−2z0
022zz¯+1+10−2zz¯−+11
=±2z+11−2−z¯−1=±kR1,γ˜R1(z)k0(γ˜R1(z)),(6.3.62)
1−22z+1
110

1−2z¯2−2¯z¯zz+1+10
011−−22zz¯02−2zz¯z¯+1+101
k0(z)kR1,γ˜R1(γ˜0(z))=1−2z02−2z+1+1=±10
2−2z+1+1
1+2−z¯−1
2z¯+102−z¯z−+110
022zz¯+1+101+2−2zz¯−+11
=±2z+11+22z+1−z¯−1=±kR1,γ˜R1(z)k∞(γ˜R1(z)),(6.3.63)
1+2+1z21−2z¯−2zz¯z¯+10
k0(z)kR∞,γ˜R∞(γ˜0(z))=1−2z0−2z+1
011−−22zz¯0−2zz¯z¯+1
z¯0−2z+1
z0=±z¯=±kR∞,γ˜R∞(z)k1(γ˜R∞(z)),(6.3.64)
z0z¯+2
0k1(z)kR∞,γ˜R∞(γ˜1(z))=z+2z¯+2
+2z1+2
z¯01+2z¯0
=±zz¯z2=±kR∞,γ˜R∞(z)k0(γ˜R∞(z)).(6.3.65)
0z01+1+z2z¯
6.4Normalizedtrinoidswithproperlyembeddedannularends,whichare
rotationallysymmetricwithrespecttoatrinoidaxis
Letl∈{0,1,∞}andφ:M→R3beatrinoidwithproperlyembeddedannularends,whichisrotationally
symmetricwithrespecttothetrinoidaxisAl.Moreover,letψ=φ◦πbetheassociatedCMC-immersion
M˜→R3.DenotebyRλthecorrespondingsymmetryofφ(andψ),i.e.therotationbytheangleπ
aroundthetrinoidaxisAl.
Wereviewtheresultsofsection6.3inthespecialcasethattheextendedframeF:M˜→ΛSU(2)σ
associatedwithψasinsection4.2is“normalized”atz∗,l∈M˜,whichwechoosedependingonlas
ws:folloz∗,0=−1+i∈M˜,(6.4.1)
z∗,1=−1+i∈M˜,(6.4.2)
2z∗,∞=i∈M˜.(6.4.3)
The“normalization”ofFisrealizedinformofthepresumptionthat
F(z∗,l,λ)=I(6.4.4)
forallλ∈S1.Moreprecisely(cf.section4.2),thenormalizationF(z∗,l,λ)=IofFisaconsequenceof
normalizingthe(conformal)CMC-immersionψ,suchthat
1ψ(z∗,l)=2He3,U(z∗,l)=G(1),(6.4.5)
whereU∈SO(3)representsthenaturalorthonormalframecorrespondingtoψ,andG(1)isgivenin
(4.2.5).Recallfromsection4.2,thatthisnormalizationofψcorrespondstorotatingandshiftingthe
(imageofthe)trinoidinR3,suchthattheconditions(6.4.5)aremet.Itturnsout(cf.corollary6.8),that
111

thecsymmetrichoiceofwithz∗,lrespasectabotovethe(foratrinoidtrinoidaxisφAl)withcorrespproperlyondsemtobeddedarranginganntheular(imageends,ofwhicthe)histrinoidrotationallyinR3,
suchthattherotationaxis3ofRlisthez-axisinR3.
respAecttotrinoidtheφ:trinoidM→axisRAwithand,propinerlyaddition,embisedded“wellannpularositioned”ends,inwhicR3hinistheserotationallynsethatsymmetheassotricciatedwith
l˜trinoidconformalwithpropCMC-immerlyemersionbψedded:Mann→ularMends,meetsthewhichisrnormalizationotationallyconditionssymmetric(6.4.5),withisrespcalledecttoathenormalizetrinoidd
.AaxisWelnowformulateamoreexplicitversionoftheorem6.4:
ψtheTheoremasso6.7.ciatedLcetMonformal=C\{0,1},CMC-immersionφ:M→onR3M˜be=aH,trinoidψ=withφ◦prπ:opM˜erly→embRe3,ddedwhereannularπdenotesendsandthe
˜respuniversalecttoctheoveringtrinoidM→axisMAlas.Mordefineedover,inlet,(3.2.2)acc.orLetdingl∈to{l0,,z1∗,,l∞}beandgivenφinber(6.4.1)otational,ly(6.4.2)orsymmetric(6.4.3),with
z∗,0=−1+i,z∗,1=−12+i,z∗,∞=i,(6.4.6)
andassumethatψhasbeennormalizedatz∗,l,suchthatψ(z∗,l)=21He3andF(z∗,l,λ)=I,where
˜cForr:espM→ondingΛSU(2)syσmmetrydenotesofφtheandbyextendeγ˜RldfrtheamecorrbiholomorphicespondingtomappingψbyM˜the→orM˜emasso4.5.ciatedDenotwithebyRlRlasthein
theorem4.9and,accordingtol,explicitlygivenin(6.2.10),(6.2.13)or(6.2.16):
−z−2−z−11
γ˜R0(z)=z+1,γ˜R1(z)=2z+1,γ˜R∞(z)=−z.(6.4.7)
Then,theextendedframeFtransformsunderγ˜Rlas
F(γ˜Rl(z),λ)=MRl(λ)F(z,λ)kRl,γR˜l(z)(6.4.8)
wherekRl,γR˜l(z)is,accordingtol,givenin(6.2.24),(6.2.25)or(6.2.26)and
MRl(λ)=0i−0i.(6.4.9)
Inparticular,MRlisactuallyindependentofλ.
Proof.Inviewoftheorem6.4,weonlyhavetoprovetheequation(6.4.9).Tothisend,wecompute
γ˜R0(z∗,0)=−((−−11++i)i)+−12=z∗,0,(6.4.10)
−−1+i−1
γ˜R1(z∗,1)=2−21+2i+1=z∗,1,(6.4.11)
γ˜R∞(z∗,∞)=−i1=z∗,∞,(6.4.12)
whichshowsthatwehaveforalll∈{0,1,∞}
γ˜Rl(z∗,l)=z∗,l.(6.4.13)
Furthermore,F(z∗,l,λ)=I.Thus,evaluatingequation(6.4.8)atz=z∗,lyields
I=F(z∗,l,λ)=F(γ˜Rl(z∗,l),λ)=MRl(λ)F(z∗,l,λ)kRl,γR˜l(z∗,l),(6.4.14)
i.e.MRl(λ)=kRl,γR˜l(z∗,l)−1.(6.4.15)
Inviewofremark4.14(forourdefinitionofthecomplexsquareroot)andequations(6.2.24),(6.2.25)
112

and(6.2.26),wehave

(−1+i)+10−i0
kR0,γR˜l(z∗,0)=(−1+i)+1(−1+i)+1=0i,(6.4.16)
0(−1+i)+1
22−−121+−ii+1+10−i0
kR1,γR˜l(z∗,1)=22−12−i+1=0i,(6.4.17)
02−21+i+1
−i0
kR∞,γR˜l(z∗,∞)=i−¯i=−0ii0,(6.4.18)
0ii.e.kRl,γR˜l(z∗,l)=−0ii0(6.4.19)
foralll∈{0,1,∞}.Altogether,(6.4.9)follows.
Corollary6.8.Weretainthenotationandtheassumptions3oftheorem6.7.Therotationaxisofthe
symmetryRlofthenormalizedtrinoidφisthez-axisinR.
Proof.Applying(thefirstpartof)theorem4.17,weknowthatthemonodromymatrixMRl(λ)explicitly
givenintheorem6.7satisfiesatλ=1
MRl(1)=±ARl,(6.4.20)
whereARl∈SU(2)denotestheconjugationmatrixrealizingtheorthogonalpartARlofthesymmetry
Rlinthesu(2)-model.Inviewofequation(6.4.9),thisyields
ARl=±0i−0i.(6.4.21)
RecallingthatARlandARlarelinkedviatheLieAlgebraisomorphismJ:R3→su(2)definedin(3.4.3)
i.e.(3.4.7),inas(J◦ARl◦J−1)(X)=ARlXAR−l1forallX∈su(2),(6.4.22)
weobtainbyadirectcomputationthat
−100
100ARl=0−10.(6.4.23)
Thus,ARldefinestherotation(inR3)bytheanglesπaroundthez-axisinR3,Re3.Consequen3tly,the
symmetryRlofthenormalizedtrinoidφisarotationbytheangleπaroundanaxisinR,whichis
paralleltothez-axis.Asthepointψ(z∗,l)∈R3(withz∗,lgivenin(6.4.1),(6.4.2)or(6.4.3)accordingto
satisfies)lRl(ψ(z∗,l))=ψ(γ˜Rl(z∗,l))=ψ(z∗,l),(6.4.24)
itliesontherotationaxisofRl.Since3byassumptionwehaveψ(z∗,l)=21He3,weinferthattherotation
axisofRlisactuallythez-axisinR.
Applyingthetheorems6.6and6.7,weobtainthefollowingresult:
3theTheoremassociated6.9.cLetonformalM=CMC\{0,1}C-immersion,φ:Mon→MR˜=beHa,ψ=trinoidφ◦πwith:M˜pr→opRerly3,embwhereeddeπddenotesannulartheendsuniversalandψ
coveringM˜→Masdefinedin(3.2.2).Letl∈{0,1,∞}andφberotationallysymmetricwithrespectto
thetrinoidaxisAl.Moreover,let,accordingtol,z∗,lbegivenin(6.4.1),(6.4.2)or(6.4.3),andassume
thatψhasbeennormalizedatz∗,l,suchthatψ(z∗,l)=21He3andF(z∗,l,λ)=I,whereF:M˜→ΛSU(2)σ
denotestheextendedframecorrespondingtoψbytheorem4.5.

113

1.Inthecasel=0,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(6.3.16)areoftheform
Mˆ0=−cos(2πµ0)I−2αicos(πµ0)cos(2πµ1)−iζ1,(6.4.25)
iζ1−cos(2πµ1)
Mˆ1=−cos(2πµ1)I−iαcos(πµ1)ζ1,(6.4.26)
ζ1−αcos(πµ1)
αcos(πµ1)−ζ1
Mˆ∞=−cos(2πµ1)I−i−ζ1−αcos(πµ1),(6.4.27)
whereα∈{±1}andζ1isanoddfunctioninλandasolutionto
ζ1ζ1=sin2(2πµ1)−cos2(πµ0).(6.4.28)
2.Inthecasel=1,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(6.3.16)areoftheform
Mˆ0=−cos(2πµ∞)I−iαcos(πµ∞)−ζ∞,(6.4.29)
−ζ∞−αcos(πµ∞)
Mˆ1=−cos(2πµ1)I−2αicos(πµ1)cos(2πµ∞)−iζ∞,(6.4.30)
iζ∞−cos(2πµ∞)
Mˆ∞=−cos(2πµ∞)I−iαcos(πµ∞)ζ∞,(6.4.31)
ζ∞−αcos(πµ∞)
whereα∈{±1}andζ∞isanoddfunctioninλandasolutionto
ζ∞ζ∞=sin2(2πµ∞)−cos2(πµ1).(6.4.32)
3.Inthecasel=∞,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(6.3.16)areoftheform
Mˆ0=−cos(2πµ0)I−iαcos(πµ0)ζ0,(6.4.33)
ζ0−αcos(πµ0)
αcos(πµ0)−ζ0
Mˆ1=−cos(2πµ0)I−i−ζ0−αcos(πµ0),(6.4.34)
Mˆ∞=−cos(2πµ∞)I−2αicos(πµ∞)cos(2πµ0)−iζ0,(6.4.35)
iζ0−cos(2πµ0)
whereα∈{±1}andζ0isanoddfunctioninλandasolutionto
ζ0ζ0=sin2(2πµ0)−cos2(πµ∞).(6.4.36)
Proof.Weprovethecasel=∞.Asbefore,wedenotethesymmetryofφbyRl=R∞andby˜γR∞the
biholomorphicmappingM˜→M˜associatedwithR∞asintheorem4.9andexplicitlygiveninlemma
6.3.Moreover,letMR∞(λ)bethecorrespondingmonodromymatrixofFasintroducedinequation
(6.2.23).Inviewoftheorem6.7,wehave
MR∞(λ)=aR∞bR∞(6.4.37)
−bR∞aR∞
whereaR∞=iandbR∞=0.(6.4.38)
Moreover,bytheorem6.6,weobtainthefollowingrelationsbetweentheunitarymonodromymatrices
MˆjoftheextendedframeF:
MR∞(λ)Mˆ0(λ)=Mˆ1(λ)MR∞(λ),(6.4.39)
MR∞(λ)Mˆ1(λ)=Mˆ0(λ)MR∞(λ),(6.4.40)
MR∞(λ)Mˆ∞(λ)=(Mˆ0(λ))−1(Mˆ1(λ))−1MR∞(λ),(6.4.41)
114

whichtranslateintothefollowingscalarequationsinvolvingthefunctionspjandqjoccurringinMˆj(cf.
(3.9.26)):p1=p0,q1=−q0.(6.4.42)
axisThA∞us,,inwetheobtaincaseoftheafollonormalizedwingequivtrinoid,alentwhicreformhisulationsrotationallyof(3.9.50)symmetricandwith(3.9.51),respcecttoharacterizingthetrinoidthe
monodromymatricesMˆj(recallfromlemma6.2thatµ0=µ1):
p0=p0andp02+q0q0=1,(6.4.43)
p02−q0q0=cos2(2πµ02)+cos(2πµ∞).(6.4.44)
sin(2πµ0)
Here,thesecondequationfollowsinviewofq0q1+q0q1=−2q0q0.
Wederivedirectlyfrom(6.4.43)that
q0q0=1−p02.(6.4.45)
Insertingthisintothesecondequation,weobtain
222p02=cos(2πµ0)+cos(22πµ∞)+sin(2πµ0)=1+cos2(2πµ∞)=cos2(πµ∞).(6.4.46)
2sin(2πµ0)2sin(2πµ0)sin(2πµ0)
Thisinturnimplies22
q0q0=1−p02=sin(2πµ02)−cos(πµ∞).(6.4.47)
sin(2πµ0)
Sofar,weconcludethat
p1=p0=αcos(sin(2ππµµ∞0)),(6.4.48)
ζ0−q1=q0=sin(2πµ0),(6.4.49)
whereα∈{±1}andζ0isanoddfunctioninλsatisfying
ζ0ζ0=sin2(2πµ0)−cos2(πµ∞).(6.4.50)
(RecallNext,that,recallbythatremarkthe3.44,monoq0dromandythusmatricesalsoζ0Mˆjaresatisfynecessarily(3.9.32),oddi.e.functionsMˆ0Mˆ1inMˆ∞λ.)=I,whichreadsin
scalarformas(3.9.33)and(3.9.34).Insertingthepreviousresultstogetherwiththeidentityµ0=µ1
fromlemma6.2into(3.9.33),weobtain
cos(2πµ∞)+isin(2πµ∞)p∞=−cos2(2πµ0)+2icos(2πµ0)sin(2πµ0)p0+sin2(2πµ0)(p02−q0q0)
=−cos2(2πµ0)+2αicos(2πµ0)cos(πµ∞)+cos2(πµ∞)−sin2(2πµ0)+cos2(πµ∞)(6.4.51)
whichinviewofcos(2πµ∞)=2cos2(πµ∞)−1transformsinto
sin(2πµ∞)p∞=2αcos(2πµ0)cos(πµ∞),(6.4.52)
i.e.)µπcos(2p∞=αsin(πµ∞0).(6.4.53)
asreads(3.9.34),Similarlyisin(2πµ∞)q∞=−2αζ0cos(πµ∞),(6.4.54)
whichimpliesζ0
q∞=αisin(πµ∞).(6.4.55)
Applyingourresultsto(3.9.26),weobtaintheclaimedformsforthemonodromymatricesMˆj.
reformTheulationscasesl=(3.9.37),0andl(3.9.38)=1andarepro(3.9.35),ved(3.9.36),analogouslyresp(bectivyely,shiftingof(3.9.33),indices)b(3.9.34),yusingandthetheequivaccordingalent
reformulationof(3.9.51)asgiveninremark3.56.
115

Theorem6.9describesthe(unitary)3monodromymatricesassociatedwiththeextendedframeF:
˜M→symmetricΛSU(2)withσofrespaecttrinoidtotheφ:Mtrinoid→RaxisAwithlforpropsomeerlylem∈b{0,edded1,∞},annandularwhicends,hhaswhicbheisenrotationallynormalized
suchthatF(z∗,l)=Iandψ(z∗,l)=21He3,wherez∗,l∈M˜isgiven,accordingtol,in(6.4.1),(6.4.2)
or(6.4.3),respectively,andψdenotestheconformalCMC-immersionM˜→R3correspondingtoφ.It
turnsoutthat,inthissetting,wecanalsoprovetheconverseresult:Atrinoidφwithproperlyembedded
annularendsandwithextendedframeFsatisfying(forsomel∈{0,1,∞})F(z∗,l)=Iatz∗,l∈M˜
thefrom,formaccordingivengintol,theorem(6.4.1),6.9is(6.4.2)ornecessarily(6.4.3),rotationrespectivallyely,symmetricandcorrespwithreondingspecttomonothedromytrinoidmatricesaxisAlof.
Thisresultisformulatedinthefollowingtheorem.
Theorem6.10.Letηbea(standardized)trinoidpotentialassociatedwiththreeoff-diagonalDelaunay
3trinoidmatriceswiDth0,prDop1,erlyD∞embewithddedeigenvaluesannularends±µ0,on±Mµ1=Cand\{±0µ,∞1},rgenerespeatectively.dbyηviaDenotethebyloopφ:grMoup→methoRd.a
Moreover,letF:M˜→ΛSU(2)σbetheextendedframeassociatedwiththemappingψ=φ◦πbytheorem
4.5.1.Letµ1=µ∞,z∗,0∈M˜givenin(6.4.1)andF(z∗,0)=I.Assumetheunitarymonodromymatrices
Mˆ0,Mˆ1,Mˆ∞∈ΛSU(2,C)σassociatedwithFaregivenby
Mˆ0=−cos(2πµ0)I−2αicos(πµ0)cos(2iζ1πµ1)−cos(2−iζπ1µ1),(6.4.56)
Mˆ1=−cos(2πµ1)I−iαcos(ζ1πµ1)−αcos(ζ1πµ1),(6.4.57)
Mˆ∞=−cos(2πµ1)I−iαcos(−ζπ1µ1)−α−cos(ζ1πµ1),(6.4.58)
whereα∈{±1}andζ1isanoddfunctioninλandasolutionto
ζ1ζ1=sin2(2πµ1)−cos2(πµ0).(6.4.59)
Then,φisrotationallysymmetricwithrespecttothetrinoidaxisA0.
2.Lˆetµˆ0=ˆµ∞,z∗,1∈M˜givenin(6.4.2)andF(z∗,1)=I.Assumetheunitarymonodromymatrices
M0,M1,M∞∈ΛSU(2,C)σassociatedwithFaregivenby
Mˆ0=−cos(2πµ∞)I−iαcos(−ζπµ∞)−α−cos(ζ∞πµ),(6.4.60)
∞∞Mˆ1=−cos(2πµ1)I−2αicos(πµ1)cos(2iζπ∞µ∞)−cos(2−iζπ∞µ∞),(6.4.61)
Mˆ∞=−cos(2πµ∞)I−iαcos(ζ∞πµ∞)−αζcos(∞πµ∞),(6.4.62)
whereα∈{±1}andζ∞isanoddfunctioninλandasolutionto
ζ∞ζ∞=sin2(2πµ∞)−cos2(πµ1).(6.4.63)
Then,φisrotationallysymmetricwithrespecttothetrinoidaxisA1.
3.Letµ0=µ1,z∗,∞∈M˜givenin(6.4.3)andF(z∗,∞)=I.Assumetheunitarymonodromymatrices
Mˆ0,Mˆ1,Mˆ∞∈ΛSU(2,C)σassociatedwithFaregivenby
Mˆ0=−cos(2πµ0)I−iαcos(ζπµ0)−αcos(ζ0πµ),(6.4.64)
00Mˆ1=−cos(2πµ0)I−iαcos(−ζπµ0)−α−cos(ζ0πµ),(6.4.65)
00Mˆ∞=−cos(2πµ∞)I−2αicos(πµ∞)cos(2iζπµ0)−cos(2−iζπ0µ),(6.4.66)
00whereα∈{±1}andζ0isanoddfunctioninλandasolutionto
ζ0ζ0=sin2(2πµ0)−cos2(πµ∞).(6.4.67)
Then,φisrotationallysymmetricwithrespecttothetrinoidaxisA∞.
116

Prtheoof.first,Wesecond,startbythirdcaseconsideringwiththle=sp0,lecial=1,forml=of∞,theresppotenectivtialely,ηanindeachdenoteofthethethreecorrespcases.ondingWepassootenciatetial
byη0,η1,η∞,respectively.
Inthefirstcase(l=0)wehaveµ1=µ∞andthus(cf.section3.6)
0λ−1
η0=−λQ0(z,λ)0dz,(6.4.68)

whereQ(z,λ)=b0(λ)+b1(λ)+b0(λ)−b0(λ)
0z2(z−1)2zz−1
b0(λ)(z−1)2+b1(λ)z2−b0(λ)z(z−1)b0(λ)(1−z)+b1(λ)z2
(6.4.69)==z2(z−1)2z2(z−1)2
andbj(λ)=41−(µj(λ))2forj=0,1.ConsideringthebiholomorphicmappingγR0:M→Mdefinedby
z→γR0(z):=z−z1andthefunctionh0:M→C\{0},z→h0(z)=−i(z−1),wecompute
2zzb0(λ)(1−z−1)+b1(λ)(z−1)2b0(λ)(1−z)+b1(λ)z24
Q0(γR0(z),λ)=z2z2=21=(h0(z))Q0(z,λ).(6.4.70)
(z−1)2(z−1−1)z(z−1)2
Recallingfromlemma4.21thatγR0correspondstothepermutationσ=(1∞)oftheset{0,1,∞},we
applylemma4.25toinferthatη0transformsunderγR0as
γR∗0η0=η0#W+,0,(6.4.71)
whereh0(z)0
W+,0=W+,0(z,λ)=−λ∂zh0(z)(h0(z))−1.(6.4.72)
Analogously,inthesecondcase(l=1)wehaveµ0=µ∞andthus(cf.section3.6)
0λ−1
η1=−λQ(z,λ)0dz,(6.4.73)
1

whereQ(z,λ)=b0(λ)+b1(λ)+b1(λ)−b1(λ)
1z2(z−1)2zz−1
b0(λ)(z−1)2+b1(λ)z2−b1(λ)z(z−1)b0(λ)(z−1)2+b1(λ)z
(6.4.74)==z2(z−1)2z2(z−1)2
andbj(λ)=41−(µj(λ))2forj=0,1.ConsideringthebiholomorphicmappingγR1:M→Mdefinedby
1z→γR1(z):=zandthefunctionh1:M→C\{0},z→h1(z)=−iz,wecompute
2121Q(γ(z),λ)=b0(λ)(z−1)+b1(λ)z=b0(λ)(z−1)+b1(λ)z=(h(z))4Q(z,λ).(6.4.75)
111R1z12(z1−1)2z12(z−1)2
Recallingfromlemma4.21thatγR1correspondstothepermutationσ=(0∞)oftheset{0,1,∞},we
applylemma4.25toinferthatη1transformsunderγR1as
γR∗1η1=η1#W+,1,(6.4.76)
whereh1(z)0
W+,1=W+,1(z,λ)=−λ∂zh1(z)(h1(z))−1.(6.4.77)
Finally,inthethirdcase(l=∞)wehaveµ0=µ1andthus(cf.section3.6)
0λ−1
η∞=−λQ(z,λ)0dz,(6.4.78)
∞117

whereQ∞(z,λ)=b0(λ)+b0(λ)+c0(λ)−c0(λ)=β0(λ)(z−1)2+b0(λ)z2−c0(λ)z(z−1),(6.4.79)
z2(z−1)2zz−1z2(z−1)2
bj(λ)=41−(µ0(λ))2forj=0,∞andc0(λ)=2b0(λ)−b∞(λ).Consideringthebiholomorphicmapping
γR∞:M→Mdefinedbyz→γR∞(z):=1−zandthefunctionh∞:M→C\{0},z→h∞(z)=−i,
computeewQ∞(γR∞(z),λ)=β0(λ)(−z)2+b0(λ)(1−2z)2−2c0(λ)(1−z)(−z)=(h∞(z))4Q∞(z,λ).(6.4.80)
(1−z)(−z)
Recallingfromlemma4.21thatγR∞correspondstothepermutationσ=(01)oftheset{0,1,∞},we
applylemma4.25toinferthatη∞transformsunderγR∞as
γ∗R∞η∞=η∞#W+,∞,(6.4.81)
whereh∞(z)0
W+,∞=W+,∞(z,λ)=−λ∂zh∞(z)(h∞(z))−1.(6.4.82)
Altogether,wehaveforalll∈{0,1,∞}therelation
γR∗lηl=ηl#W+,l.(6.4.83)
Applyingthepullbackconstructionwithrespecttothecoveringmappingπ:M˜→Mto(6.4.83),we
obtainπ∗(γR∗lηl)=π∗(ηl#W+,l)=η˜l#W˜+,l,(6.4.84)
whereη˜l=π∗ηldenotesthepullbackpotentialofthetrinoidpotentialηl(cf.section2.3)andW˜+,l=
W+,l◦π.Moreover,recallthatthebiholomorphicmappingsγ˜Rl:M˜→M˜,
−z−2−z−11
γ˜R0:z→z+1,γ˜R1:z→2z+1,γ˜R∞:z→−z(6.4.85)
fromlemma6.3satisfyγRl◦π=π◦γ˜Rl.Thus,thelefthandsideof(6.4.84)canbetransformedas
ws:follo1−π∗(γR∗lηl)=π∗−λQl(γ0Rl(z),λ)λ0dγRl(z)
0λ−10λ−1
=−λQl((γRl◦π)(z),λ)0d(γRl◦π)(z)=−λQl((π◦γ˜Rl)(z),λ)0d(π◦γ˜Rl)(z)
0λ−1
=γ˜R∗l−λQl(π(z),λ)0dπ(z)=γ˜R∗l(π∗ηl)=γ˜R∗lη˜l.(6.4.86)
yields(6.4.84)Altogether,γ˜R∗lη˜l=η˜l#W˜+,l.(6.4.87)
ConsideringtheextendedframeFassociatedwiththetrinoidφ,weobtain(forl∈{0,1,∞})a
solutionΨl=FB+,ltothedifferentialequationdΨl=Ψlη˜l.NotethatΨlpossessesthesame(unitary)
monodromymatricesasFat∗thesingularitiesofthepotentialηl,namelyMˆ0,Mˆ1andMˆ∞.∗
(γ˜∗Ψl)(Naturally˜γ∗η,˜l),thewhichmappinginviewγ˜RlofΨl(6.=Ψ4.87)l◦γ˜Rreadslasdefinesasolutiontothedifferentialequationd(γ˜RlΨl)=
lRlRd(γ˜R∗lΨl)=(γ˜R∗lΨl)(η˜l#W˜+,l).(6.4.88)
SincethisdifferentialequationisalsosolvedbythemappingΨlW˜+,l,i.e.
d(ΨlW˜+,l)=(ΨlW˜+,l)(η˜l#W˜+,l),(6.4.89)
themappingsγ˜R∗lΨlandΨlW˜+,lonlydifferbyaλ-dependentmatrixρl=ρl(λ):
γ˜R∗lΨl=ρlΨlW˜+,l.(6.4.90)
118

Considerthecasel=0.Applying˜therelationγ˜R0◦γ˜1=γ˜∞◦γ˜R0from(6.3.3),involvingthecovering
transformationsγ˜1andγ˜∞onMasgiveninsection3.3,wecompute
ρ0(λ)Mˆ1(λ)Ψ0(z,λ)W˜+,0(γ˜1(z),λ)=ρ0(λ)Ψ0(γ˜1(z),λ)W˜+,0(γ˜1(z),λ)=γR˜0∗Ψ0(γ˜1(z),λ)
=Ψ0((γ˜R0◦γ˜1)(z),λ)=Ψ0((γ˜∞◦γ˜R0)(z),λ)=Mˆ∞(λ)Ψ0(γ˜R0(z),λ)
=Mˆ∞(λ)(γ˜R∗0Ψ0(z,λ))=Mˆ∞(λ)ρ0(λ)Ψ0(z,λ)W˜+,0(z,λ).(6.4.91)
˜˜isAsW+,0holomorphicdefinesontheM˜pullbacandkofthereforethedomappingesnotW+pic,0k,upwhichaniysmonoholomorphicdromyonunderMγ˜1(with,i.e.respW˜+ect,0(γ˜to1(zz),),λW)+,=0
W˜+,0(z,λ).Thus,weconcludethat
ρ0(λ)Mˆ1(λ)=Mˆ∞(λ)ρ0(λ).(6.4.92)
Analogously,applyingγ˜R0◦γ˜∞=γ˜1◦γ˜R0from(6.3.3),wehave
ρ0(λ)Mˆ∞(λ)Ψ0(z,λ)W˜+,0(γ˜∞(z),λ)=ρ0(λ)Ψ0(γ˜∞(z),λ)W˜+,0(γ˜∞(z),λ)=γR˜0∗Ψ0(γ˜∞(z),λ)
=Ψ0((γ˜R0◦γ˜∞)(z),λ)=Ψ0((γ˜1◦γ˜R0)(z),λ)=Mˆ1(λ)Ψ0(γ˜R0(z),λ)
=Mˆ1(λ)(γ˜R∗0Ψ0(z,λ))=Mˆ1(λ)ρ0(λ)Ψ0(z,λ)W˜+,0(z,λ).(6.4.93)
UsingtheholomorphicityofW˜+,0onM˜,weknowthatW˜+,0(γ˜1(z),λ)=˜W+,0(z,λ),whichyields
ρ0(λ)Mˆ∞(λ)=Mˆ1(λ)ρ0(λ).(6.4.94)
seteWρ0(λ)=ca00((λλ))db00((λλ)),(6.4.95)
wherea0,b0,c0andd0definecomplexvˆaluedfunctionsˆofλsatisfyinga0(λ)d0(λ)−b0(λ)c0(λ)=1.
Comparingtheupperleftentriesofρ0(λ)M1(λ)andM∞(λ)ρ0(λ),weobtain
b0(λ)ζ1=−c0(λ)ζ1.(6.4.96)
Then,bycomparingtheupperrightentriesofρ0(λ)Mˆ1(λ)andMˆ∞(λ)ρ0(λ)(resp.ofρ0(λ)Mˆ∞(λ)and
Mˆ1(λ)ρ0(λ)),weinferthat
a0(λ)ζ1−b0(λ)αcos(2πµ1)=b0(λ)αcos(2πµ1)−d0(λ)ζ1,(6.4.97)
−a0(λ)ζ1−b0(λ)αcos(2πµ1)=b0(λ)αcos(2πµ1)+d0(λ)ζ1,(6.4.98)
whicinsertingh(bythisinsummingtoup(6.4.97),thesewetwoinferthatequations)a0(λ)di=rec−tlyd0(λ),implieswhicb0h(λ)=together0andwiththus,thebyrelation(6.4.96),a0(cλ0)(dλ0)(λ=)−0.
b0(λ)c0(λ)=1impliesa0(λ)=−d0(λ)=±i.Therefore,wehave
ρ0(λ)=±0i−0i,(6.4.99)
inparticularCarryingρout0(λ)∈exactlyΛSU(2)theσ.samecomputationsasaboveforthecasesj=1andj=∞(onlyshifting
obtainewappropriately),indices0iρ1(λ)=±0−i,(6.4.100)
ρ∞(λ)=±i0−0i.(6.4.101)
Consequentlywehaveforalll∈{0,1,∞}inparticularρl(λ)∈ΛSU(2)σ.Thus,(ρlFρl−1)(ρlB+,lW˜+,l)
definesanIwasawa-decompositionofρlΨlW˜+,l(pointwiseforallz∈M˜)withρlFρl−1∈ΛSU(2)σ,
ρB+,lW˜+,l∈Λ+SL(2,C)σand(ρlFρl−1)(z∗,l)=I.Therefore,wecanwrite
(F◦γ˜Rl)(B+,l◦γ˜Rl)=γ˜R∗lΨl=ρlΨlW˜+,l=(ρlFρl−1)(ρlB+,lW˜+,l).(6.4.102)
119

Thisimpliesthat,usingtheloopgroupmethod,γ˜R∗lΨlproducesontheonehandthetrinoidJ(ψ◦γ˜Rl)=
SymBob(F◦γRl)|λ=1andontheotherhandtherotatedtrinoidρlJ(ψ)ρl−1=SymBob(ρlFρl−1)|λ=1.
Consequently,thesetwosurfacescoincide,i.e.
J(ψ◦y˜Rl)(M˜)=(ρlJ(ψ)ρl−1)(M˜).(6.4.103)
UsingtheidentityρlJ(ψ)ρl−1=J◦ARl◦ψ,where
−100
100ARl=0−10,(6.4.104)
fromtheproofofcorollary6.8,weswitchintotheR3modelandobtainψ◦γ˜Rl=ARl◦ψ.Asγ˜Rl(M˜)=M˜,
yieldsthisψ(M˜)=ARl(ψ(M˜)).(6.4.105)
Thismeansthatψ(andthusalsoφ)issymmetricwithrespecttotheEuclideanmotionARl∈Iso(R3)
definingtherotationbytheangleπaroundthez-axisinR3.Thus,φisnecessarilyrotationallysymmetric
withrespecttothetrinoidaxisAl.(Inviewoftheorem4.31,whichlistsallpossibletrinoidsymmetries,
onlytherotationbytheangleπaroundthetrinoidaxisAlshowsthebehaviourofARl.Inparticular,
ARlisassociatedwiththebiholomorphicmappingγ˜Rl:M˜→M˜keepingthetrinoidendcorresponding
tothesingularityzlfixed.Thus,weinferthatthez-axisinR3coincideswiththetrinoidaxisAl,and
thatφisrotationallysymmetricwithrespecttothetrinoidaxisAl,coincidingwiththez-axisinR3.)

120

7Reflectionalsymmetrywithrespecttothetrinoidplane
Definition7.1Inerty,thisnamelysectionwtrinoidseturnwithtoproptrinoidserlywithembpropeddederlyannembulareddedendsannwhicularhareendsswithymmetricanotherwithrespsymmetryecttoprop-the
B0,(orienB1tationandBrev∞ersing)accordingtoreflectionthepSerminsomeutation(trinoid).RecallplaneEthat,.Inthoughparticular,thereS“pexistermaprioriutes”ptheossiblytrinoidsevendseral
trinoidplanesofφ,EisuniquelydeterminedbythesymmetryS.Throughoutthissection,wewill-by
aslighsymmetrytabuseS,ofsimplynotationasof-thespeaktrinoidoftheplaneoftrinoidφ.planeE,whichistheplaneofreflectionofthetrinoid
3LetM˜Definition=H7.1.andψLet=φM◦=π:CM\˜{0→,1R}3andtheφ:Mconformal→RbeaCMC-immersiontrinoidwithassopropciatederlyemwithbφeddedviaanntheularunivends.ersal
coveringπ:M˜→Mgivenin(3.2.2).Then,ifφ(or,equivalently,ψ)issymmetricwithrespecttothe
reflectionSinthetrinoidplaneE,
S(φ(M))=φ(M),(7.1.1)
φ(orψ)iscalledreflectionallysymmetricwithrespecttothetrinoidplane.
propLikerteyin(7.1.1)theincasetooffurthertrinoidsconstrainwithtsotheronthemsymmetries,onodromwyearematricesinterestedassociatedinwithtranslatingthetheextendedsymmetryframe
.ψofF7.2Implicationsofreflectionalsymmetrywithrespecttothetrinoidplane
3LetMconformal=C\{0,1},CMC-immersionφ:M→onRM˜be=aH,trinoidψ=φwith◦πprop:M˜erly→emR3b,eddedwhereannπulardenotesendstheandunivψtheersalassocovciatedering
M˜→Mgivenin(3.2.2).Letφ(or,equivalently,ψ)bereflectionallysymmetricwithrespecttothe
obtaintrinoidbyplanetheoremEand4.9letaSpairdenoteofbi-anthecorresptiholomorphicondingmappings,symmetry.γS:inceM→SMrevandersesγ˜:orienM˜→tationM˜onR3satisfying,we
SSS◦φ=φ◦γS,(7.2.1)
S◦ψ=ψ◦˜γS,(7.2.2)
π◦γ˜Rl=γS◦π.(7.2.3)
ThemappingγScanbeexplicitlycomputed,asdoneinlemma4.21:
γS(z)=z¯.(7.2.4)
Thephismmappinggroupγ˜SAut(isM˜/Muniquely)ofπ.Thedeterminedfolloupwingtocomplemmaositionexplicitlyfromthstateseleftavalidwithcanhoiceelemenfortγ˜Sof:theautomor-
Lemma7.2.LetM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversalcoveringasgivenin(3.2.2).
LetγS:M→Mbegivenby(7.2.4).Then,themapping
γ˜S:M˜→M˜,γ˜S(z)=−z¯(7.2.5)
satisfiesandbi-antiholomorphicisπ◦γ˜S=γS◦π,(7.2.6)
S◦ψ=ψ◦γ˜S.(7.2.7)
Pro(3.2.12)of.ofDefinelemmaγ˜Sas3.4,inwe(7.2.5).obtainObforviouslyallz,∈γ˜SM˜isabi-antiholomorphicfunction.Byapplyingtherelations
π◦γ˜S(z)=π(−z¯)=π(z)=γS◦π(z).(7.2.8)
,ytlconsequenand,S◦ψ=S◦φ◦π=φ◦γS◦π=φ◦π◦γ˜S=ψ◦γ˜S.(7.2.9)
121

Bytheabovelemma,wehaveexplicitlydeterminedamappingγ˜Scorrespondingtothetrinoid
symmetrySinthesenseoftheorem4.9.Thus,wecanapplytheorem4.17toobtain
Theorem7.3.LetM=C\{0,1},φ:M→R3˜beatrinoidwithprop˜erlyembe3ddedannularendsand
ψtheassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotesthe
trinoiduniversalcplaneoveringE.M˜Denote→Mtheascorrdefineespdondingin(3.2.2)symmetry.LetφbybSerandeflebyctionalγ˜lythesymmetricbi-antiholomorphicwithrespecttomappingthe
S˜˜Mextende→MdfrassoameciateF:dM˜with→SasΛSU(2)inσthecororremesp4.9ondingandtoψexplicitlybytheordefineemd4.5intrlemmaansforms7.2,γ˜Sunder(z)γ˜=S(−zz¯).asThenthe
F(γ˜S(z),λ−1)=MS(λ)F(z,λ)kS,γ˜S(z),(7.2.10)
ewherkS,γ˜S(z)=−0ii0(7.2.11)
andMS(λ)denotesanelementofΛSU(2)σ,whichisindependentofz.
Proapplyof.theWeprosecondceedpartasinofttheheorproemof4.17oftotheoremobtain5.6.AsS∈O(3)\SO(3)reversesorientationonR3,we
F(γ˜S(z),λ−1)=MS(λ)F(z,λ)kS,γ˜S(z),(7.2.12)
˜Mwhereγ˜S(λ)F:Mdenotes→anΛSU(2)elemenσtofdenotesΛSU(2)theσ,whicextendedhisindepframeendencorresptofz.ondingkS,γ˜Sto(z)ψisbgyiventheoremby4.5equationandM(4.4.118)S:=
fromlemma4.18.Bycomputing
∂z¯γ˜(z)=−1(7.2.13)
weinferthat∂γ˜(z)
|∂z¯z¯γ˜(z)|=−1(7.2.14)
andthusobtainfrom(4.4.118)(inviewofourdefinitionofthecomplexsquarerootonthez-planegiven
4.14)remarkinkS,γ˜S(z)=−0ii0.(7.2.15)
7.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,
whicharereflectionallysymmetricwithrespecttothetrinoidplane
Wemebnoweddedstudyannularthe(uendsnitary)withmonoreflectionaldromysymmmatricesetryMˆin0,theMˆ1,Mˆtrinoid∞assoplaneciatedE.Owithuratrinoidconsiderationswitharepropbaseerlyd
coonvertheingrelationstransformationsbetweenγ˜jtheonbi-anM˜generatingtiholomorphicthemonomappingdromγ˜SyassomatricesciatedMˆj.withRecallthethesymmlatteetryrSonesandfromthe
3.3:sectionγ˜0(z)=−2zz+1,γ˜1(z)=2zz−−23,γ˜∞(z)=z+2.(7.3.1)
Thecorrespondinginversefunctionsaregivenby
γ˜0−1(z)=2zz+1,γ˜1−1(z)=−−23zz++12,γ˜∞−1(z)=z−2.(7.3.2)
Therelationsmentionedabovearestatedinthefollowinglemma.
Lemma7.4.LetM˜=Handγ˜0,γ˜1,γ˜∞:M˜→M˜begivenasabove.Forγ˜S:M˜→M˜,γ˜S(z)=−z¯,the
hold:identitieslowingfolγ˜S◦γ˜0=γ˜0−1◦γ˜S,γ˜S◦γ˜1=γ˜1−1◦γ˜S,γ˜S◦γ˜∞=γ˜1◦γ˜0◦γ˜S.(7.3.3)
122

Proof.Forz∈M˜wehavebydirectcomputation
γ˜S◦γ˜0(z)=−2z¯−z¯+1=γ˜0−1◦γ˜S(z),(7.3.4)
γ˜S◦γ˜1(z)=−z¯−2=γ˜1−1◦γ˜S(z),(7.3.5)
andthusγ˜S◦γ˜0=γ˜0−1◦γ˜Sandγ˜S◦γ˜1=γ˜1−1◦γ˜S.Usingthisandrecallingγ˜0◦γ˜1◦γ˜∞=idonM˜,we
obtainγ˜S◦γ˜∞(z)=γ˜S◦γ˜1−1◦γ˜0−1(z)=γ˜1◦γ˜0◦γ˜S(z)(7.3.6)
forallz∈M˜,whichprovestheremainingidentity.
Inviewofthis,weareabletoprovethefollowingtheorem:
Theorem7.5.LetM=C\{0,1},φ:M→R3beatrinoidwithproperlyembeddedannularendsandψ
theassociatedconformalCMC-immersiononM˜=H,ψ=φ◦π:M˜→R3,whereπdenotestheuniversal
coveringM˜→Masdefinedin(3.2.2).Letφbereflectionallysymmetricwithrespecttothetrinoidplane
˜E.associateDenotedwiththecψorrbyesptheonordinemg4.5.symmetryDenotebybyS.MˆF0,Mˆ1,urthermorMˆ∞e,∈letΛSU(2F:,MC)σ→theΛSU(2)unitaryσbemonothedromyextendedmatricframees
Mˆj=−cos(2πµj)0110+isin(2πµj)qpj−qpj(7.3.7)
jjassociatedwithFasin(4.5.13)by
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),j=0,1,∞,(7.3.8)
whereαj∈{±1}andγ˜jdenotethecoveringtransformationsonM˜fromsection3.3.Finally,letγ˜S
bethebi-antiholomorphicmappingM˜→M˜associatedwithSasintheorem4.9andexplicitlygivenin
lemma7.2,andab
MS(λ):=−bSSaSS(7.3.9)
thecorrespondingtransformationmatrixofFsatisfying(7.2.10).Then,themonodromymatricessatisfy
MS(λ)Mˆ0(λ)=(Mˆ0(λ−1))−1MS(λ),(7.3.10)
MS(λ)Mˆ1(λ)=(Mˆ1(λ−1))−1MS(λ),(7.3.11)
MS(λ)Mˆ∞(λ)=Mˆ1(λ−1)Mˆ0(λ−1)MS(λ).(7.3.12)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(7.3.10)to(7.3.12)areequivalentto
aSpj(λ)+bSqj(λ)=aSpj(λ−1)−bSqj(λ−1),(7.3.13)
aSqj(λ)−bSpj(λ)=bSpj(λ−1)+aSqj(λ−1),(7.3.14)
forj∈{0,1}.
Proof.Westartwiththefollowingobservation,whichisimpliedby(7.3.8):
F(z,λ)=F(γ˜j◦γ˜j−1(z),λ)=αjMˆj(λ)F(γ˜j−1(z),λ)kj(γ˜j−1(z))(7.3.15)
andthuswehave
F(γ˜j−1(z),λ)=αjMˆj(λ)−1F(z,λ)(kj(γ˜j−1(z)))−1.(7.3.16)
Additionally,byequation(7.2.10)fromtheorem7.3,
F(γ˜S(z),λ−1)=MS(λ)F(z,λ)kS,γ˜S(z),(7.3.17)
wherekS,γ˜S(z)=−i0.(7.3.18)
i0

123

(7.3.15)(7.3.16)(7.3.17)(7.3.18)

Combiningtheseresultswiththeidentities(7.3.3)fromtheabovelemma,weobtainforj∈{0,1}:
MS(λ)αjMˆj(λ)F(z,λ)kj(z)−0ii0=MS(λ)F(γ˜j(z),λ)kS,γ˜S(γ˜j(z))
=F(γ˜S◦γ˜j(z),λ−1)=F(γ˜j−1◦γ˜S(z),λ−1)=αjMˆj(λ−1)−1F(γ˜S(z),λ−1)(kj(γ˜j−1(γ˜S(z))))−1
=αjMˆj(λ−1)−1MS(λ)F(z,λ)−i0(kj(γ˜j−1(γ˜S(z))))−1.(7.3.19)
i0Computing−i1−2z¯0
k0(z)−0ii0=1−2z1−2z¯
0i1−2z
−i1−2−2−−zz¯z+10
21−=1−2−2z¯+1−z=−0ii0(k0(γ˜0−1(γ˜S(z))))−1(7.3.20)
0i1−2−−22−z¯zz¯+1+1
and
k1(z)−0ii0=−0ii0=−0ii0(k1(γ˜1−1(γ˜S(z))))−1,(7.3.21)
weconcludeforj∈{0,1}that
MS(λ)Mˆj(λ)=(Mˆj(λ−1))−1MS(λ).(7.3.22)
Whatremainstoproveis(7.3.11).Weshowthatthisequationisadirectconsequenceofequations
(7.3.10)and(7.3.12),whichcanbeequivalentlyformulatedas
1−MS(λ)Mˆj(λ)=Mˆj(λ−1)MS(λ),j=0,1.(7.3.23)
Togetherwiththeidentity(3.9.32)weobtain
MS(λ)Mˆ∞(λ)=MS(λ)Mˆ1(λ)−1Mˆ0(λ)−1=Mˆ1(λ−1)Mˆ0(λ−1)MS(λ),(7.3.24)
claimed.asAsequation(7.3.12)isimpliedbyequations(7.3.10)and(7.3.11),thesethreeequationsareequivalent
tothescalarreformulationsoftheˆequations(7.3.10)and(7.3.11),whichareobtainedasfollows:First
recallthatthemonodromymatricesMj(λ)areoftheform
10pj(λ)qj(λ)
Mˆj(λ)=−cos(2πµj)01+isin(2πµj)qj(λ)−pj(λ)(7.3.25)
withpj2+qjqj=1andpj=pj,(7.3.26)
whichimpliesthat
(Mˆj(λ−1))−1=−cos(2πµj)10−isin(2πµj)pj(λ−−11)qj(λ−−11)(7.3.27)
01qj(λ)−pj(λ)
and
Mˆj(λ)=−cos(2πµj)0110−isin(2πµj)qpj((λλ))−qpj((λλ)).(7.3.28)
jjThescalarequationsassociatedwith(7.3.10)and(7.3.11),respectively,arethen(omittingredundant
ones)givenby(j∈{0,1})
−cos(2πµj)aS+isin(2πµj)(aSpj(λ)+bSqj(λ))=−cos(2πµj)aS+isin(2πµj)(aSpj(λ−1)−bSqj(λ−1)),
(7.3.29)−cos(2πµj)bS+isin(2πµj)(aSqj(λ)−bSpj(λ))=−cos(2πµj)bS+isin(2πµj)(bSpj(λ−1)+aSqj(λ−1)).
(7.3.30)

124

(7.3.31)(7.3.32)

Theseequationssimplifyto(j∈{0,1})
aSpj(λ)+bSqj(λ)=aSpj(λ−1)−bSqj(λ−1),(7.3.31)
aSqj(λ)−bSpj(λ)=bSpj(λ−1)+aSqj(λ−1),(7.3.32)
whichfinishestheproof.
7.4Normalizedtrinoidswithproperlyembeddedannularends,whichare
reflectionallysymmetricwithrespecttothetrinoidplane
Letφ:M→R3beatrinoidwithproperlyembeddedannularends,whichisreflectionallysymmetric
M˜with→Rresp3.ecttoDenotethebyStrinoidtheplane.correspMoreoondingver,symletψmetry=φof◦φπb(andetheψ),assoi.e.theciatedreflectionconformalinthetrinoidCMC-immersionplane.
NormalizingtheextendedframeF:M˜→ΛSU(2)σassociatedwithψasinsection4.2,suchthat
F(z∗∗,λ)=Iat
(7.4.1)i,=z∗∗wecanformulateamoreexplicitversionoftheorem7.3(seebelow).ThenormalizationF(z∗∗,λ)=Iof
Fisaconsequenceofnormalizingthe(conformal)CMC-immersionψ,suchthat
1ψ(z∗∗)=2He3,U(z∗∗)=G(1),(7.4.2)
where(4.2.5).U∈RecallSO(3)fromrepresensectionts4.2,thenthataturalthisorthonormalnormalizationframeofψcorrespcorrespondsondingtotoψ,rotatingandGand(1)isshiftinggiventhein
(imageofthe)trinoidinR3,suchthattheconditions(7.4.2)aremet.Itturnsout(cf.corollary7.7),that
thechoiceofz∗∗asabove(foratrinoidφwithproperlyembeddedannularends,whichisreflectionally3
suchsymmetricthatthewithtrinoidrespectplanetoisthetheytrinoid-z-planeplane)inR3corresp.ondstoarrangingthe(imageofthe)trinoidinR,
Atrinoidφ:M→R3withproperlyembeddedannularends,which3isreflectionallysymmetricwith
respecttothetrinoidplaneand,inaddition,is“wellpositioned”inRinthesensethattheassociated
˜trinoidconformalwithpropCMC-immerlyemersionbedψded:Mann→ularMends,meetswhicthehisnormalizationreflectionallyconditionssymmetric(7.4.2),withisrespcalledecttoathenormalizetrinoidd
plane.Wenowformulateamoreexplicitversionoftheorem7.3:
Theorem7.6.LetM=C\{0,1},φ:M→R3˜beatrinoidwithprop˜erlyembe3ddedannularendsand
ψtheuniversalassocciateoveringdcM˜onformal→MasdefineCMC-immersiondin(3.2.2)on.MLet=φH,beψr=efleφ◦ctionalπ:lyM→symmetricR,wherwitherπespedenotescttothethe
trinoidplane.Moreover,letz∗∗begivenin(7.4.1),
(7.4.3)i,=z∗∗andassumethatψhasbeennormalizedatz∗∗,suchthatψ(z∗,l)=21He3andF(z∗,l,λ)=I,where
˜cForr:Mesp→ondingΛSU(2)σsymmetrydenotesofφtheandbyextendeγ˜Sdtheframebi-antiholcorrespomondingorphictoψmappingbytheM˜or→emM˜4.5.associateDenotedbywithSStheas
intheorem4.9andexplicitlygivenin(7.2.5):
γ˜S(z)=−z¯.(7.4.4)
Then,theextendedframeFtransformsunderγ˜Sas
F(γ˜S(z),λ−1)=MS(λ)F(z,λ)−i0,(7.4.5)
i0wherei0
MS(λ)=0−i.(7.4.6)
Inparticular,MSisactuallyindependentofλ.

125

Proof.Inviewoftheorem7.3,weonlyhavetoproveequation(7.4.6).Notethatγ˜S(z∗)=−z∗=z∗.
Furthermore,F(z∗,λ)=Iforallλ∈S1.Keepingthisinmind,weevaluateequation(7.4.5)atz=z∗to
obtainI=F(z∗,λ−1)=F(γ˜S(z∗),λ−1)=MS(λ)F(z∗,λ)0i−0i=MS(λ)I−0ii0(7.4.7)
andequation(7.4.6)follows.
Corollary7.7.Weretainthenotationandtheassumptionsoftheorem7.6.Thereflectionplaneofthe
symmetrySofthenormalizedtrinoidφ,i.e.thetrinoidplane,isthey-z-planeinR3.
Proof.Applying(thesecondpartof)theorem4.17,weknowthatthemonodromymatrixMS(λ)explicitly
givenintheorem7.6satisfiesatλ=1
10MS(1)=±AS−10,(7.4.8)
whereAS∈SU(2)denotestheconjugationmatrixrealizingtheorthogonalpartASofthesymmetryS
inthesu(2)-model.Inviewofequation(7.4.6),thisyields
AS=±i00i.(7.4.9)
RecallingthatASandASarelinkedviatheLieAlgebraisomorphismJ:R3→su(2)definedin(3.4.3)
i.e.(3.4.8),inas(J◦AS◦J−1)(X)=−ASXAS−1forallX∈su(2),(7.4.10)
weobtainbyadirectcomputationthat
−100
100AS=010.(7.4.11)
Thus,ASdefinesthereflection(inR3)inthey-z-planeinR3.Consequently,thesymmetrySofthe
normalizedtrinoidφisareflectioninsomeplaneinR3,whichisparalleltothey-z-plane.Sincethe
pointψ(z∗∗)∈R3(withz∗∗givenin(7.4.1))satisfies
S(ψ(z∗∗))=ψ(γ˜S(z∗∗))=ψ(z∗∗),(7.4.12)
itliesinthereflectionplaneofS.Sincebyassumptionwehaveψ3(z∗∗)=21He3,weinferthatthereflection
planeofS(i.e.thetrinoidplane)isactuallythey-z-planeinR.
Applyingthetheorems7.5and7.6,weobtainthefollowingresult:
3andψTheoremtheasso7.8.ciateLetdMc=onformalC\{0,1},φCMC-immersion:M→RonbMe˜a=H,trinoidψ=withφ◦prπop:Merly˜→embRe3,ddedwhereπannulardenotesends
theuniversalcoveringM˜→Masdefinedin(3.2.2).Letφbereflectionallysymmetricwithrespect
tothetrinoidplane.Moreover,letz∗∗begivenin(7.4.1)andassumethatψhasbeennormalizedat
z∗∗,suchthatψ(z∗∗)=21He3andF(z∗∗,λ)=I,whereF:M˜→ΛSU(2)σdenotestheextendedframe
correspondingtoψbytheorem4.5.
Then,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwithFasin(7.3.8)
satisfyequations(7.3.10)to(7.3.12)fromtheorem7.5.Intermsofthefunctionspjandqjoccurringin
Mˆj,theseequationsareequivalentto
pj(λ)=pj(λ−1),(7.4.13)
qj(λ)=−qj(λ−1),(7.4.14)

forj∈{0,1}.

126

Proof.Asbefore,wedenotethesymmetryofφbySandbyγ˜Sthebi-antiholomorphicmappingM˜→M˜
associatedwithSasintheorem4.9andexplicitlygiveninlemma7.2.Moreover,letMS(λ)bethe
correspondingmonodromymatrixofFasintroducedinequation(7.2.10).Keepinginmindthatby
theorem7.6
MS(λ)=aSbS=i0,(7.4.15)
−bSaS0−i
thegivenidentitiesfollowdirectlyfromtheorem7.5.
Theorem7.8describesthe(unitary)monodromymatricesassociatedwiththeextendedframeF:
M˜→ΛSU(2)σofatrinoidφ:M→R3withproperlyembeddedannularends,whichisreflectionally
symmetricwithrespecttothetrinoidplane,andwhichhasbeennormalizedsuchthatF(z∗∗)=Iand
ψ(z∗∗)=21He3,wherez∗∗∈M˜isgivenin(7.4.1)andψdenotestheconformalCMC-immersionM˜→R3
correspondingtoφ.Itturnsoutthat,inthissetting,wecanalsoprovetheconverseresult:Atrinoid
φwithproperlyembeddedannularendsandwithextendedframeFsatisfyingF(z∗∗)=Iatz∗∗∈M˜
from(7.4.1)andcorrespondingmonodromymatricesoftheformgivenintheorem7.8isnecessarily
reflectionallysymmetricwithrespecttothetrinoidplane.Thisresultisformulatedinthefollowing
theorem.Theorem7.9.Letηbea(standardized)trinoidpotentialassociatedwiththreeoff-diagonalDelaunay
matricesD0,D1,D∞witheigenvalues±µ0,±µ1and±µ∞,respectively.Denotebyφ:M→R3a
trinoidwithproperlyembeddedannularendsonM=C\{0,1}generatedbyηviatheloopgroupmethod.
Moreover,letF:M˜→ΛSU(2)σbetheextendedframeassociatedwiththemappingψ=φ◦πby
theorem4.5,satisfyingF(z∗∗)=Iatz∗∗∈M˜givenin(7.4.1).Assumetheunitarymonodromymatrices
Mˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwithFareoftheform
pj(λ)qj(λ)
Mˆj=−cos(2πµj)I−isin(2πµj)qj(λ)−pj(λ),(7.4.16)
withfunctionspjandqjsatisfying(3.9.51)and(3.9.50)and,additionally,forj∈{0,1},
pj(λ)=pj(λ−1),(7.4.17)
qj(λ)=−qj(λ−1).(7.4.18)
Then,φisreflectionallysymmetricwithrespecttothetrinoidplane.
Proof.Considerthestandardizedtrinoidpotential(cf.section3.6)
0λ−1
η=−λQ(z,λ)0dz,(7.4.19)

(7.4.19)

whereb0(λ)b1(λ)c0(λ)c1(λ)b0(λ)(z−1)2+b1(λ)z2−c0(λ)z(z−1)
Q(z,λ)=z2+(z−1)2+z+z−1=z2(z−1)2(7.4.20)
andb0,b1,b∞,c0,c1areobtainedfrom
bj(λ)=1−µj2forj=0,1,∞,(7.4.21)
4b0(λ)+b1(λ)+0∙c0(λ)+1∙c1(λ)=b∞(λ),(7.4.22)
c0(λ)+c1(λ)=0.(7.4.23)
Then,forthebi-antiholomorphicmappingγS:M→M,z→γS(z)=z¯andthefunctionh:M→C\{0},
z→h0(z)=1,wecomputeforλ∈S1
b0(λ)(z¯−1)2+b1(λ)z¯2−c0(λ)z¯(z¯−1)
Q(γS(z),λ)=z¯2(z¯−1)2=
b0(λ−1)(z¯−1)2+b1(λ−1)z¯2−c0(λ−1)z¯(z¯−1)4−1
z¯2(z¯−1)2=(h(z))Q((z),λ),(7.4.24)

127

whereweusedthefactthatforλ∈S1theidentities
bj(λ−1)=bj(λ)forj=0,1,∞and(7.4.25)
c0(λ−1)=c0(λ)(7.4.26)
hold.Recallingfromlemma4.21thatγScorrespondstothepermutationσ=()oftheset{0,1,∞},we
applylemma4.25toinferthatηtransformsunderγSas
γS∗η(z,λ)=η(z,λ−1)#W+,(7.4.27)
whereh(z)0
W+=W+(z,λ)=−λ∂z¯h(z)(h(z))−1.(7.4.28)
Applyingthepullbackconstructionwithrespecttothecoveringmappingπ:M˜→Mto(7.4.27),we
obtainπ∗(γS∗η(z,λ))=π∗(η(z,λ−1)#W+)=η˜(z,λ−1)#W˜+,(7.4.29)
whereη˜=π∗ηdenotesthepullbackpotentialofthetrinoidpotentialη(cf.section2.3)andW˜+=W+◦π.
Moreover,recallthatthebi-antiholomorphicmappingγ˜S:M˜→M˜,z→−z¯,fromlemma7.2satisfies
γS◦π=π◦γ˜S.Thus,thelefthandsideof(7.4.29)canbetransformedasfollows:
1−π∗(γS∗η)=π∗−λQ(γ0S(z),λ)λ0dγS(z)
0λ−10λ−1
=−λQ((γS◦π)(z),λ)0d(γS◦π)(z)=−λQ((π◦γ˜S)(z),λ)0d(π◦γ˜S)(z)
1−=γ˜S∗−λQ(π0(z),λ)λ0dπ(z)=γ˜S∗(π∗η)=γ˜S∗η˜.(7.4.30)
yields(7.4.29)Altogether,γ˜S∗η˜(z,λ)=η˜(z,λ−1)#W˜+.(7.4.31)
ConsideringtheextendedframeFassociatedwiththetrinoidφ,weobtainasolutionΨ=FB+to
thedifferentialequationdΨ=Ψη˜.NotethatΨˆpˆossessestheˆsame(unitary)monodromymatricesasF
atthesingularitiesofthepotentialη,namelyM0,M1andM∞.
∗Naturally∗,themappingγ˜S∗Ψ=Ψ◦γ˜Sdefinesasolutiontothedifferentialequationd(γ˜S∗Ψ)=
(γ˜SΨ)(γ˜Sη˜),whichinviewof(7.4.31)readsas
d(γ˜S∗Ψ(z,λ))=(γ˜S∗Ψ(z,λ))(η˜(z,λ−1)#W˜+).(7.4.32)
SincethisdifferentialequationisalsosolvedbythemappingΨ(z,λ−1)W˜+,i.e.
d(Ψ(z,λ−1)W˜+)=(Ψ(z,λ−1)W˜+)(η˜(z,λ−1)#W˜+),(7.4.33)
themappingsγ˜S∗Ψ(z,λ)andΨ(z,λ−1)W˜+onlydifferbyaλ-dependentmatrixρ=ρ(λ):
γ˜S∗Ψ(z,λ)=ρΨ(z,λ−1)W˜+.(7.4.34)
Applying(forj=0,1)therelationγ˜S◦γ˜j=γ˜j−1◦γ˜Sfrom(7.3.3),involvingthecoveringtransfor-
mationsγ˜j,j=0,1,onM˜asgiveninsection3.3,wecompute
ρ(λ)Mˆj(λ−1)Ψ(z,λ−1)W˜+(γ˜j(z),λ)=ρ(λ)Ψ(γ˜j(z),λ−1)W˜+(γ˜j(z),λ)=γ˜S∗Ψ(γ˜j(z),λ)
=Ψ((γ˜S◦γ˜j)(z),λ)=Ψ((γ˜j−1◦γ˜S)(z),λ)=(Mˆj(λ))−1Ψ(γ˜S(z),λ)
=(Mˆj(λ))−1(γ˜S∗Ψ(z,λ))=(Mˆj(λ))−1ρ(λ)Ψ(z,λ−1)W˜+(z,λ),(7.4.35)
wherewehavemadeuseoftheidentity
Ψ(γ˜j−1(z),λ)=(Mˆj(λ))−1Ψ(z,λ),(7.4.36)
adirectconsequenceoftherelation
Ψ(z,λ)=Ψ(γ˜j(γ˜j−1(z)),λ)=Mˆj(λ)Ψ(γ˜j−1(z),λ).(7.4.37)

128

(7.4.37)

˜W˜+AsisanW+definestiholomorphictheonpullbacM˜kandofthethereforemappingdoesWnot+,pickwhicuphisanyanmonotiholomorphicdromyunderonγ˜M0,i.e.(withW˜+(respγ˜0(ectz),toλ)z=),
W˜+(z,λ).Thus,weconcludethat
ρ(λ)Mˆj(λ−1)=(Mˆj(λ))−1ρ(λ).(7.4.38)
Settingρ(λ)=ca((λλ))db((λλ)),(7.4.39)
wherea,b,candddefinecomplexvaluedfunctionsofλsatisfyinga(λ)d(λ)−b(λ)c(λ)=1,andcomparing
theupperleftentries(resp.theupperrightentries)ofρ(λ)Mˆj(λ−1)and(Mˆj(λ))−1ρ(λ),weobtain
a(λ)pj(λ−1)+b(λ)qj(λ−1)=a(λ)pj(λ)+c(λ)qj(λ),(7.4.40)
a(λ)qj(λ−1)−b(λ)pj(λ−1)=b(λ)pj(λ)+d(λ)qj(λ).(7.4.41)
Inviewof(3.9.50),(7.4.17)and(7.4.18),theseequationssimplifyinto
−b(λ)qj(λ)=c(λ)qj(λ),(7.4.42)
−(a(λ)+d(λ))qj(λ)=2b(λ)pj(λ).(7.4.43)
Sinceingeneral(i.e.forallλinS1excludingafinitesubsetofS1)pj,qj=0,wecansolveforc(λ)and
b(λ),respectively:
c(λ)=−b(λ)qj(λ)qj(λ)−1,(7.4.44)
b(λ)=−1(a(λ)+d(λ))qj(λ)(pj(λ))−1.(7.4.45)
2again)(3.9.50)(usingyieldsThis11=a(λ)d(λ)−b(λ)c(λ)=a(λ)d(λ)+4(a(λ)+d(λ))2qj(λ)qj(λ)(pj(λ))−2
=a(λ)d(λ)+1(a(λ)+d(λ))2(pj(λ))−2−1(a(λ)+d(λ))2=1(a(λ)+d(λ))2(pj(λ))−2−1(a(λ)−d(λ))2,
4444(7.4.46),tlyalenequivor,(pj(λ))2(4+(a(λ)−d(λ))2)=(a(λ)+d(λ))2.(7.4.47)
manyλAssume∈S1no).wWethat,inferinthatgeneral,4+(a(λ)−d(λ))2=0(i.e.4+(a(λ)−d(λ))2=0foratmostfinitely
(a(λ)+d(λ))2
(p0(λ))2=(p1(λ))2=4+(a(λ)−d(λ))2(7.4.48)
forallbut(atmost)finitelymanyλ∈S1andthus
p0(λ)=αp1(λ)(7.4.49)
forsomeα∈{±1}andallbut(atmost)finitelymanyλ∈S1.Consequently,by(7.4.45),thisimplies
q0(λ)=αq1(λ),q0(λ)=αq1(λ)(7.4.50)
usthandp0(λ)p1(λ)+q0(λ)q1(λ)+2q0(λ)q1(λ)=α((p0(λ))2+q0(λ)q0(λ))=α(7.4.51)
forallbut(atmost)finitelymanyλ2∈S1,whichclearly1isacontradictiontoequation2(3.9.51).Therefore,1
Tweogether,concludethesethat4+relations(a(λ)yield−d(aλ())λ)==0−dfor(λ)all=λ±∈iSandand(by(by(7.4.45)(7.4.47))and(a(λ(7.4.44)))+d(bλ())λ)==c0(λfor)=all0.λ∈ThSus,.
ρ(λ)=±0i−0i,(7.4.52)
129

+inparticularConsequenρ(tlyλ,)(∈ρF(z,ΛSU(2)λ−1σ)ρ∩−Λ1)(ρSL(2B+,(zC,)λσ.−1)W˜+(z,λ))definesanIwasawa-decompositionof
ρΨ(z,λ−1)W˜+(z,λ)(7.4.53)
+1−ρ(pF(oinzt,wiseλ−1)forρ−1all=zI.∈M˜)Therefore,withρwFe(zcan,λ−1write)ρ∈ΛSU(2)σ,ρB+(z,λ−1)W˜+(z,λ)∈ΛSL(2,C)σand
∗∗F(γ˜S(z),λ)B+(γ˜S(z),λ)=γ˜S∗Ψ(z,λ)=ρ(λ)Ψ(z,λ−1)W˜+(z,λ)
=(ρ(λ)F(z,λ−1)(ρ(λ))−1)(ρ(λ)B+(z,λ−1)W˜+(z,λ)).(7.4.54)
∗Thandus,onγ˜StheΨprootherhducesand(btheythetrinoidloopgroupSymBob(ρmetho(λ)Fd)(zon,λ−the1)(ρone(λ))−hand1)|λthe=1.trinoidConsequenSymBob(tly,Fthese(γ˜St(wzo),λ))surfaces|λ=1
coincide,and,usingthestraightforwardidentities
SymBob(F(z,λ−1))|λ=1=−SymBob(F(z,λ))|λ=1(7.4.55)
and010−1
X=−10X10forallX∈su(2),(7.4.56)
computeewJ(ψ◦γ˜S)=SymBob(F(γ˜S(z),λ))|λ=1=SymBob(ρ(λ)F(z,λ−1)(ρ(λ))−1)|λ=1
=ρ(λ)SymBob(F(z,λ−1))|λ=1(ρ(λ))−1=−ρ(λ)SymBob(F(z,λ))|λ=1(ρ(λ))−1=−ρ(λ)J(ψ)(ρ(λ))−1
=−ρ(λ)−0101J(ψ)10−01(ρ(λ))−1=−i00iJ(ψ)−0i−0i.(7.4.57)
Usingtheidentity
−i00iX−0i−0i=(J◦AS◦J−1)(X)forallX∈su(2),(7.4.58)
where−100
AS=010.(7.4.59)
100fromtheproofofcorollary7.7,weswitchintotheR3modelandobtainψ◦γ˜S=AS◦ψ.Asγ˜S(M˜)=M˜,
yieldsthisψ(M˜)=AS(ψ(M˜)).(7.4.60)
Thismeansthatψ(andthusalsoφ)is3symmetricwithrespecttotheEuclideanmotionAS∈Iso(R3)
definingthereflectioninthey-z-planeinR.Thus,φisnecessarilyreflectionallysymmetricwithrespect
tothetrinoidplane.(Inviewoftheorem4.31,whichlistsallpossibletrinoidsymmetries,onlythe
anreflectiontiholomorphicinthetrinoimappingdγ˜planeS:shoM˜ws→theM˜bkehaeepingviourallofAthreeS.Intrinoidparticular,endsfixed.ASisThassous,ciwateedinferwiththatthethebi-
y-z-planeinR3coincideswiththetrinoidplane,and3thatφisreflectionallysymmetricwithrespectto
thetrinoidplane,coincidingwiththey-z-planeinR.)
130

8Reflectionalsymmetrywithrespecttoatrinoidnormalplane
Definition8.1Inthissection,weconsiderthepossibletrinoidsymmetrieswithrespecttothe(orientationreversing)
reflectionsS0,S1andS∞inR3thatfixoneaxisofthetrinoidwhileinterchangingtheothertwo,i.e.
withrespecttothereflectionsinsometrinoidnormalplanesE0,E1andE∞alongthetrinoidaxesA0,
A1andA∞,respectively.Moreprecisely(cf.theorem4.31),wedenotebyS0,S1andS∞theorientation
reversingEuclideanmotionsinR3whichpermutethetrioidendsaccordingtothepermutations(1∞),
(0∞)and(01)oftheset{0,1,∞},respectively.
Recallthat,thoughthereexistaprioripossiblyseveraltrinoidnormalplanesofφalongeachtrinoid
axisAj,thetrinoidnormalplanesEjweconsiderareuniquelydeterminedbytherespectivesymmetry
Sj.Throughoutthissection,wewill-byaslightabuseofnotation-speakofthetrinoidnormalplane
Ej,whichistheplaneofreflectionofthetrinoidsymmetrySj,simplyasofthetrinoidnormalplaneof
φ(alongthetrinoidaxisAj).
Definition8.1.LetM=C\{0,1}3andφ:M→R3beatrinoidwithproperlyembeddedannularends.
LetM˜=Handψ=φ◦π:M˜→RtheconformalCMC-immersionassociatedwithφviatheuniversal
coveringπ:M˜→Mgivenin(3.2.2).Then,ifφ(or,equivalently,ψ)issymmetricwithrespecttothe
reflectionSlinthetrinoidnormalplaneEl,i.e.if
Sl(φ(M))=φ(M),Sl(ψ(M˜))=ψ(M˜),(8.1.1)
φiscalledreflectionallysymmetricwithrespecttothetrinoidnormalplaneEl.
Again,weareinterestedintranslatingthesymmetryproperty(8.1.1)intoconstraintsonthemon-
odromymatricesassociatedwiththeextendedframeFofψ.

8.2Implicationsofreflectionalsymmetrywithrespecttoatrinoidnormal
planeAsadirectconsequenceofdefinition8.1,westatethefollowinglemma:
Lemma8.2.LetM=C\{0,1}andφ:M→R3beatrinoidwithproperlyembeddedannu-
larendsproducedfromatrinoidpotentialηasintheorem3.14.DenotebyD0,D1,D∞thecorre-
spondingDelaunaymatriceswitheigenvalues±µ0,±µ1,±µ∞,respectively,where,forj∈{0,1,∞},
µj=XjXj=41+wj(λ−λ−1)2andwj=sjtjasinsection3.5.Moreover,denotebyB0,B1and
B∞thetrinoidendsandbyE0,E1andE∞thetrinoidnormalplanes(alongthetrinoidaxes).Then,
holds:lowingfolthe1.IfφisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE0,wehave
µ1=µ∞.(8.2.1)
2.IfφisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE1,wehave
µ0=µ∞.(8.2.2)
3.IfφisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE∞,wehave
µ0=µ1.(8.2.3)
Proof.Wecarryouttheproofforthefirstcase,i.e.supposeφisreflectionallysymmetricwithrespectto
thetrinoidnormalplaneE0.Bytheorem4.31,thecorrespondingsymmetryS0preservesthetrinoidend
B0,whileitmapsthetrinoidendsB1andB∞ontoeachother.ThismeansthattheasymptoticDelaunay
DelaunasurfacesysassourfacesciatedonlywithdifferthebyendsaatrigidB1motionandBon∞R3are.Inmappedparticular,ontothiseachimpliesotherasthatwtheell.correspHence,ondingthese
DelaunaymatricesD1andD∞possessthesameeigenvalues,i.e.µ1=µ∞.
Theothertwocasesareprovedanalogously.

131

LetM=C\{0,1},φ:M→R3be˜atrinoidwithprop˜erlyemb3eddedannularendsandψthe
associatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotestheuniversal
coveringM˜→Mgivenin(3.2.2).Supposeφ(or,equivalently,ψ)isreflectionallysymmetricwith
resporienecttationtotheonR3trinoid,wenormalobtainbplaneythEelor,emand4.9denoteapairtheofcorrespbiholomorphicondingsymmetrymappings,byγSl.:SinceMS→lMreversesand
lSγ˜Sl:M˜→M˜satisfying
Sl◦φ=φ◦γSl,(8.2.4)
Sl◦ψ=ψ◦γ˜Sl,(8.2.5)
π◦γ˜Sl=γSl◦π.(8.2.6)
ThemappingsγSl,l=0,1,∞,areuniquelydeterminedandexplicitlygivenbylemma4.21:
z¯γS0(z)=z¯−1,(8.2.7)
1γS1(z)=z¯,(8.2.8)
γS∞(z)=1−z¯.(8.2.9)
oftheThemappingsautomorphismγ˜Sl,lgroup=0,1,Aut(∞,M˜/areM)ofuniquelyπ.Thefollodeterminedwinguptolemmacompexplicitlyositionfromstatesvthealidleftcwithhoiceselemforγ˜ents,
lSl=0,1,∞:
˜˜LetLemmaγSl:M8.3.→LMet,Ml==0C,1\,{∞0,,1b}e,Mgiven=Hbyand(8.2.7)π:,M→(8.2.8)Mbandethe(8.2.9)uni,rversalespecctivelyovering.asThen,giventheinfol(3.2.2)lowing.
holds:1.Themappingγ˜S0:M˜→M˜,z¯
γ˜S0(z)=−z¯−1,(8.2.10)
satisfiesandbi-antiholomorphicisπ◦γ˜S0=γS0◦π,(8.2.11)
S0◦ψ=ψ◦γ˜S0.(8.2.12)
2.Themappingγ˜S1:M˜→M˜,γ˜(z)=−z¯−1,(8.2.13)
1Ssatisfiesandbi-antiholomorphicisπ◦γ˜S1=γS1◦π,(8.2.14)
S1◦ψ=ψ◦γ˜S1.(8.2.15)
3.Themappingγ˜S∞:M˜→M˜,
1γ˜S∞(z)=z¯,(8.2.16)
satisfiesandbi-antiholomorphicisπ◦γ˜S∞=γS∞◦π,(8.2.17)
S∞◦ψ=ψ◦γ˜S∞.(8.2.18)
˜˜PrMoreooof.ver,Directbyapplyingcomputationstheshorelationswthatγ˜(3.2.10),Sl,l=(3.2.11)0,1,∞anddefine(3.2.12)bi-anoflemmatiholomorphic3.4,weobtamappingsinforMallz→∈MM˜.
z¯z11
ππ◦γ˜S0(z)=π−z¯−1=πz+1=π1−z+1=1
+1z−11π(z)
=1−π(z+1)=1−1=π(z)−1=γS0◦π(z),(8.2.19)
)z(π132

1π◦γ˜S1(z)=π(−z¯−1)=π(z+1)=π(z)=γS1◦π(z),(8.2.20)
π◦γ˜S∞(z)=π1=π−1=1−π(z)=γS∞◦π(z),(8.2.21)
zz¯i.e.π◦γ˜Sl=γSl◦πforl=0,1,∞.Consequently,
Sl◦ψ=Sl◦φ◦π=φ◦γSl◦π=φ◦π◦γ˜Sl=ψ◦γ˜Sl,(8.2.22)
i.e.Sl◦ψ=ψ◦γ˜Slforl=0,1,∞.
trinoidBystheymabovmetrieseSllemma,,l=w0e,1,ha∞ve,respexplicitlyectively,indeterminedthesenseofmappingstheoremγ˜Sl,l4.9.=0,Th1,us,∞,wecorrespcanapplyondingtotheoremthe
obtainto4.17Theorem8.4.LetM=C\{0,1},φ:M→R3b˜eatrinoidwithproperly˜embe3ddedannularendsand
ψtheassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotesthe
trinoiduniversalcnormaloveringplaneM˜E→.MDenoteasdefinethedcorrinesp(3.2.2)onding.LetφsymmetrybereflebySctionalandlybyγ˜symmetricthewithrespbi-antiholomorphicecttothe
˜˜llSl
extendemappingdfrMame→FM:M˜asso→ciatedΛSU(2)withσcSlorrasespinondingtheortoemψ4.9byandtheoremexplicitly4.5trdefineansformsdinunderlemmaγ˜Sl8.3.asThen,the
F(γ˜Sl(z),λ−1)=MSl(λ)F(z,λ)kSl,γ˜Sl(z),(8.2.23)
whereMSl(λ)denotesanelementofΛSU(2)σ,whichisindependentofz,and
0−z−1
kS0,γ˜S0(z)=0z¯+1−z−1inthecasel=0,(8.2.24)
+1z¯kS1,γ˜S1(z)=−0ii0inthecasel=1,(8.2.25)
z−z−kS∞,γ˜S∞(z)=z¯0inthecasel=∞.(8.2.26)
0z¯Proof.AsSlreversesorientation,weapplythesecondpartoftheorem4.17toobtain
F(γ˜Sl(z),λ−1)=MSl(λ)F(z,λ)kSl,γS˜l(z),(8.2.27)
wheredenotesFan:M˜elemen→tΛSU(2)ofσΛSU(2)denotesσ,whicthehisextendedindependenframetofzcorresp.Moreoondingver,tokSψ,γb˜yisgivtheoremenby4.5andequationMSl:=(4.4.118)Mγ˜Sl
Sllfromlemma4.18.Recallingfromlemma8.3thatγ˜S0(z)=−z¯z¯−1,γ˜R1(z)=−z¯−1andγ˜R∞(z)=z¯1,we
compute1−∂z¯γ˜S0(z)=(z¯+1)2,(8.2.28)
∂z¯˜γS1(z)=−1,(8.2.29)
∂z¯γ˜S∞(z)=z¯−21.(8.2.30)
impliesThis2|∂∂z¯z¯γ˜γ˜SS00((zz))|=−|(z¯z¯++11)|2=−z¯z+−11,(8.2.31)
|∂∂z¯z¯γ˜γ˜SS1((zz))|=−1,(8.2.32)
1∂z¯γ˜S∞(z)|z¯|2−z
|∂z¯γ˜S∞(z)|=−z¯2=z¯.(8.2.33)
andhenceweobtainfrom(4.4.118)theclaimedexplicitformsforkSl,l∈{0,1,∞}.
133

8.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,
whicharereflectionallysymmetricwithrespecttoatrinoidnormalplane
Wenowstudythe(unitary)monodromymatricesMˆ0,Mˆ1,Mˆ∞associatedwithatrinoidwithproperly
embeddedannularendsandwithreflectionalsymmetrywithrespecttooneofthetrinoidnormalplanes.
Ourconsiderationsarebasedontherelationsbetweenthebi-antiholomorphicmappingsγ˜Slassociated
withthesymmetriesSlandthecoveringtransformationsγ˜jonM˜generatingthemonodromymatrices
Mˆj.Recallthelatteronesfromsection3.3:
γ˜0(z)=−2zz+1,˜γ1(z)=z+2,γ˜∞(z)=−2z3z+−12.(8.3.1)
Thecorrespondinginversefunctionsaregivenby
γ˜0−1(z)=z,γ˜1−1(z)=z−2,γ˜∞−1(z)=z+2.(8.3.2)
2z+1−2z−3
Therelationsmentionedabovearestatedinthefollowinglemma.
Lemma8.5.LetM˜=Handγ˜0,γ˜1,γ˜∞:M˜→M˜begivenasabove.
1.Forγ˜S0:M˜→M˜,γ˜S0(z)=−¯zz¯−1,thefollowingidentitieshold:
γ˜S0◦γ˜0=γ˜0−1◦γ˜S0,˜γS0◦γ˜1=γ˜∞−1◦γ˜S0,γ˜S0◦γ˜∞=γ˜1−1◦γ˜S0.(8.3.3)
2.Forγ˜S1:M˜→M˜,γ˜S1(z)=−¯z−1,thefollowingidentitieshold:
γ˜S1◦γ˜0=γ˜∞−1◦γ˜S1,γ˜S1◦γ˜1=γ˜1−1◦γ˜S1,γ˜S1◦γ˜∞=γ˜0−1◦γ˜S1.(8.3.4)
3.Forγ˜S∞:M˜→M˜,γ˜S∞(z)=¯1z,thefollowingidentitieshold:
γ˜S∞◦γ˜0=γ˜1−1◦γ˜S∞,γ˜S∞◦γ˜1=γ˜0−1◦γ˜S∞,γ˜S∞◦γ˜∞=γ˜∞−1◦γ˜S∞.(8.3.5)
Proof.Thisisprovedbydirectcomputation:Letz∈M˜,then
z¯z¯γ˜S0◦γ˜0(z)=−z¯2z¯+1=z¯=−z¯z¯−1=γ˜0−1◦γ˜S0(z),(8.3.6)
−−2z¯+1−1z¯−12−z¯−1+1
z¯+2z¯+2z¯+2
γ˜S0◦γ˜1(z)=−z¯−2−1=−z¯−3=−2−z¯−z¯1−3=γ˜∞−1◦γ˜S0(z),(8.3.7)
1z¯−−−3z¯−2−3¯z−2z¯
γ˜S0◦γ˜∞(z)=3z¯2+2z¯+1−1=z¯+1=−z¯−1−2=γ˜1−1◦γ˜S0(z),(8.3.8)
+1z¯2γ˜S1◦γ˜0(z)=−z¯−1=−z¯+2z¯−1=−z¯−1+2=γ˜∞−1◦γ˜S1(z),(8.3.9)
−2z¯+1−2z¯+12z¯+2−3
γ˜S1◦γ˜1(z)=−(z¯+2)−1=−z¯−3=−z¯−1−2=γ˜1−1◦γ˜S1(z),(8.3.10)
γ˜S1◦γ˜∞(z)=3z¯+2−1=3z¯+2−2z¯−1=−z¯−1=γ˜0−1◦γ˜S1(z),(8.3.11)
2z¯+12z¯+1−2z¯−2+1
γ˜S∞◦γ˜0(z)=−2z¯+1=1−2=γ˜1−1◦γ˜S∞(z),(8.3.12)
z¯z¯1γ˜S∞◦γ˜1(z)=z¯+12=21z¯+1=γ˜0−1◦γ˜S∞(z),(8.3.13)
z¯1γ˜S∞◦γ˜∞(z)=−23z¯z¯+−12=z¯1+2=γ˜∞−1◦γ˜S∞(z).(8.3.14)
−2z¯−3

Inviewofthis,weareabletoprovethefollowingtheorem:

134

(8.3.3)(8.3.4)(8.3.5)(8.3.6)(8.3.7)(8.3.8)(8.3.9)(8.3.10)(8.3.11)(8.3.12)(8.3.13)(8.3.14)

3ψtheTheoremassociate8.6.dLcetMonformal=C\{0,1},CMC-immersionφ:M→onRM˜be=aH,trinoidψ=withφ◦prπop:Merly˜→embRe3,ddedwhereannularπdenotesendsandthe
universalcoveringM˜→Masdefinedin(3.2.2).Letφbereflectionallysymmetricwithrespecttothe
betrinoidtheextendenormaldfrplaneameElasso.ciateDenotedwiththeψcorrbyesptheorondingem4.5.symmetryDenotebybySlMˆ.,FMˆ,urthermorMˆ∈e,letΛSU(2F,:CM˜)→theΛSU(2)unitaryσ
monodromymatrices01∞σ
Mˆj=−cos(2πµj)0110+isin(2πµj)qpjj−qpjj(8.3.15)
associatedwithFasin(4.5.13)by
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),j=0,1,∞,(8.3.16)
whereαj∈{±1}andγ˜jdenotethe˜covering˜transformationsonM˜fromsection3.3.Finally,letγ˜Sl,
belemmathe8.3,andbi-antiholomorphicmappingM→MassociatedwithSlasintheorem4.9andexplicitlygivenin
MSl(λ):=−abSSllabSSll(8.3.17)
thecorrespondingmonodromymatrixofFsatisfying(8.2.23).
1.Inthecasel=0,themonodromymatricessatisfy
MS0(λ)Mˆ0(λ)=(Mˆ0(λ−1))−1MS0(λ),(8.3.18)
MS0(λ)Mˆ1(λ)=(Mˆ∞(λ−1))−1MS0(λ),(8.3.19)
MS0(λ)Mˆ∞(λ)=(Mˆ1(λ−1))−1MS0(λ).(8.3.20)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(8.3.18)to(8.3.20)areequivalent
toaS0p0(λ)+bS0q0(λ)=aS0p0(λ−1)−bS0q0(λ−1),(8.3.21)
aS0q0(λ)−bS0p0(λ)=bS0p0(λ−1)+aS0q0(λ−1),(8.3.22)
aS0p1(λ)+bS0q1(λ)=aS0p∞(λ−1)−bS0q∞(λ−1),(8.3.23)
aS0q1(λ)−bS0p1(λ)=bS0p∞(λ−1)+aS0q∞(λ−1).(8.3.24)
2.Inthecasel=1,themonodromymatricessatisfy
MS1(λ)Mˆ0(λ)=(Mˆ∞(λ−1))−1MS1(λ),(8.3.25)
MS1(λ)Mˆ1(λ)=(Mˆ1(λ−1))−1MS1(λ),(8.3.26)
MS1(λ)Mˆ∞(λ)=(Mˆ0(λ−1))−1MS1(λ).(8.3.27)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(8.3.25)to(8.3.27)areequivalent
toaS1p0(λ)+bS1q0(λ)=aS1p∞(λ−1)−bS1q∞(λ−1),(8.3.28)
aS1q0(λ)−bS1p0(λ)=bS1p∞(λ−1)+aS1q∞(λ−1),(8.3.29)
aS1p1(λ)+bS1q1(λ)=aS1p1(λ−1)−bS1q1(λ−1),(8.3.30)
aS1q1(λ)−bS1p1(λ)=bS1p1(λ−1)+aS1q1(λ−1).(8.3.31)
3.Inthecasel=∞,themonodromymatricessatisfy
MS∞(λ)Mˆ0(λ)=(Mˆ1(λ−1))−1MS∞(λ),(8.3.32)
MS∞(λ)Mˆ1(λ)=(Mˆ0(λ−1))−1MS∞(λ),(8.3.33)
MS∞(λ)Mˆ∞(λ)=(Mˆ∞(λ−1))−1MS∞(λ).(8.3.34)
135

IntermsofthefunctionspjandqjoccurringinMˆj,equations(8.3.32)to(8.3.34)areequivalent
to

aS∞p0(λ)+bS∞q0(λ)=aS∞p1(λ−1)−bS∞q1(λ−1),(8.3.35)
1−aS∞q0(λ)−bS∞p0(λ)=bS∞p1(λ)+aS∞q1(λ−1),(8.3.36)
1−aS∞p∞(λ)+bS∞q∞(λ)=aS∞p∞(λ)−bS∞q∞(λ−1),(8.3.37)
aS∞q∞(λ)−bS∞p∞(λ)=bS∞p∞(λ−1)+aS∞q∞(λ−1).(8.3.38)
Proof.Likeintheproofoftheorem7.5,wemakeuseofthefollowingfact,adirectconsequenceof(8.3.16):
F(γ˜j−1(z),λ)=αjMˆj(λ)−1F(z,λ)(kj(γ˜j−1(z)))−1.(8.3.39)
Weconsidertheproofofthefirstcase:l=0.Combining(8.2.23)fromtheorem8.4,equations(8.3.16)
and(8.3.39)andtheidentities(8.3.3)fromtheabovelemma,weobtain
MS0(λ)α0Mˆ0(λ)F(z,λ)k0(z)kS0,γ˜S0(γ˜0(z))=MS0(λ)F(γ˜0(z),λ)kS0,γ˜S0(γ˜0(z))
=F(γ˜S0(γ˜0(z)),λ−1)=F(γ˜0−1(γ˜S0(z)),λ−1)=α0(Mˆ0(λ−1))−1F(γ˜S0(z),λ−1)(k0(γ˜0−1(γ˜S0(z))))−1
=α0(Mˆ0(λ−1))−1MS0(λ)F(z,λ)kS0,γ˜S0(z)(k0(γ˜0−1(γ˜S0(z))))−1,(8.3.40)

MS0(λ)α1Mˆ1(λ)F(z,λ)k1(z)kS0,γ˜S0(γ˜1(z))=MS0(λ)F(γ˜1(z),λ)kS0,γ˜S0(γ˜1(z))
=F(γ˜S0(γ˜1(z)),λ−1)=F(γ˜∞−1(γ˜S0(z)),λ−1)=α∞(Mˆ∞(λ−1))−1F(γ˜S0(z),λ−1)(k∞(γ˜∞−1(γ˜S0(z))))−1
−1−1−1−1
=α∞(Mˆ∞(λ))MS0(λ)F(z,λ)kS0,γ˜S0(z)(k∞(γ˜∞(γ˜S0(z)))),(8.3.41)
andMS0(λ)α∞Mˆ∞(λ)F(z,λ)k∞(z)kS0,γ˜S0(γ˜∞(z))=MS0(λ)F(γ˜∞(z),λ)kS0,γ˜S0(γ˜∞(z))
=F(˜γS0(γ˜∞(z)),λ−1)=F(γ˜1−1(γ˜S0(z)),λ−1)=α1(Mˆ1(λ−1))−1F(γ˜S0(z),λ−1)(k1(γ˜1−1(γ˜S0(z))))−1
=α1(Mˆ1(λ−1))−1MS0(λ)F(z,λ)kS0,γ˜S(z)(k1(γ˜1−1(γ˜S0(z))))−1.(8.3.42)
0Wecontinuebycomputing(duetotheoccurringcomplexrootsuptosign)
1−2z¯−−2zz+1−1
1−2z0−2z¯z¯+1+10
011−−22zz¯0−−z¯2z+1−1
k0(z)kS0,γ˜S0(γ˜0(z))=z
−2z¯+1+1
(1−2z¯)(−−2z¯z¯+1−1)z¯−1
(1−2z)(−2zz+1+1)0−z+10
=±=±
0(1−2z)(−2zz+1+1)−z+1
(1−2z¯)(−−2z¯z¯+1−1)=z¯−1
0(−z¯−1)(z−1−2z)(z¯−1)
=±0(−z¯−1)(z−1−2z)(z¯−1)0
(z+1)(2z¯−1−2z¯)(z−1)
(z+1)(2z¯−1−2z¯)(z−1)
1−2z−z1
−z−101−2z¯−z¯10
0−z−11−2z−1
=±z¯+1z=±kS0,γ˜S0(z)(k0(γ˜0−1(γ˜S0(z))))−1,(8.3.43)
z¯+101−2z¯z¯−1

136

−z¯z+3−30
0−¯zz+3−3
k1(z)kS0,γ˜S0(γ˜1(z))=
(−z−1)(z¯+3−2z¯−4)(z+3)0
0(−z−1)(z¯+3−2z¯−4)(z+3)
=±(z+1)(z+3−2z−4)(z¯+3)
(z+1)(z+3−2z−4)(z¯+3)
1+2−zz+3−2
−z¯z+1−101+2−z¯z¯+3−20−1−1
0z¯+101+2−zz¯+3−2
=±−z−1−z−2=±kS0,γ˜S0(z)(k∞(γ˜∞(γ˜S0(z))))(8.3.44)
1+2+3z¯and−−3z−2−1
+1z¯21+21+2zz¯0−32z¯z−+12+10
01+2z0−−32z¯z−+12−1
k∞(z)kS0,γ˜S0(γ˜∞(z))=1+2z¯−3z−2
+1(1+2z)(3z+2−2z−1)(2z¯+1)
2z¯+1
(1+2z)(3z+2−2z−1)(2z¯+1)
=±(1+2z¯)(−3z¯−2+2z¯+1)(2z+1)0
0(1+2z¯)(−3z¯−2+2z¯+1)(2z+1)
0−z−1
0−z¯z+1−1
=±z¯+1=±kS0,γ˜S0(z)(k1(γ˜1−1(γ˜S0(z))))−1.(8.3.45)
Combiningtheseresultswiththeequationsabove,weobtain
MS0(λ)Mˆ0(λ)=β0(Mˆ0(λ−1))−1MS0(λ),(8.3.46)
MS0(λ)Mˆ1(λ)=β1α1α∞(Mˆ∞(λ−1))−1MS0(λ),(8.3.47)
MS0(λ)Mˆ∞(λ)=β∞α∞α1(Mˆ1(λ−1))−1MS0(λ).(8.3.48)
withβ0,β1,β∞∈{±1}.Thiscanbereformulatedas
(Mˆ0(λ−1))−1=β0MS0(λ)Mˆ0(λ)(MS0(λ))−1,(8.3.49)
(Mˆ∞(λ−1))−1=β1α1α∞MS0(λ)Mˆ1(λ)(MS0(λ))−1,(8.3.50)
(Mˆ1(λ−1))−1=β∞α∞α1MS0(λ)Mˆ∞(λ)(MS0(λ))−1.(8.3.51)
Comparingtheupperleftentriesaswellasthelowerrightentriesofbothsidesineachoftheseequations,
obtainew−cos(2πµ0)+isin(2πµ0)p0(λ−1)=
β0−cos(2πµ0)+isin(2πµ0)(aS0aS0p0(λ)+aS0bS0q0(λ)+aS0bS0q0(λ)−bS0bS0p0(λ)),(8.3.52)
−cos(2πµ0)−isin(2πµ0)p0(λ−1)=
β0−cos(2πµ0)−isin(2πµ0)(aS0aS0p0(λ)+aS0bS0q0(λ)+aS0bS0q0(λ)−bS0bS0p0(λ)),(8.3.53)
−cos(2πµ∞)+isin(2πµ∞)p∞(λ−1)=
β1α1α∞−cos(2πµ1)+isin(2πµ1)(aS0aS0p1(λ)+aS0bS0q1(λ)+aS0bS0q1(λ)−bS0bS0p1(λ)),(8.3.54)
−cos(2πµ∞)−isin(2πµ∞)p∞(λ−1)=
β1α1α∞−cos(2πµ1)−isin(2πµ1)(aS0aS0p1(λ)+aS0bS0q1(λ)+aS0bS0q1(λ)−bS0bS0p1(λ)),(8.3.55)
137

−cos(2πµ1)+isin(2πµ1)p1(λ−1)=
β∞α∞α1−cos(2πµ∞)+isin(2πµ∞)(aS0aS0p∞(λ)+aS0bS0q∞(λ)+aS0bS0q∞(λ)−bS0bS0p∞(λ)),
(8.3.56)

−cos(2πµ1)−isin(2πµ1)p1(λ−1)=
β∞α∞α1−cos(2πµ∞)−isin(2πµ∞)(aS0aS0p∞(λ)+aS0bS0q∞(λ)+aS0bS0q∞(λ)−bS0bS0p∞(λ)),
(8.3.57)respectively.Bysummingupthefirsttwoequations,weconcludethatβ0necessarilyequals+1.Analo-
gously,bysummingupthenexttwo(resp.thelasttwo)equationsandrecallingthatµ∞=µ1,wededuce
β1α1α∞=+1(resp.β∞α∞α1=+1).Therefore,
MS0(λ)Mˆ0(λ)=(Mˆ0(λ−1))−1MS0(λ),(8.3.58)
MS0(λ)Mˆ1(λ)=(Mˆ∞(λ−1))−1MS0(λ),(8.3.59)
MS0(λ)Mˆ∞(λ)=(Mˆ1(λ−1))−1MS0(λ),(8.3.60)
asclaimed.Notethatinviewof(3.9.32)equation(8.3.20)isimpliedbyequations(8.3.18)and(8.3.19).
Thus,allthreeequationsareequivalenttothescalarreformulationsoftheequations(8.3.18)and(8.3.19),
readhwhic−cos(2πµ0)aS0+isin(2πµ0)(aS0p0(λ)+bS0q0(λ))
=−cos(2πµ0)aS0+isin(2πµ0)(aS0p0(λ−1)−bS0q0(λ−1)),(8.3.61)
−cos(2πµ0)bS0+isin(2πµ0)(aS0q0(λ)−bS0p0(λ))
=−cos(2πµ0)bS0+isin(2πµ0)(bS0p0(λ−1)+aS0q0(λ−1)(8.3.62)
and−cos(2πµ1)aS0+isin(2πµ1)(aS0p1(λ)+bS0q1(λ))
=−cos(2πµ∞)aS0+isin(2πµ∞)(aS0p∞(λ−1)−bS0q∞(λ−1)),(8.3.63)
−cos(2πµ1)bS0+isin(2πµ1)(aS0q1(λ)−bS0p1(λ))
=−cos(2πµ∞)bS0+isin(2πµ∞)(bS0p∞(λ−1)+aS0q∞(λ−1),(8.3.64)
respectively.Astraightforwardsimplificationoftheseequationsyieldstheclaimedonesandfinishesthe
proofforl=0.
Theclaimsinthecasesl=1andl=∞areprovedanalogously.
8.4Normalizedtrinoidswithproperlyembeddedannularends,whichare
refletionallysymmetricwithrespecttoatrinoidnormalplane
Letl∈{0,1,∞}andφ:M→R3beatrinoidwithproperlyembeddedannularends,whichisreflection-
allysymmetricwith˜resp3ecttothetrinoidnormalplaneEl.Moreover,letψ=φ◦πbetheassociated
CMC-immersionM→R.DenotebySλthecorrespondingsymmetryofφ(andψ),i.e.thereflectionin
thetrinoidnormalplaneEl.
Wereviewtheresultsofsection8.3inthespecialcasethattheextendedframeF:M˜→ΛSU(2)σ
associatedwithψasinsection4.2is“normalized”atz∗∈M˜,whichwechooseindependentoflas
ws:follo√z∗=−1+i3∈M˜.(8.4.1)
2The“normalization”ofFisrealizedinformofthepresumptionthat
F(z∗,λ)=I(8.4.2)
138

forallλ∈S1.Moreprecisely(cf.section4.2),thenormalizationF(z∗,λ)=IofFisaconsequenceof
normalizingthe(conformal)CMC-immersionψ,suchthat
1ψ(z∗)=2He3,U(z∗)=G(1),(8.4.3)
whereU∈SO(3)representsthenaturalorthonormalframecorrespondingtoψ,andG(1)isgivenin
(4.2.5).Recallfromsection4.2,thatthisnormalizationofψcorrespondstorotatingandshiftingthe
3the(imagechoiceofthofe)z∗astrinoidaboinveR,(forsuachthattrinoidtheφwithconditionsproperly(8.4.3)embareeddedmet.annItularturnsendouts,(cf.whichcorollaryis8.8),reflectionallythat
trinoidsymmetricinR3,withsucrhespthatectthetoretheflectiontrinoidplanenormalElofSplanelconEl)tainscorrespthezonds-axistoinR3.arrangingthe(imageofthe)
Atrinoidφ:M→R3withproperlyembeddedannularends,whichisreflectionallysymmetricwith
3respassoectciatedtotheconformaltrinoidnormalCMC-immersionplaneElψ:and,M˜in→Maddition,meetsisthe“wellpnormalizationositioned”inconditionsRinthe(8.4.3),senseisthatcalledthea
normalizedtrinoidwithproperlyembeddedannularends,whichisreflectionallysymmetricwithrespect
tothetrinoidnormalplaneE.
Wenowformulateamorelexplicitversionoftheorem8.4:
Theorem8.7.LetM=C\{0,1},φ:M→R˜3beatrinoidwith˜properly3embeddedannularendsandψ
theassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotestheuniversal
ctotheoveringtrinoidM˜→MnormalasdplaneefinedElin.Mor(3.2.2)e.over,Letletl∈z∗{b0e,1,given∞}inandφ(8.4.1)bereflectionallysymmetricwithrespect
√z∗=−1+i3∈M˜,(8.4.4)
2andassumethatψhasbeennormalizedatz∗,suchthatψ(z∗)=21He3andF(z∗,λ)=I,whereF:M˜→
ΛSU(2)symmetryσofdenotesφandthebyγ˜extendethedframecorrbi-antiholomorphicespondingtomappψingbyM˜the→oremM˜4.5.associateDenotedwithbySSlastheincorrtheesporemonding4.9
llSand,accordingtol,explicitlygivenin(8.2.10),(8.2.13)or(8.2.16):
γ˜S0(z)=−z¯z¯−1,γ˜S1(z)=−z¯−1,γ˜S∞(z)=z¯1.(8.4.5)
Then,theextendedframeFtransformsunderγ˜Slas
F(γ˜Sl(z),λ−1)=MSl(λ)F(z,λ)kSl,γS˜l(z)(8.4.6)
wherekSl,γS˜l(z)is,accordingtol,givenin(8.2.24),(8.2.25)or(8.2.26)and
iπMS0(λ)=e−6π0i,(8.4.7)
6e0MS1(λ)=0i−0i,(8.4.8)
iπMS∞(λ)=e6−0π6i.(8.4.9)
e0Inparticular,thematricesMSl,l=0,1,∞,areactuallyindependentofλ.
Proof.Inviewoftheorem8.4,weonlyhavetoprovetheequations(8.4.7),(8.4.8)and(8.4.9).Tothis
computeewend,√γ˜S0(z∗)=1−+1i−√3i−32=z∗,(8.4.10)
√γ˜S1(z∗)=−−1−i3−1=z∗,(8.4.11)
2γ˜S∞(z∗)=−1−2i√3=z∗,(8.4.12)
139

whichshowsthatwehaveforalll∈{0,1,∞}
γ˜Sl(z∗)=z∗.(8.4.13)
Furthermore,F(z∗,λ)=I.Thus,evaluatingequation(8.4.6)atz=z∗yields
I=F(z∗,λ−1)=F(γ˜Sl(z∗),λ−1)=MSl(λ)F(z∗,λ)kSl,γ˜Sl(z∗),(8.4.14)
i.e.1−MSl(λ)=kSl,γS˜l(z∗).(8.4.15)
Inviewofremark4.14(forourdefinitionofthecomplexsquareroot)andequations(8.2.24),(8.2.25)
and(8.2.26),wehave
√√
−11−−ii√3−3+2202−24i30eπ6i0
kS0,γS˜0(z∗)=1−i√√3−2=2−2i√3=0e−π6i,(8.4.16)
0−1−i3+204
0i−kS1,γS˜1(z∗)=0i,(8.4.17)
61−i√√302+2i√3−πi
kS∞,γS˜∞(z∗)=−1−i3√=40√=eπ0i.(8.4.18)
0−11−−ii√3302+24i30e6
Inviewofequation(8.4.15),theclaimedidentities(8.4.7),(8.4.8)and(8.4.9)follow.
Corollary8.8.Weretainthenotationandtheassumptionsoftheorem8.7.Thereflectionplaneofthe
symmetrySlofthenormalizedtrinoidφcontainsthez-axisinR3.
Proof.Applying(thesecondpartof)theorem4.17,weknowthatthemonodromyMSl(λ)explicitlygiven
10intheorem8.7satisfiesatλ=1
MSl(1)=±ASl−10,(8.4.19)
whereASl∈SU(2)denotestheconjugationmatrixrealizingtheorthogonalpartASlofthesymmetrySl
inthesu(2)-model.Inviewoftheequations(8.4.7),(8.4.8)and(8.4.9),thisyields
iπ−AS0=±0πi−e6,(8.4.20)
06ei0−AS1=±−i0,(8.4.21)
0−eπ6i
AS∞=±e−π6i0.(8.4.22)
RecallingthatASlandASlarelinkedviatheLieAlgebraisomorphismJ:R3→su(2)definedin(3.4.3)
i.e.(3.4.8),inas(J◦ASl◦J−1)(X)=−ASlXAS−l1forallX∈su(2),(8.4.23)
weobtainbyadirectcomputationthat
√√21230
100AS0=23−210,(8.4.24)
−100
AS1=010,(8.4.25)
001√
21√−230
100AS∞=−23−210.(8.4.26)
140

3RTh3,us,Re3for.alllConsequen∈{0,1tly,,∞}for,AallSll∈{defines0,1,a∞},reflectionthe(insymmetryR),Slwhoseofthereflectionnormalizedplanecontrinoidtainsφistheaz-axisreflectionin
insomeplaneinR3,whichisparalleltothez-axis.Asthepointψ(z∗)∈R3(withz∗givenin(8.4.1))
satisfiesSl(ψ(z∗))=ψ(γ˜Sl(z∗))=ψ(z∗),(8.4.27)
itliesinthereflectionplaneofSl.Sincebyass3umptionwehaveψ(z∗)=21He3,weinferthatthereflection
planeofSlactuallycontainsthez-axisinR.
Applyingthetheorems8.6and8.7,weobtainthefollowingresult:
Theorem8.9.LetM=C\{0,1},φ:M→R3b˜eatrinoidwithproperly˜embe3ddedannularendsand
ψtheassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotesthe
universalcoveringM˜→Masdefinedin(3.2.2).Letl∈{0,1,∞}andφbereflectionallysymmetric
bewithenrespnormectalizetodtheatz∗,trinoidsuchnormalthatψ(z∗plane)=El1.e3Moreandover,F(z∗let,λz)∗=beI,givenwhereinF:(8.4.1)M˜→andΛSU(2)assumeσthatdenotesψthehas
extendedframecorrespondingtoψbytheor2Hem4.5.
1.Inthecasel=0,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(8.3.16)satisfyˆequations(8.3.18)to(8.3.20)fromtheorem8.6.Intermsofthefunctions
pjandqjoccurringinMj,theseequationsareequivalentto
p0(λ)=p0(λ−1),(8.4.28)
q0(λ)=eπ3iq0(λ−1),(8.4.29)
p1(λ)=p∞(λ−1),(8.4.30)
q1(λ)=eπ3iq∞(λ−1).(8.4.31)
2.Inthecasel=1,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(8.3.16)satisfyˆequations(8.3.25)to(8.3.27)fromtheorem8.6.Intermsofthefunctions
pjandqjoccurringinMj,theseequationsareequivalentto
p0(λ)=p∞(λ−1),(8.4.32)
q0(λ)=−q∞(λ−1),(8.4.33)
p1(λ)=p1(λ−1),(8.4.34)
q1(λ)=−q1(λ−1).(8.4.35)
3.Inthecasel=∞,theunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associatedwith
Fasin(8.3.16)satisfyˆequations(8.3.32)to(8.3.34)fromtheorem8.6.Intermsofthefunctions
pjandqjoccurringinMj,theseequationsareequivalentto
p0(λ)=p1(λ−1),(8.4.36)
q0(λ)=e−π3iq1(λ−1),(8.4.37)
p∞(λ)=p∞(λ−1),(8.4.38)
q∞(λ)=e−π3iq∞(λ−1).(8.4.39)
Proof.Keepinginmindthatbytheorem8.7
−π6i
MS0(λ)=−abSS00abSS00=e0eπ06i,(8.4.40)
aS1bS1i0
MS1(λ)=−bS1aS10−i,(8.4.41)
π6i
MS∞(λ)=−abSS∞∞abSS∞∞=e0e−0π6i,(8.4.42)
theclaimedidentitiesfollowdirectlyfromtheorem8.6.
141

M˜→TheoremΛSU(2)σ8.9ofadescribtrinoidestheφ:M(unitary)→R3monowithdpropromeyrlyemmatricesbeddedassoannciatedularwithends,thewhichextendedisframereflectionallyF:
symmetricwithrespecttothetrinoidnormalplaneElforsomel∈{0,1,∞},andwhichhasbeen
normalizedsuchthatF(z∗)=Iandψ(z∗)=21He3,wherez∗∈M˜isgivenin(8.4.1)andψdenotesthe
conformalCMC-immersionM˜→R3correspondingtoφ.Itturnsoutthat,inthissetting,wecanalso
provsatisfyingetheFcon(zv∗)erse=Iatresult:z∗A∈M˜trinoidfromφwith(8.4.1)propanderlycorreemspbeddedondingannmonoulardromendsyandmatriceswithoftheextendedformframegivenF
intheorem8.9isnecessarilyreflectionallysymmetricwithrespecttothetrinoidnormalplaneEl.This
resultisformulatedinthefollowingtheorem.
Theorem8.10.Letηbea(standardized)trinoidpotentialassociatedwiththreeoff-diagonalDelaunay3
trinoidmatriceswiDth0,prDop1,erlyD∞embewithddedeigenvaluesannularends±µ0,on±Mµ1=Cand\{±0,µ1∞},rgenerespeatedctively.byηviaDenotethelobyopφgr:oupM→methoRd.a
Moreover,letF:M˜→ΛSU(2)σbetheextendedframeassociatedwiththemappingψ=φ◦πbytheorem
4.5,satisfyingF(z∗)=Iatz∗∈M˜givenin(8.4.1).
1.Letµ1=µ∞.AssumetheunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associated
withFareoftheform
Mˆj=−cos(2πµj)I−isin(2πµj)pj(λ)qj(λ),(8.4.43)
qj(λ)−pj(λ)
withfunctionspjandqjsatisfying(3.9.51)and(3.9.50)and,additionally,
p0(λ)=p0(λ−1),(8.4.44)
q0(λ)=eπ3iq0(λ−1),(8.4.45)
p1(λ)=p∞(λ−1),(8.4.46)
q1(λ)=eπ3iq∞(λ−1).(8.4.47)
Then,φisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE0.
2.Letµ0=µ∞.AssumetheunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associated
withFareoftheform
Mˆj=−cos(2πµj)I−isin(2πµj)qpj((λλ))−qpj((λλ)),(8.4.48)
jjwithfunctionspjandqjsatisfying(3.9.51)and(3.9.50)and,additionally,
p0(λ)=p∞(λ−1),(8.4.49)
q0(λ)=−q∞(λ−1),(8.4.50)
p1(λ)=p1(λ−1),(8.4.51)
q1(λ)=−q1(λ−1).(8.4.52)
Then,φisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE1.
3.Letµ0=µ1.AssumetheunitarymonodromymatricesMˆj∈ΛSU(2,C)σ,j=0,1,∞,associated
withFareoftheform
Mˆj=−cos(2πµj)I−isin(2πµj)qpjj((λλ))−qpjj((λλ)),(8.4.53)
withfunctionspjandqjsatisfying(3.9.51)and(3.9.50)and,additionally,
p0(λ)=p1(λ−1),(8.4.54)
q0(λ)=e−π3iq1(λ−1),(8.4.55)
p∞(λ)=p∞(λ−1),(8.4.56)
q∞(λ)=e−π3iq∞(λ−1).(8.4.57)
Then,φisreflectionallysymmetricwithrespecttothetrinoidnormalplaneE∞.
142

Prtheoof.first,Wesecond,startbythirdcaseconsideringwiththle=sp0,lecial=1,forml=of∞,theprespotenectivtialelyη,inandeachdenoteofthethethreecorrespcases.ondingWepassootenciatetial
byη0,η1,η∞,respectively.
Inthefirstcase(l=0)wehaveµ1=µ∞andthus(cf.section3.6)
0λ−1
η0=−λQ0(z,λ)0dz,(8.4.58)

whereQ0(z,λ)=b0(λ)+b1(λ)+b0(λ)−b0(λ)
z2(z−1)2zz−1
=b0(λ)(z−1)2+b1(λ)z2−b0(λ)z(z−1)=b0(λ)(1−z)+b1(λ)z2(8.4.59)
z2(z−1)2z2(z−1)2
andbj(λ)=1−(µj(λ))2forj=0,1.Consideringthebi-antiholomorphicmappingγS:M→Mdefined
byz→γS0(z4):=z¯−z¯1andthefunctionh0:M→C\{0},z→h0(z)=−i(z¯−1),we0compute
2z¯1−Q0(γS0(z),λ)=b0(λ)(z¯−z¯12)+b11(λ)(z¯−1)2=(z¯−1)4b0(λ)(12−z¯)+b21(λ)z¯2=(h0(z))4Q0(z,λ−1),
(z¯−1)2(z¯−1)2z¯(z¯−1)
(8.4.60)whereweusedthefactthatforλ∈S1theidentity
bj(λ−1)=bj(λ)forj=0,1,∞(8.4.61)
holds.Recallingfromlemma4.21thatγS0correspondstothepermutationσ=(1∞)oftheset{0,1,∞},
weapplylemma4.25toinferthatη0transformsunderγS0as
γS∗0η0(z,λ)=η0(z,λ−1)#W+,0,(8.4.62)
whereh0(z)0
W+,0=W+,0(z,λ)=−λ∂z¯h0(z)(h0(z))−1.(8.4.63)
Analogously,inthesecondcase(l=1)wehaveµ0=µ∞andthus(cf.section3.6)
0λ−1
η1=−λQ1(z,λ)0dz,(8.4.64)

whereQ1(z,λ)=b0(λ)+b1(λ)+b1(λ)−b1(λ)
z2(z−1)2zz−1
b0(λ)(z−1)2+b1(λ)z2−b1(λ)z(z−1)b0(λ)(z−1)2+b1(λ)z
=z2(z−1)2=z2(z−1)2(8.4.65)
andbj(λ)=41−(µj(λ))2forj=0,1.Consideringthebi-antiholomorphicmappingγS1:M→Mdefined
byz→γS1(z):=z¯1andthefunctionh1:M→C\{0},z→h1(z)=−iz¯,wecompute
(1−z¯)212
Q(γ(z),λ)=b0(λ)z¯2+b1(λ)z¯=z¯4b0(λ)(z¯−1)+b1(λ)z¯=(h(z))4Q(z,λ−1),(8.4.66)
1S11(1−z¯)2z¯2(z¯−1)211
z¯2z¯2
wherewehaveagainusedtheidentity(8.4.61).Recallingfromlemma4.21thatγS1correspondstothe
permutationσ=(0∞)oftheset{0,1,∞},weapplylemma4.25toinferthatη1transformsunderγS1
asγS∗1η1(z,λ)=η1(z,λ−1)#W+,1,(8.4.67)
whereh1(z)0
W+,1=W+,1(z,λ)=−λ∂z¯h1(z)(h1(z))−1.(8.4.68)

143

Finally,inthethirdcase(l=∞)wehaveµ0=µ1andthus(cf.section3.6)
1−η∞=−λQ∞0(z,λ)λ0dz,(8.4.69)
whereQ∞(z,λ)=b0(λ)+b0(λ)+c0(λ)−c0(λ)=β0(λ)(z−1)2+b0(λ)z2−c0(λ)z(z−1),(8.4.70)
z2(z−1)2zz−1z2(z−1)2
21bj(λmapping)=4γS−(µ:0(Mλ))→forMj=defined0,∞byandzc→0(λγ)S=(z2)b0(:=λ)1−−b∞z¯(λand).theConsideringfunctiontheh∞:bi-anM→Ctiholomorphic\{0},
∞∞z→h∞(z)=−i,wecompute
Q∞(γS(z),λ)=β0(λ)z¯2+b0(λ)(z¯−1)2−c0(λ)(1−z¯)(−z¯)=(h∞(z))4Q∞(z,λ−1),(8.4.71)
∞(z¯−1)2(z¯)2
wherewehaveused(8.4.61)togetherwiththeidentity
c0(λ−1)=c0(λ)forallλ∈S1.(8.4.72)
applyRecallinglemmafrom4.25lemtomainfer4.21thatthatη∞γS∞transformscorrespondsundertoγSthepasermutationσ=(01)oftheset{0,1,∞},we
∞γS∗∞η∞(z,λ)=η∞(z,λ−1)#W+,∞,(8.4.73)
whereh(z)0
W+,∞=W+,∞(z,λ)=−λ∂z¯∞h∞(z)(h∞(z))−1.(8.4.74)
Altogether,wehaveforalll∈{0,1,∞}therelation
γS∗lηl(z,λ)=ηl(z,λ−1)#W+,l(z,λ).(8.4.75)
Applyingthepullbackconstructionwithrespecttothecoveringmappingπ:M˜→Mto(8.4.75),we
obtainπ∗(γS∗lηl(z,λ))=π∗(ηl(z,λ−1)#W+,l)=η˜l(z,λ−1)#W˜+,l,(8.4.76)
∗Wwhere◦η˜πl.=πMoreoηlver,denotesrecallthethatpullbacthekbi-anpotentialtiholomorphicofthemaptrinoidpingspγ˜oten:tialM˜ηl→(cf.M˜,section2.3)andW˜+,l=
,l+lSγ˜S0:z→−z¯z¯−1,γ˜S1:z→−z¯−1,γ˜S∞:z→z¯1(8.4.77)
fromlemma8.3satisfyγSl◦π=π◦γ˜Sl.Thus,thelefthandsideof(8.4.76)canbetransformedas
ws:follo1−π∗(γS∗lηl)=π∗−λQl(γ0Sl(z),λ)λ0dγSl(z)
0λ−10λ−1
=−λQl((γSl◦π)(z),λ)0d(γSl◦π)(z)=−λQl((π◦γ˜Sl)(z),λ)0d(π◦γ˜Sl)(z)
∗0λ−1∗∗∗
=γ˜Sl−λQl(π(z),λ)0dπ(z)=γ˜Sl(πηl)=γ˜Slη˜l.(8.4.78)
yields(8.4.76)Altogether,γ˜S∗lη˜l(z,λ)=η˜l(z,λ−1)#W˜+,l.(8.4.79)
solutionΨConsideringl=FBthe+,ltoextendedthedifferenframetialFassoequationciateddΨlwith=Ψltheη˜l.Notetrinoidthatφ,wΨelpobtainossesses(forthel∈same{0,1,(unitary)∞})a
monodromymatricesasFatthesingularitiesofthepotentialηl,namelyMˆ0,Mˆ1andMˆ∞.
(γ˜∗Ψ)(Naturallyγ˜∗η˜,),thewhichmappiinngviewγ˜S∗oflΨl=(8.4.79)Ψl◦γ˜readsSlasdefinesasolutiontothedifferentialequationd(γ˜S∗lΨl)=
SllSll
d(γ˜S∗lΨl(z,λ))=(γ˜S∗lΨl(z,λ))(η˜l(z,λ−1)#W˜+,l).(8.4.80)
144

SincethisdifferentialequationisalsosolvedbythemappingΨl(z,λ−1)W˜+,l,i.e.
d(Ψl(z,λ−1)W˜+,l)=(Ψl(z,λ−1)W˜+,l)(η˜l(z,λ−1)#W˜+,l),(8.4.81)
themappingsγ˜S∗lΨ(z,λ)andΨ(z,λ−1)W˜+,lonlydifferbyaλ-dependentmatrixρl=ρl(λ):
γ˜S∗lΨl(z,λ)=ρlΨl(z,λ−1)W˜+,l.(8.4.82)
Considerthecasel=0.Applyingtherelationγ˜S0◦γ˜0=γ˜0−1◦˜γS0from(8.3.3),involvingthecovering
transformationγ˜0onM˜asgiveninsection3.3,wecompute
ρ0(λ)Mˆ0(λ−1)Ψ0(z,λ−1)W˜+,0(γ˜0(z),λ)=ρ0(λ)Ψ0(γ˜0(z),λ−1)W˜+,0(γ˜0(z),λ)=γS˜0∗Ψ0(γ˜0(z),λ)
=Ψ0((γ˜S0◦γ˜0)(z),λ)=Ψ0((γ˜0−1◦γ˜S0)(z),λ)=(Mˆ0(λ))−1Ψ0(γ˜S0(z),λ)
=(Mˆ0(λ))−1(γ˜S∗0Ψ0(z,λ))=(Mˆ0(λ))−1ρ0(λ)Ψ0(z,λ−1)W˜+,0(z,λ),(8.4.83)
wherewehavemadeuseoftheidentity
Ψ0(γ˜0−1(z),λ)=(Mˆ0(λ))−1Ψ0(z,λ),(8.4.84)
whichisadirectconsequenceoftherelation
Ψ0(z,λ)=Ψ0(γ˜0(γ˜0−1(z)),λ)=Mˆ0(λ)Ψ0(γ˜0−1(z),λ).(8.4.85)
˜tozAs),W˜W++,,00isdefinesanthetiholomorphicpullbackonofM˜theandmappingthereforeW+,0do,eswhicnothispickanupantiholomorphicymonoondromMy(withunderγ˜resp0,ecti.e.
W˜+,0(γ˜0(z),λ)=W˜+,0(z,λ).Thus,weconcludethat
ρ0(λ)Mˆ0(λ−1)=(Mˆ0(λ))−1ρ0(λ).(8.4.86)
Analogously,applyingγ˜S0◦γ˜1=γ˜∞−1◦γ˜S0from(8.3.3),wehave
ρ0(λ)Mˆ1(λ−1)Ψ0(z,λ−1)W˜+,0(γ˜1(z),λ)=ρ0(λ)Ψ0(γ˜1(z),λ−1)W˜+,0(γ˜1(z),λ)=γS˜0∗Ψ0(γ˜1(z),λ)
=Ψ0((γ˜S0◦γ˜1)(z),λ)=Ψ0((γ˜∞−1◦γ˜S0)(z),λ)=(Mˆ∞(λ))−1Ψ0(γ˜S0(z),λ)
=(Mˆ∞(λ))−1(γ˜S∗0Ψ0(z,λ))=(Mˆ∞(λ))−1ρ0(λ)Ψ0(z,λ−1)W˜+,0(z,λ).(8.4.87)
Usingtheanti-holomorphicityofW˜+,0onM˜,weknowthatW˜+,0(γ˜1(z),λ)=W˜+,0(z,λ),whichyields
ρ0(λ)Mˆ1(λ−1)=(Mˆ∞(λ))−1ρ0(λ).(8.4.88)
Settingρ0(λ)=ca00((λλ))db00((λλ)),(8.4.89)
wherea0,b0,c0andd0definecomplexvaluedfunctionsofλsatisfyinga0(λ)d0(λ)−b0(λ)c0(λ)=1,and
comparingtheupperleftentries(resp.theupperrightentries)ofρ0(λ)Mˆ0(λ−1)and(Mˆ0(λ))−1ρ0(λ),we
obtaina0(λ)p0(λ−1)+b0(λ)q0(λ−1)=a0(λ)p0(λ)+c0(λ)q0(λ),(8.4.90)
a0(λ)q0(λ−1)−b0(λ)p0(λ−1)=b0(λ)p0(λ)+d0(λ)q0(λ).(8.4.91)
Inviewof(3.9.50)andtheassumption,theseequationssimplifyinto
b0(λ)e−π3iq0(λ)=c0(λ)q0(λ),(8.4.92)
(a0(λ)eπ3i−d0(λ))q0(λ)=2b0(λ)p0(λ).(8.4.93)
ˆSimilarly−1,comparingtheupperleftentries(resp.theupperrightentries)ofρ0(λ)Mˆ1(λ−1)and
(M∞(λ))ρ0(λ),weinfer(byusingµ1=µ∞)that
a0(λ)p1(λ−1)+b0(λ)q1(λ−1)=a0(λ)p∞(λ)+c0(λ)q∞(λ),(8.4.94)
a0(λ)q1(λ−1)−b0(λ)p1(λ−1)=b0(λ)p∞(λ)+d0(λ)q∞(λ).(8.4.95)
145

Inviewof(3.9.50)andtheassumption,theseequationssimplifyinto
b0(λ)e−π3iq∞(λ)=c0(λ)q∞(λ),(8.4.96)
(a0(λ)eπ3i−d0(λ))q∞(λ)=2b0(λ)p∞(λ).(8.4.97)
Together,wehaveforj=0,∞:
b0(λ)e−π3iqj(λ)=c0(λ)qj(λ),(8.4.98)
(a0(λ)eπ3i−d0(λ))qj(λ)=2b0(λ)pj(λ).(8.4.99)
Sinceingeneral(i.e.forallλinS1excludingafinitesubsetofS1)pj,qj=0,wecansolveforc0(λ)and
b0(λ),respectively:
c0(λ)=e−3piib0(λ)qj(λ)qj(λ)−1,(8.4.100)
piib0(λ)=21(e3a0(λ)+d0(λ))qj(λ)(pj(λ))−1.(8.4.101)
again)(3.9.50)(usingyieldsThis1=a0(λ)d0(λ)−b0(λ)c0(λ)=a0(λ)d0(λ)−41qj(λ)qj(λ)(pj(λ))−2(e3piia0(λ)−d0(λ))(a0(λ)−e−3piid0(λ))
=a0(λ)d0(λ)−1(pj(λ))−2(e6piia0(λ)−e−6piid0(λ))2+1(e6piia0(λ)−e−6piid0(λ))2
4piipii4piipii
=−41(pj(λ))−2(e6a0(λ)−e−6d0(λ))2+41(e6a0(λ)+e−6d0(λ))2,(8.4.102)
,tlyalenequivor,(pj(λ))2(4−(e6piia0(λ)+e−6piid0(λ))2)=−(e6piia0(λ)−e−6piid0(λ))2.(8.4.103)
Assumenowthat,ingeneral,4−(e6piia0(λ)+e−6piid0(λ))2=0(i.e.4−(e6piia0(λ)+e−6piid0(λ))2=0
foratmostfinitelymanyλ∈S1).Weinferthat
piipii22(e6a0(λ)−e−6d0(λ))2
(p0(λ))=(p∞(λ))=−4−(e6piia0(λ)+e−6piid0(λ))2(8.4.104)
forallbut(atmost)finitelymanyλ∈S1andthus
p0(λ)=αp∞(λ)(8.4.105)
forsomeα∈{±1}andallbut(atmost)finitelymanyλ∈S1.Consequently,by(8.4.101),thisimplies
q0(λ)=αq∞(λ),q0(λ)=αq∞(λ)(8.4.106)
usthandp0(λ)p∞(λ)+q0(λ)q∞(λ)+q0(λ)q∞(λ)=α((p0(λ))2+q0(λ)q0(λ))=α(8.4.107)
2forallbut(atmost)finitelymanyλ∈S1,whichinviewofremark3.56clearlyisacontradictionto
equation(3.9.51).piiTherefore,wpiieconcludethat4−(e6piia0(λ)+e−6piid0(λ))2=0forallλ∈S1and
(bypii(8.4.103))(e6piia0(λ)−e−6d0(λ))2=0forallλ∈S1.Together,theserelationsyielda0(λ)=
e−3d0(λ)=±e−6and(by(8.4.101)and(8.4.100))b0(λ)=c0(λ)=0.Thus,
pii6−ρ0(λ)=±e0e06pii,(8.4.108)
inparticularρ0(λ)∈ΛSU(2)σ∩Λ+SL(2,C)σ.

146

1−andWγ˜e◦proγ˜ceed=γ˜−1◦analogouslyγ˜frominthe(8.3.4),casewsel=deduce1andinlthe=∞case:l=Applying1thetheidentitiesrelationsγ˜S1◦γ˜0=γ˜∞◦γ˜S1
S111S1
ρ1(λ)Mˆ0(λ−1)=(Mˆ∞(λ))−1ρ1(λ),(8.4.109)
ρ1(λ)Mˆ1(λ−1)=(Mˆ1(λ))−1ρ1(λ),(8.4.110)
which(analogouslyasabove)inviewof(3.9.50)andtheassumptiontranslateinto
−b1(λ)q1(λ)=c1(λ)q1(λ),(8.4.111)
(−a1(λ)−d1(λ))q1(λ)=2b1(λ)p1(λ),(8.4.112)
−b1(λ)q∞(λ)=c1(λ)q∞(λ),(8.4.113)
(−a1(λ)−d1(λ))q∞(λ)=2b1(λ)p∞(λ),(8.4.114)
wherea1,b1,c1andd1definethecomplexvaluedfunctionsofλsatisfyinga1(λ)d1(λ)−b1(λ)c1(λ)=1
andoccurringin
ρ1(λ)=ca1((λλ))db1((λλ)).(8.4.115)
11Applyingtheargumentasinthecasel=0(basicallyonlyadjustingindices)weconcludethat
ρ1(λ)=±0i−0i,(8.4.116)
inparticularρ1(λ)∈ΛSU(2)σ∩Λ+SL(2,C)σ.
Applyingtherelationsγ˜S∞◦γ˜0=γ˜1−1◦γ˜S∞andγ˜S∞◦γ˜∞=γ˜∞−1◦γ˜S∞from(8.3.5),wededucein
thecasel=∞theidentities
ρ∞(λ)Mˆ0(λ−1)=(Mˆ1(λ))−1ρ∞(λ),(8.4.117)
ρ∞(λ)Mˆ∞(λ−1)=(Mˆ∞(λ))−1ρ∞(λ),(8.4.118)
which(analogouslyasabove)inviewof(3.9.50)andtheassumptiontranslateinto
b∞(λ)eπ3iq∞(λ)=c∞(λ)q∞(λ),(8.4.119)
(a∞(λ)e−π3i−d∞(λ))q∞(λ)=2b∞(λ)p∞(λ),(8.4.120)
iπb∞(λ)e3q1(λ)=c∞(λ)q1(λ),(8.4.121)
(a∞(λ)e−π3i−d∞(λ))q1(λ)=2b∞(λ)p1(λ),(8.4.122)
wherea∞,b∞,c∞andd∞definetheλ-dependent,complexvaluedfunctionssatisfyinga∞(λ)d∞(λ)−
b∞(λ)c∞(λ)=1andoccurringin
ρ∞(λ)=ca∞((λλ))db∞((λλ)).(8.4.123)
∞∞Applyingthesameargumentasinthecasel=0(basicallyonlyadjustingindices)weconcludethat
piiρ∞(λ)=±e6−06pii,(8.4.124)
e0inparticularρ∞(λ)∈ΛSU(2)σ∩Λ+SL(2,C)σ.
Altogether,foralll∈{0,1,∞},wehaveρl∈ΛSU(2)σ∩Λ+SL(2,C)σ.Consequently,
(ρlF(z,λ−1)ρl−1)(ρlB+,l(z,λ−1)W˜+,l(z,λ))(8.4.125)
definesanIwasawa-decompositionofρlΨl(z,λ−1)W˜+,l(z,λ)(pointwiseforallz∈M˜)with
ρlF(z,λ−1)ρl−1∈ΛSU(2)σ,ρlB+,l(z,λ−1)W˜+,l(z,λ)∈Λ+SL(2,C)σ(8.4.126)

147

andρlF(z∗,λ−1)ρl−1=I.Therefore,wecanwrite
F(γ˜Sl(z),λ)B+,l(γ˜Sl(z),λ)=γ˜S∗lΨ(z,λ)=ρl(λ)Ψl(z,λ−1)W˜+,l(z,λ)
=(ρl(λ)F(z,λ−1)(ρl(λ))−1)(ρ(λ)B+(z,λ−1)W˜+(z,λ)).(8.4.127)
Thus,γ˜S∗lΨlproduces(bytheloopgroupmethod)ontheonehandthetrinoidSymBob(F(γ˜Sl(z),λ))|λ=1
andontheotherhandthetrinoidSymBob(ρl(λ)F(z,λ−1)(ρl(λ))−1)|λ=1.Consequently,thesetwosur-
facescoincide,and,usingthestraightforwardidentities
SymBob(F(z,λ−1))|λ=1=−SymBob(F(z,λ))|λ=1(8.4.128)
andX=01X0−1forallX∈su(2),(8.4.129)
0101−computeewJ(ψ◦γ˜Sl)=SymBob(F(γ˜Sl(z),λ))|λ=1=SymBob(ρl(λ)F(z,λ−1)(ρl(λ))−1)|λ=1
=ρl(λ)SymBob(F(z,λ−1))|λ=1(ρl(λ))−1=−ρl(λ)SymBob(F(z,λ))|λ=1(ρl(λ))−1
=−ρl(λ)J(ψ)(ρl(λ))−1=−ρl(λ)−0101J(ψ)10−01(ρl(λ))−1.(8.4.130)
Weobtainpiipii
J(ψ◦γ˜S0)=−0piie−6J(ψ)0pii−e−6(8.4.131)
−e60e60
i0i0−inthecasel=0,
J(ψ◦γ˜S0)=−i0J(ψ)−i0(8.4.132)
inthecasel=1and
0e6pii0−e6pii
J(ψ◦γ˜S∞)=−−e−6pii0J(ψ)e−6pii0(8.4.133)
inthecasel=∞.
titiesidentheUsingpiipii
−0piie−6X0pii−e−6=(J◦AS0◦J−1)(X)forallX∈su(2),(8.4.134)
−e60e60
−i00iX−0i−0i=(J◦AS1◦J−1)(X)forallX∈su(2),(8.4.135)
piipii
−−0piie6X−0pii−e6=(J◦AS∞◦J−1)(X)forallX∈su(2),(8.4.136)
−e60e60

where

√31022AS0=√23−210,
100−100
AS1=010,
1001−√30
22AS∞=−2√3−120,
100148

(8.4.134)(8.4.135)(8.4.136)

(8.4.137)(8.4.138)(8.4.139)

fromtheproofofcorollary8.8,weswitchintotheR3modelandobtainψ◦γ˜Sl=ASl◦ψ.Asγ˜Sl(M˜)=M˜,
thisyieldsineachcase(l∈{0,1,∞})

ψ(M˜)=ASl(ψ(M˜)).(8.4.140)
EuclideanThismeansmotionthatA,for∈aIso(givRen3)l∈defining{0,1,a∞},ψreflection(andinthRus3.alsoInφview)isofsymmtheoremetric4.31,withrespwhichectliststotheall
lSbpehaossibleviouroftrinoidthereflectionsymmetries,AwSleobserv(concerningethattheonlypermtheutationreflectionofinthethetrinoidtrinoidends),normalwhichplanecanElbeshoreadwstheoff
ofthetheassotrinoidciatedsinbi-angularities.tiholomorphicThus,φismappingγ˜necessarilySl:M˜→M˜,reflectionallyor,moresymmetricprecisely,withitsprespermectutationtothebehatrinoidviour
normalplaneEl,asclaimed.

149

9Rotoreflectionalsymmetrywithrespecttothetrinoidnormal
Definition9.1Finally,inthissectionwediscusstrinoidsφ:M→R3withproperlyembeddedannularendsonM=
Cˆ\{0,1,∞}whicharesymmetricinthe2πsenseofdefinition4.2withrespecttotherotoreflectionSˆ
composedoftherotationRbytheangle±3aroundthetrinoidnormal(asstudiedinsection5)and
thereflectionSinthe3trinoidplane(asstudiedinsection7),Sˆ=S◦R.Recallthat,inthecasethat
atrEuclideaninoidφ:motionM→Sˆ,Rtherewithexistspropaerlyuniqueembtrinoideddedannplaneularandaendsisuniquesymmetrictrinoidwinormalthrespofφ,ectwhictohtheenablesgiven
ustospeakofthetrinoidplaneandthetrinoidnormalofφ,respectively.)
Sˆreversesorientationandpermutesthetrinoidendsaccordingtothepermutation(01∞)oftheset
{0,1,∞}.Moreover,sincewehave
Sˆ(φ(M))=φ(M)⇐⇒Sˆ−1(φ(M))=φ(M),(9.1.1)
wenoterightawˆaythatagiventrinoidφ:M→R3withproperlyˆ−1embedded−1ann−1ularendsis−1symmetric
withrespecttoSifandonlyifitissymmetric−1withrespectto2πS=R◦S=S◦R,defining
inthesectionrotoreflection5)andthecomposedreflectionoftheSinrotationtheRtrinoidbypthelanean(asglestudied3inaroundsectionthe7).trinoid(Notenormalthe(asfactstudiedthat
S−1=SandR−1◦S=S◦R−1.)Sˆ−1reversesorientationandpermutesthetrinoidendsaccordingto
thepermutation(0∞1)oftheset{0,1,∞}.
Definition˜9.1.LetM=C\˜{0,1}3andφ:M→R3beatrinoidwithproperlyembeddedannularends.
LetM=H˜andψ=φ◦π:M→RtheconformalCMC-immersionassociatedwithφviatheuniversal
coandverinngaπ:normalM→vMectorgivofentheintrinoid(3.2.2).planeLetAE,nb=e{theC+trinoiλn;dλ∈Rnormal.},whereThen,Cifφdenotes(or,theequivalentrinoidtly,cenψ)teris
symmetricwithrespecttotherotoreflectionSˆ,composedoftherotationRbytheangle±23πaround
ˆtheaccordingtrinoidtothenormalpermandtheutationσreflection=(01S∞in)theofthetrinoidset{0,plane,1,∞}S,=S◦R,andpermutingthetrinoidends
Sˆ(φ(M))=φ(M),Sˆ(ψ(M˜))=ψ(M˜),(9.1.2)
or,equivalently,ifφ(or,equivalently,ψ)issymmetricwithrespecttotheinverserotoreflectionSˆ−1,
Sˆ−1(φ(M))=φ(M),Sˆ−1(ψ(M˜))=ψ(M˜),(9.1.3)
φ(orψ)iscalledrotoreflectionallysymmetricwithrespecttothetrinoidnormal.
Again,weareinterestedintranslatingthissymmetrypropertyintofurtherconstraintsonthemon-
odromymatricesassociatedwiththeextendedframeFofψ.
9.2Implicationsofrotoreflectionalsymmetrywithrespecttothetrinoid
normalThefollowingresultisanimmediateconsequenceofdefinition9.1:
3larLemmaendspro9.2.duceLdetfrMom=aCtrinoid\{0,p1}otandentialφη:asMin→theRorembea3.14.trinoidDenotewithbyprDop,erlDy,embDeddethedcorrannu-e-
10∞spondingDelaunaymatriceswitheigenvalues±µ0,±µ1,±µ∞,respectively,where,forj∈{0,1,∞},
µj=XjXj=41+wj(λ−λ−1)2andwj=sjtjasinsection3.5.Then,ifφisrotoreflectionally
symmetricwithrespecttothetrinoidnormal,wehave
1µ:=µ0=µ1=µ∞=4+w(λ−λ−1)2,(9.2.1)
ewherw:=w0=w1=w∞.(9.2.2)

150

Proof.Bydefinition9.1,theendsofatrinoidwithproperlyembeddedannularends,whichisrotore-
Sˆ(resp.flectionallySˆ−1)insymmetrictoeachwithotherrespaccectortodingthetothetrinoidpermnormal,utationareσ=mapp(01ed∞b)y(rtheesp.σcorresp−1=onding(0∞1)).symmetrThisy
well.meansthatHence,thetheseasymptoticDelaunaysurDelaunafaceysonlysurfacesdifferassobyciatedarigidwithmotiontheendsonRare3.Inmappedparticular,ontoeacthishotherimpliesas
thatthecorrespondingDelaunaymatricesDj,j=0,1,∞,(seesection3.5formoredetails)allpossess
thesameeigenvalues.Thisyieldsµ0=µ1=µ∞andallowsfordefiningµ:=µ0=µ1=µ∞.Using
lemmaB.6,weinferthatw0=w1=w∞,whencewgivenin(9.2.2)iswelldefined.Consequently,
µ=41+w(λ−λ−1)2holds.Thisfinishestheproof.
LetM=C\{0,1},φ:M→R3be˜atrinoidwithprop˜erlyemb3eddedannularendsandψthe
associatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotestheuniversal
coveringM˜→Mgivenin(3.2.2).Supposeφ(or,equivalently,ψ)isˆrotoreflectionallyˆsymmetricwith
respecttothetrinoidnormal,anddenotethecorrespondingsymmetrybyS.SinceSreversesorientation
onR3,weobtainbytheorem4.9apairofbi-antiholomorphicmappings,γSˆ:M→Mandγ˜Sˆ:M˜→M˜
satisfyingSˆ◦φ=φ◦γSˆ,(9.2.3)
Sˆ◦ψ=ψ◦γ˜Sˆ,(9.2.4)
π◦γ˜Sˆ=γSˆ◦π.(9.2.5)
Analogously,weobtainforSˆ−1apairofbi-antiholomorphicmappings,γSˆ−1:M→Mandγ˜Sˆ−1:M˜→
˜satisfyingMSˆ−1◦φ=φ◦γSˆ−1,(9.2.6)
Sˆ−1◦ψ=ψ◦γ˜Sˆ−1,(9.2.7)
π◦γ˜Sˆ−1=γSˆ−1◦π.(9.2.8)
ThemappingsγSˆandγSˆ−1areuniquelydeterminedandexplicitlygivenbylemma4.21:
γSˆ(z)=1−1z¯,(9.2.9)
γSˆ−1(z)=z¯−1.(9.2.10)
z¯Themappingsγ˜Sˆandγ˜Sˆ−1are˜uniquelydetermineduptocompositionfromtheleftwithanelement
oftheautomorphismgroupAut(M/M)ofπ.Thefollowinglemmaexplicitlystatesapairofvalidchoices
forγ˜Sˆandγ˜Sˆ−1:
Lemma9.3.LetM=C\{0,1},M˜=Handπ:M˜→Mbetheuniversalcoveringasgivenin(3.2.2).
LetγSˆ:M→MandγSˆ−1:M→Mbegivenby(9.2.9)and(9.2.10),respectively.Then,thefollowing
holds:1.Themappingγ˜Sˆ:M˜→M˜,
˜γSˆ(z)=z¯z¯+1,(9.2.11)
satisfiesandbi-antiholomorphicisπ◦γ˜Sˆ=γSˆ◦π,(9.2.12)
Sˆ◦ψ=ψ◦γ˜Sˆ.(9.2.13)
2.Themappingγ˜Sˆ−1:M˜→M˜,
γ˜Sˆ−1(z)=z¯−11,(9.2.14)
satisfiesandbi-antiholomorphicisπ◦γ˜Sˆ−1=γSˆ−1◦π,(9.2.15)
Sˆ−1◦ψ=ψ◦γ˜Sˆ−1.(9.2.16)
151

Proof.Westartwithprovingthefirstclaim.Adirectcomputationshowsthatγ˜Sˆisabi-antiholomorphic
mappingM˜→M˜.Moreover,byapplyingtherelations(3.2.10),(3.2.11)and(3.2.12)oflemma3.4,we
obtainforallz∈M˜
z¯+11111
π◦γ˜Sˆ(z)=πz¯=π1+z¯=πz¯1=1−π(−z¯)=1−π(z)=γSˆ◦π(z),(9.2.17)
i.e.π◦γ˜Sˆ=γSˆ◦π.Consequently,
Sˆ◦ψ=Sˆ◦φ◦π=φ◦γSˆ◦π=φ◦π◦γ˜Sˆ=ψ◦γ˜Sˆ,(9.2.18)
i.e.Sˆ◦ψ=ψ◦γ˜Sˆ.
Nowweturntothesecondclaim.Adirectcomputationshowsthatγ˜Sˆ−1isabi-antiholomorphic
mappingM˜→M˜.Moreover,byapplyingtherelations(3.2.10),(3.2.11)and(3.2.12)oflemma3.4,we
obtainforallz∈M˜
111π(z)−1
π◦γ˜Sˆ−1(z)=πz¯−1=1−π(1−z¯)=1−π(−z¯)=1−π(z)=π(z)=γSˆ−1◦π(z),(9.2.19)
i.e.π◦γ˜Sˆ−1=γSˆ−1◦π.Consequently,
Sˆ−1◦ψ=Sˆ−1◦φ◦π=φ◦γSˆ−1◦π=φ◦π◦γ˜Sˆ−1=ψ◦γ˜Sˆ−1,(9.2.20)
i.e.Sˆ−1◦ψ=ψ◦γ˜Sˆ−1.
Remark9.4.Notethat,sinceγ˜Sˆ◦γ˜Sˆ−1=γ˜Sˆ−1◦γ˜Sˆ=idforthemappingsγ˜Sˆandγ˜Sˆ−1definedin
(9.2.11)and(9.2.14),respectively,wehave
γ˜Sˆ−1=γ˜S−ˆ1.(9.2.21)
Bytheabovelemma,we−1haveexplicitlydeterminedmappingsγ˜Sˆandγ˜Sˆ−1correspondingtothe
trinoidsymmetriesSˆandSˆ,respectively,inthesenseoftheorem4.9.Thus,wecanapplytheorem
obtainto4.173theTheoremassociated9.5.cLetonformalM=CMC\{0,1}C-immersion,φ:Mon→MR˜=beHa,ψ=trinoidφ◦πwith:M˜pr→opRerly3,embwhereeddeπddenotesannulartheendsuniversalandψ
cnormal.overingM˜Denote→MbyasSˆanddefineSˆd−1inthe(3.2.2)corr.espLetφondingberotorsymmetrieseflectionalplyermutingsymmetricthetrinoidwithrendsespectacctoorthedingtotrinoidthe
permutationsσ=(01∞)andσ−1=(0∞1),respectively.Mor−1eover,denotebyγ˜Sˆandbyγ˜Sˆ−1the
bi-antiholomorphicmappingsM˜→M˜associatedwithSˆandSˆ,respectively,asintheorem4.9and
explicitlygiveninlemma9.3.Then,thefollowingholds:
1.TheextendedframeF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜Sˆas
F(γ˜Sˆ(z),λ−1)=MSˆ(λ)F(z,λ)kSˆ,γ˜Sˆ(z),(9.2.22)
where−z0
kSˆ,γ˜Sˆ(z)=0z¯−z¯z.(9.2.23)
andMSˆ(λ)denotesanelementofΛSU(2)σ,whichisindependentofz.
2.TheextendedframeF:M˜→ΛSU(2)σcorrespondingtoψbytheorem4.5transformsunderγ˜Sˆ−1
asF(γ˜Sˆ−1(z),λ−1)=MSˆ−1(λ)F(z,λ)kSˆ−1,γ˜Sˆ−1(z),(9.2.24)
wherez−1
kSˆ−1,γ˜Sˆ−1(z)=z¯−10z−1,(9.2.25)
01z¯−andMSˆ−1denotesanelementofΛSU(2)σ,whichisindependentofz.

152

Proof.Westartwiththeproofofthefirstpart.Letγ˜(z)=γ˜Sˆ(z)=z¯z¯+1forallz∈M˜=H.(For
convenienceweomittheindexSˆthroughoutthisproof.)AsSˆreversesorientationonR3,weapplythe
secondpartoftheorem4.17toobtain
F(γ˜(z),λ−1)=Mγ˜(λ)F(z,λ)kSˆ,γ˜(z),(9.2.26)
˜anwhereelemenF:tMof→ΛSU(2)ΛSU(2)σ,σwhicdenoteshisithendepextendedendentofframez.kSˆ,γ˜corresp(z)isondinggiventobyψbyequationtheorem4.5(4.4.118)andMfromγ˜lemmdenotesa
computingBy4.18.∂z¯γ˜(z)=−z¯12(9.2.27)
weinferthat∂z¯γ˜(z)|z¯|2z
|∂z¯γ˜(z)|=−z¯2=−z¯(9.2.28)
andthusobtainfrom(4.4.118)
0kSˆ,˜γ(z)=−z¯z0z.(9.2.29)
z¯−Asγ˜=γ˜Sˆ,wedenoteMγ˜byMSˆ.Thisfinishestheproofofequation(9.2.22).
thenTodoneprovetheanalogouslysecond.Weparthaofvethetheorem,wedefineγ˜(z)=γ˜Sˆ−1(z)=z¯−11onM˜=H.Everythingis
1∂z¯γ˜(z)=(z¯−1)2(9.2.30)
andthus∂γ˜(z)|z¯−1|2z−1
|∂z¯z¯γ˜(z)|=(z¯−1)2=z¯−1.(9.2.31)
Formula(4.4.118)fromlemma4.18thenyields
0z−1
kSˆ−1,γ˜(z)=0z¯−1z−1,(9.2.32)
1z¯−andbysettingMSˆ−1(λ):=Mγ˜(λ),thesecondpartoftheorem4.17implies(9.2.24).
9.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,
whicharerotoreflectionallysymmetricwithrespecttothetrinoidnormal
Mˆ0With,Mˆ1the,Mˆ∞resultsassoofciatedthewithpreviousatrinoidsectionwithwearepropnoerlywemablebtoeddeddescribanneulartheends,(unitary)whichismonodromyrotoreflectionallymatrices
symmetricwithrespecttothetrinoidnormal.Asastart,recallfromsection3.3thecoveringtransfor-
mationsγ˜j,j=0,1,∞,onM˜generatingthemonodromymatricesMˆj,j=0,1,∞,respectively:
z−3z−2
γ˜0(z)=−2z+1,γ˜1(z)=z+2,γ˜∞(z)=2z+1.(9.3.1)
Thecorrespondinginversefunctionsaregivenby
2+zzγ˜0−1(z)=2z+1,γ˜1−1(z)=z−2,γ˜∞−1(z)=−2z−3.(9.3.2)
Lemma9.6.L˜etM˜˜=Handγ˜z¯0,+1γ˜1,γ˜∞:M˜→M˜begivenasabove.Then,forthebi-antiholomorphic
mapping˜γSˆ:M→M,γ˜Sˆ(z)=z¯thefollowingidentitieshold:
γ˜Sˆ◦γ˜0=γ˜1−1◦γ˜Sˆ,γ˜Sˆ◦γ˜1=γ˜1◦γ˜0◦γ˜Sˆ,γ˜Sˆ◦γ˜∞=γ˜0−1◦γ˜Sˆ.(9.3.3)
153

Proof.Theclaimisprovedbystraightforwardcomputation:Forz∈M˜wehave
z¯γ˜Sˆ◦γ˜0(z)=−2z¯+1z¯+1=z¯−2z¯z¯+1=z¯z¯+1−2γ˜1−1◦γ˜Sˆ(z)(9.3.4)
−2z¯+1z¯+1
γ˜Sˆ◦γ˜1(z)=z¯z¯++23=−−z¯z¯−−23=γ˜1−z¯z¯+−12=γ˜1−2z¯+1z¯+1=γ˜1◦γ˜0◦γ˜Sˆ(z)(9.3.5)
z¯γ˜Sˆ◦γ˜∞(z)=−23z¯−z¯3+1−z¯2−2+1=3z¯+2−2z¯−1=z¯z+1¯z¯+1γ˜0−1◦γ˜Sˆ(z).(9.3.6)
2z¯+13z¯+22z¯+1
ontheTheabmonoovedromlemmayismatricesneededoftotheproveextendedthefolloframewingFassotheorem,ciatedwhicwithhastatestrinoidfurtherwithnepropcessaryerlyembconditionsedded
annularends,whichisrotoreflectionallysymmetricwithrespecttothetrinoidnormal.
Theorem9.7.LetM=C\{0,1},φ:M→R3˜beatrinoidwithprop˜erlyembe3ddedannularendsand
ψtheassociatedconformalCMC-immersiononM=H,ψ=φ◦π:M→R,whereπdenotesthe
theuniversaltrinoidcnoroveringmal.M˜Denote→MbyasSˆthedefinecdorrinesp(3.2.2)onding.Letsymmetryφberpotorermuefletingctionalthelytrinoidsymmetricendsacwcitorhrdingespetocttheto
˜pbytheermutationoremσ4.5.=(0Denote1∞)by.FMˆ0,Mˆurthermor1,Mˆ∞e,∈letFΛSU(2:M,C)→σtheΛSU(2)unitaryσbethemonodrextendeomydfrmatricameesassociatedwithψ
Mˆj=−cos(2πµj)0110+isin(2πµj)qpjj−qpjj(9.3.7)
associatedwithFasin(4.5.13)by
F(γ˜j(z),λ)=αjMˆj(λ)F(z,λ)kj(z),j=0,1,∞,(9.3.8)
whereαj∈{±1}andγ˜jdenotethecoveringtransformationsonM˜fromsection3.3.Finally,letγ˜Sˆ
belemmathe9.3,andbi-antiholomorphicletMˆ(λ)bemappingthecM˜orr→espM˜ondingassociatemonoddrwithomySˆasmatrixinoftheForemas4.9givenandineexplicitlyquationgiven(9.2.22)in.
SsetWeMSˆ(λ)=±(MSˆ−1(λ))−1=:−abSˆabSˆ.(9.3.9)
ˆˆSSThen,themonodromymatricessatisfy
MSˆ(λ)Mˆ0(λ)=(Mˆ1(λ−1))−1MSˆ(λ),(9.3.10)
MSˆ(λ)Mˆ∞(λ)=(Mˆ0(λ−1))−1MSˆ(λ),(9.3.11)
MSˆ(λ)Mˆ1(λ)=(Mˆ1(λ−1))(Mˆ0(λ−1))MSˆ(λ).(9.3.12)
IntermsofthefunctionspjandqjoccurringinMˆj,equations(9.3.10)to(9.3.12)areequivalentto
aSˆp0(λ)+bSˆq0(λ)=aSˆp1(λ−1)−bSˆq1(λ−1),(9.3.13)
aSˆq0(λ)−bSˆp0(λ)=bSˆp1(λ−1)+aSˆq1(λ−1),(9.3.14)
aSˆp∞(λ)+bSˆq∞(λ)=aSˆp0(λ−1)−bSˆq0(λ−1),(9.3.15)
aSˆq∞(λ)−bSˆp∞(λ)=bSˆp0(λ−1)+aSˆq0(λ−1).(9.3.16)
Proof.Likeintheproofoftheorem7.5,wemakeuseofthefollowingfact,adirectconsequenceof(9.3.8):
F(γ˜j−1(z),λ)=αjMˆj(λ)−1F(z,λ)(kj(γ˜j−1(z)))−1.(9.3.17)
Considerthebi-antiholomorphicmappingγ˜Sˆ:M˜→M˜givenin(9.2.11):γ˜Sˆ(z)=−zz−1.Applying
obtainew9.5,theoremF(γ˜Sˆ(z),λ−1)=MSˆ(λ)F(z,λ)kSˆ,γ˜Sˆ(z),(9.3.18)
154

wherez0−z¯kSˆ,γ˜Sˆ(z)=0z¯−z.(9.3.19)
Combiningthiswiththemonodromyequations(9.3.8)and(9.3.17)andapplyingtheidentities(9.3.3)
fromtheabovelemma,wededuce
MSˆ(λ)α0Mˆ0(λ)F(z,λ)k0(z)kSˆ,γ˜Sˆ(γ˜0(z))=MSˆ(λ)F(γ˜0(z),λ)kSˆ,γ˜Sˆ(γ˜0(z))
=F(γ˜Sˆ(γ˜0(z)),λ−1)=F(γ˜1−1(γ˜Sˆ(z)),λ−1)=α1(Mˆ1(λ−1))−1F(γ˜Sˆ(z),λ−1)(k1(γ˜1−1(γ˜Sˆ(z))))−1
=α1(Mˆ1(λ−1))−1MSˆ(λ)F(z,λ)kSˆ,γ˜ˆ(z)(k1(γ˜1−1(γ˜Sˆ(z))))−1,(9.3.20)
S

ˆMSˆ(λ)α∞M∞(λ)F(z,λ)k∞(z)kSˆ,γ˜Sˆ(γ˜∞(z))=MSˆ(λ)F(γ˜∞(z),λ)kSˆ,γ˜Sˆ(γ˜∞(z))
=F(γ˜ˆ(γ˜∞(z)),λ−1)=F(γ˜0−1(˜γˆ(z)),λ−1)=α0(Mˆ0(λ−1))−1F(γ˜ˆ(z),λ−1)(k0(γ˜0−1(γ˜ˆ(z))))−1
SSSSˆ−1−1−1−1
=α0(M0(λ))MSˆ(λ)F(z,λ)kSˆ,γ˜Sˆ(z)(k0(γ˜0(γ˜Sˆ(z)))).(9.3.21)
Wecontinuebycomputing(duetotheoccurringcomplexrootsuptosign)
1−2z¯−2z+1
z
1−2z0−−2z¯z¯+10
z¯21−01−2z0−−2z¯z+1
k0(z)kSˆ,γ˜Sˆ(γ˜0(z))=z
+1z¯2−0z¯−(1−2z)1−z2z
(1−2z¯)1−2z¯−z0−1−1
(1−2z)1−2z0−z¯S
=±z=±z¯z=±kSˆ,γ˜ˆ(z)(k1(γ˜1(γ˜Sˆ(z))))
0−(1−2z¯)1−z¯2z¯
(9.3.22)and1+2z¯−2z3z+1−2
+1z¯21+2z0−−3z¯−20
z¯1+2S01+2z0−−23zz¯+1−2
k∞(z)kSˆ,γ˜ˆ(γ˜∞(z))=−3z−2
+1z¯22z3−−(1+2z)−31+2z−z2
−(1+2z¯)−31+2z¯−z¯203z¯+20
=±=±
0−(1+2z¯)−3z¯−23z¯+2
(1+2z)−31+2z−z20−3z−2
z¯1+20
z3z¯+23z¯−+22z¯−21−23zz+1+2
−z¯3z+23z−+22z−20−z01−23z¯z¯+1+2
z¯=±=±z¯
0−z¯3z+23z−+22z−201−23z¯z¯+1+2
z3z¯+23z¯−2+2z¯−20−z1−23zz+1+2
11−−=±kSˆ,˜γSˆ(z)(k0(γ˜0(γ˜Sˆ(z)))).(9.3.23)
Combiningtheseresultswiththeequationsabove,weobtain
11−−MSˆ(λ)Mˆ0(λ)=β0(Mˆ1(λ))MSˆ(λ),(9.3.24)
MSˆ(λ)Mˆ∞(λ)=β1(Mˆ0(λ−1))−1MSˆ(λ)(9.3.25)
withβ0,β1∈{±1}.Thiscanbereformulatedas
(Mˆ1(λ−1))−1=β0Mˆ(λ)Mˆ0(λ)(Mˆ(λ))−1,(9.3.26)
SS(Mˆ0(λ−1))−1=β1MSˆ(λ)Mˆ∞(λ)(MSˆ(λ))−1.(9.3.27)
155

Comparingtheupperleftentriesaswellasthelowerrightentriesofbothsidesineachofthese
obtainewequations,−cos(2πµ1)+isin(2πµ1)p1(λ−1)=
β0−cos(2πµ0)+isin(2πµ0)(aSˆaSˆp0(λ)+aSˆbSˆq0(λ)+aSˆbSˆq0(λ)−bSˆbSˆp0(λ)),(9.3.28)
−cos(2πµ1)−isin(2πµ1)p1(λ−1)=
β0−cos(2πµ0)−isin(2πµ0)(aSˆaSˆp0(λ)+aSˆbSˆq0(λ)+aSˆbSˆq0(λ)−bSˆbSˆp0(λ)),(9.3.29)
−cos(2πµ0)+isin(2πµ0)p0(λ−1)=
β1−cos(2πµ∞)+isin(2πµ∞)(aSˆaSˆp∞(λ)+aSˆbSˆq∞(λ)+aSˆbSˆq∞(λ)−bSˆbSˆp∞(λ)),(9.3.30)
−cos(2πµ0)−isin(2πµ0)p0(λ−1)=
β1−cos(2πµ∞)−isin(2πµ∞)(aSˆaSˆp∞(λ)+aSˆbSˆq∞(λ)+aSˆbSˆq∞(λ)−bSˆbSˆp∞(λ)),(9.3.31)
respectively.Bysummingupthefirsttwoequationsandrecallingµ1=µ0,weconcludethatβ0necessarily
equals+1.Analogously,bysumminguptheothertwoequationsandrecallingthatµ0=µ∞,wededuce
Therefore,+1.=β1MSˆ(λ)Mˆ0(λ)=(Mˆ1(λ−1))−1MSˆ(λ),(9.3.32)
MSˆ(λ)Mˆ∞(λ)=(Mˆ0(λ−1))−1MSˆ(λ),(9.3.33)
proving(9.3.10)and(9.3.11).Finally,wecomputeinviewof(3.9.32)
MSˆ(λ)Mˆ1(λ)=MSˆ(λ)(Mˆ0(λ))−1(Mˆ∞(λ))−1=Mˆ1(λ−1)Mˆ0(λ−1)MSˆ(λ),(9.3.34)
allprothrvingee(9.3.12).equationsareSincebequivyusealenofttothe(3.9.32)scalarequationreform(9.3.12)ulationsisoftheimpliedbequationsyequations(9.3.10)and(9.3.10)and(9.3.11),(9.3.11),which
read−cos(2πµ0)aSˆ+isin(2πµ0)(aSˆp0(λ)+bSˆq0(λ))
=−cos(2πµ1)aSˆ+isin(2πµ1)(aSˆp1(λ−1)−bSˆq1(λ−1)),(9.3.35)
−cos(2πµ0)bSˆ+isin(2πµ0)(aSˆq0(λ)−bSˆp0(λ))
=−cos(2πµ1)bSˆ+isin(2πµ1)(bSˆp1(λ−1)+aSˆq1(λ−1)(9.3.36)
and−cos(2πµ∞)aSˆ+isin(2πµ∞)(aSˆp∞(λ)+bSˆq∞(λ))
=−cos(2πµ0)aSˆ+isin(2πµ0)(aSˆp0(λ−1)−bSˆq0(λ−1)),(9.3.37)
−cos(2πµ∞)bSˆ+isin(2πµ∞)(aSˆq∞(λ)−bSˆp∞(λ))
=−cos(2πµ0)bSˆ+isin(2πµ0)(bSˆp0(λ−1)+aSˆq0(λ−1),(9.3.38)
respectively.Astraightforwardsimplificationoftheseequations(usingagainµ0=µ1=µ∞)yieldsthe
claimedonesandfinishestheproof.
Remark9.8.Sincethebi-antiholomorphicmappingγ˜Sˆ:M˜→M˜,z→γ˜Sˆ(z)=z¯z¯+1doesnotpossess
anyfixedpointsinM˜=H,thereexists–unlikeintheprecedingsectionsdealingwiththeotherpossible
trinoidsymmetries–nosimilarchoiceforaspecial“basepoint”fortheextendedframeofagiventrinoid
withproperlyembeddedannularends,whichisrotoreflectionallysymmetricwithrespecttothetrinoid
normal,matriceswhicassohwciatedouldwithleadtosucmhoraeexptrinoid.licitconstrainConsequentsontly,thetherefunctionswillbpej,noqjofurtherccurringinsectionthemonodealingdromwithy
“normalized”trinoidswithproperlyembeddedannularends,whicharerotoreflectionallysymmetricwith
respecttothetrinoidnormal.
156

AAppendix:BasicTopology
Themaingoalofthisappendixistoexplicatetherelationbetweenthefundamentalgroupπ1(X,x)ofa
π:Y→(path-connected)XofX.WtopebrieflyologicalreviewspaceXsomeandbasicthetopologyautomorphisminsectiongroupA.1,Aut(bYefore/Xin)oftrotheducingunivtheersalcoconceptsveringof
thefundamentalgroupandtheautomorphismgroupinsectionsA.2andA.3,respectively.Bothconcepts
arethenrelatedtoeachotherinsectionA.4.
Throughoutthisappendix,wefollowthebookofFulton([20]),towhichwereferthereaderformore
details.

A.1Topologicalspaces,continuousmappingsandpaths
AtopologicalspaceisasetXtogetherwithacollectionMofsubsetsofXwiththefollowingproperties:
McontainstheemptysetandXitself,aswellasanyunionandanyfiniteintersectionofelementsof
M.Miscalledatopology(onX)anditselementsarecalledopensets(inX).AsubsetNofXiscalled
aneighborhoodofapointxinXifitincludesanopensetMcontainingx.
AtopologicalspaceXiscalledHausdorff(space)ifanytwodifferentpointsinXcanbeseperated
bytwodisjointopensets,onecontainingoneofthepoints,theotheronecontainingtheother.
Amappingf:X→YbetweentwotopologicalspacesXandYiscalledcontinuousifthepreimage
underfofanyopensetinYisopeninX.Obviously,thecompositiong◦foftwocontinuousmappings
f:X→Yandg:y→Zisagaincontinuous.Anequivalentcharacterizationofacontinuousmapping,
whichitisoftenmoreconvenienttoworkwith,isthefollowing:Amappingf:X→Ybetween
twotopologicalspacesXandYiscontinuousifandonlyifthereexistsforeveryx∈Xandevery
neighborhoodUoff(x)inYaneighborhoodUofxinX,suchthatf(U)⊆U.
Acontinuousmappingγ:[0,1]→Xdefinesapathfromγ(0)toγ(1)(inX).Apathγ:[0,1]→X
withγ(0)=γ(1)iscalledaloop(basedatγ(0)).Ifγ:[0,1]→Xsatifiesγ(t)=xforsomefixed
x∈Xandforallt∈[0,1],wespeakofaconstantloop(basedatx),whichweoftendenotebyx.
Moreover,forapathγ:[0,1]→Xwedenotebyγ−1theinversepathwhichtraversesγ“backwards”
fromγ(1)toγ(0).Strictlyspeaking,wedefineγ−1:[0,1]→Xbyγ−1(t):=γ(1−t).Finally,given
twopathsγ1,γ2:[0,1]→Xsatisfyingγ1(1)=γ2(0),itisappropriatetodefinebyγ1∙γ2:[0,1]→X,
(γ1∙γ2)(t)=γ1(2t)fort∈[0,21]and(γ1∙γ2)(t)=γ2(2t−1)fort∈[21,1]theproductpathγ1∙γ2traversing
firstγ1fromγ1(0)toγ1(1)andγ2fromγ2(0)=γ1(1)toγ2(1)afterwards,bothwithtwicetheoriginal
eed”.“sp

grouptalfundamenTheA.2Fortwopathsγ1,γ2:[0,1]→XinatopologicalspaceXwithcommonendpointsγ1(0)=γ2(0)and
γ1(1)=γ2(1)acontinuousmappingH:[0,1]×[0,1]→XsatisfyingH(t,0)=γ1(t)forallt∈[0,1],
H(t,1)=γ2(t)forallt∈[0,1],H(0,s)=γ1(0)=γ2(0)foralls∈[0,1]andH(1,s)=γ1(1)=γ2(1)for
alls∈[0,1]iscalledahomotopyfromγ1toγ2(withfixedendpoints).Ifsuchahomotopyexists,γ1and
γ2arecalledhomotopic.
LemmaA.1.LetXbeatopologicalspaceandx,x∈X.Thehomotopyrelationon{γ:[0,1]→
Xpath;γ(0)=x,γ(1)=x}givenby
γ1∼γ2:⇐⇒γ1andγ2arehomotopic(A.2.1)
isanequivalencerelation.
Proof.Foranypathγ:[0,1]→Xwithγ(0)=xandγ(1)=x,themappingH:[0,1]×[0,1]→X,
H(t,s):=γ(t)definesahomotopyfromγtoγitself,whichprovesthereflexivityof∼.Supposenow
γ1∼γ2fortwopathsγ1,γ2:[0,1]→Xfromxtox.ThenthereexistsahomotopyH:[0,1]×[0,1]→X
fromγ1toγ2.BydefiningHˆ:[0,1]×[0,1]→X,Hˆ(t,s):=H(t,1−s)weobtainahomotopyfrom
γ2toγ1.Thuswehaveγ2∼γ1,whichprovesthesymmetryoftherelation∼.Finally,letγ1∼γ2
andγ2∼γ3forpathsγ1,γ2,γ3fromxtoxandletH1:[0,1]×[0,1]→XandH2:[0,1]×[0,1]→X
denotethehomotopiesfromγ1toγ2andfromγ2toγ3,respectively.ThenH:[0,1]×[0,1]→X,
H(t,s):=H1(t,2s)fors∈[0,21]andH(t,s):=H2(t,2s−1)fors∈[21,1]definesahomotopyfromγ1to
γ3,whichmeansγ1∼γ3andprovesthetransitivityoftherelation.Altogether,∼definesanequivalence
relationon{γ:[0,1]→Xpath;γ(0)=x,γ(1)=x}.

157

xbyDefinition[γ]theA.2.equivLetalenceXbclasseaoftopγwithologicalrespspaceecttoandthex∈Xhomotop.Fuyrthermore,relationondenotethesetforofanyallloloopopsγbasedbasedatat
settheThen,.xπ1(X,x):={[γ];γloopbasedatx}(A.2.2)
iscalledthefundamentalgroupofXwithbasepointx.
Theuseoftheterm“group”forthefundamentalgroupisjustifiedbythefollowingresult:
LemmaA.3.LetXbeatopologicalspaceandx∈X.Thegroupoperationofthefundamentalgroup
π1(X,x)isgivenby[γ1]∙[γ2]:=[γ1∙γ2],(A.2.3)
whereγ1∙γ2denotesthepathproductoftheloopsγ1andγ2.
Proof.Thisisexplicatedindetailinsection12aof[20].
AtopologicalspaceXiscalledconnectedifitcannotbewrittenasaunionoftwodisjointnon-empty
bopyaenpathsetsγ(in:X[0,).1]F→Xurthermore,withγX(0)is=xcalledandpγath-c(1)=onnex.ctedAsifweanywilltwoneepdointsthemxsoandon,xwineXgivecansomebejoinedmore
path-cdefinitionsonnectedifconcerninganytheneighborhoo‘connectedness”dofanyofpaointtopxinologicalXconspace:tainsAatoppath-cologicalonnectedspaceXneighisborhocalledodloofcalxly.
Furthermore,apath-connectedtopologicalspaceXiscalledsimplyconnectedifitsfundamentalgroupis
semilotrivial,cial.ely.ifansimplyylocoponnebasedctedifataanpypointoinxtxininXXisphomotopicossessesatoneightheborhoconstanodtNlo,opsuchbasedthatatallx.loXopsisincalledN
areInthehomotopiccasetoofaaconstanpath-connectedtloop.topologicalspaceX,wehavethefollowingresult:
LemmaA.4.LetXbeapath-connectedtopologicalspaceandx,x∈X.Then,thefundamentalgroups
ofXwithbasepointsxandx,respectively,areisomorphicvia
π1(X,x)→π1(X,x),[γ]→[δ−1∙γ∙δ],(A.2.4)
whereδdenotesapathfromxtox.
Proof.Alsothisproofcanbefoundinsection12aof[20].
Aspath-connectedaconsequencetopologicalofthespaceabovXelemma,withoutitspisecifyingconvaenientbaseptooinspteakinXof.“thefundamentalgroup”ofa

groupautomorphismTheA.3Abijectivemappingf:X→YbetweentopologicalspacesXandY,suchthatfanditsinverse
functionf−1arecontinuous,iscalledahomeomorphism.Ifsuchamappingexists,XandYarecalled
.omorphichomeDefinitionA.5.LetXandYbetopologicalspaces.Acontinuousmappingπ:Y→Xiscalleda
coveringofX,ifthereexistsforanypointinXaneighborhoodN,suchthatthepreimageπ−1(N)of
Ncanbewrittenasa(possiblyinfinite)disjointunionofopensetsMiinYwiththepropertythatthe
restrictionofπtoanyoftheMidefinesahomeomorphismMi→N.InthiscaseYiscalledthecovering
spaceofX(withregardtoπ).TheneighborhoodNinvolvediscalledevenlycoveredbyπ.Ifπ:Y→X
isacoveringofXandYissimplyconnected,πiscalledtheuniversalcoveringofXandYiscalledthe
universalcoverofX.
Westatethefollowingresultconcerningtheuniquenessandexistenceoftheuniversalcoverofa
connectedandlocallypath-connectedtopologicalspace.
TheoremA.6.(Corollary13.6andTheorem13.20of[20])LetXbeaconnectedandlocallypath-
connectedtopologicalspace.Theuniversalcoveringπ:Y→XofX,ifitexists,isuniquelydetermined
(uptoanisomorphicchangeofthecoveringspace).Furthermore,auniversalcoveringdoesexistifand
onlyifXissemilocallysimplyconnected.
Nowwearesettodefinetheso-calledautomorphismgroupassociatedwithacoveringofatopological
.Xspace

158

DefinitionA.7.GivenatopologicalspaceXandacoveringπ:Y→X,theset
Aut(Y/X):={σ˜:Y→Y;σ˜homeomorphism,π◦σ˜=π}(A.3.1)
iscalledtheautomorphismgroupofπ.(Thegroupoperationisgivenbythecompositionoffunctions.)
TheelementsofAut(Y/X)arecalledcoveringtransformations.
Weendthissectionbyrecordinganotherusefulresult:
iscTheoremonnected,A.8.the(Propautomorphismostion11.38groupofAut([20])YL/Xet)Xofbeπaactstopproloopgicerlyalspdiscaceandontinuouslyπ:Y→onXY,aci.e.anyovering.pointIfYy
inYpossessesaneighborhoodNinY,suchthatσ˜1(N)∩σ˜2(N)=∅foranytwocoveringtransformations
σ˜1,σ˜2∈Aut(Y/X)withσ˜1=σ˜2.
A.4Themonodromyactionofthefundamentalgroup
Wearenowinterestedinrelatingthefundamentalgroupπ1(X,x)ofa(connected,locallypath-connected
andsemilocallysimplyconnected)topologicalspaceXtotheautomorphismgroupAut(Y/X)ofthe
[γ]univ∈πersal1(X,coxv)aeringcorrespπ:Yon→dingXcoofvXering.Moretransformationprecisely,onweYwantdenotedtobassoyzciate→[γ]with∙z.anyhomotopyclass
Westartbycollectingsomeresultsfrom[20]:
TheoremA.9.(Proposition11.6of[20])LetXbeatopologicalspaceandπ:Y→Xacovering.
Furthermore,letγ:[0,1]→XbeapathandyapointinYwithπ(y)=γ(0).Thenthereexistsaunique
pathγ˜:[0,1]→Y,suchthatπ◦γ˜=γandγ˜(0)=y.γ˜iscalledthe(path)liftofγ.
MoreTheoremover,letγA.10.:[0,1](Pr→opXositionbeap11.8athofwith[20])liftγ˜L:et[0,X1]be→aY.topSuppologicosealHsp:ac[0e,1]and×[0π,:1]Y→→XXisaachomotopyovering.
˜frsuchomγthattoπ◦H˜another=HpathandinH˜X(t,.0)=Then,γ˜(t)therforeallexistst∈a[0,1]unique.H˜ciscalontinuousledthemapping(homotopy)H:lift[0,1]of×H.[0,1]→Y,
Forapathγ:[0,1]→XinatopologicalspaceXwithcoveringπ:Y→Xweintroducethefollowing
notation:IfyisapointinYwithπ(y)=γ(0),theendpointofthe(unique)liftofγstartingatyis
denotedbyy∗γ.
Given[γ]∈π1(X,x)andapointzinY,weconstructapoint[γ]∙zasfollows:First,wefixy∈Y,
suchthatπ(y)=xandchooseapathδinYconnectingyandz.(AsYissimplyconnectedandthus
theorempath-connected,A.9,thissucpathhaδcandobeesalw(uniquely)aysexist.)liftedtoNext,awpatheinYconsiderstartingtheatpathyproandductendingγ∙(atπy◦∗δ)(γin∙(πX◦.δ)).By
Wedefinethedesiredpoint[γ]∙ztobethisendpoint,i.e.weset
[γ]∙z:=y∗(γ∙(π◦δ)).(A.4.1)
Notethatreplacingγbyanyotherloopγ∈[γ]willnotchangetherighthandsideintheabovedefinition,
i.e.y∗(γ∙(π◦δ))=y∗(γ∙(π◦δ)).Thereasonforthisisthatfortheconstructionofy∗(γ∙(π◦δ))–
asfarasγisconcerned–onlytheendpointoftheliftofγisimportant.But,bytheoremA.10,thelifts
ofthehomotopicloopsγandγarehomotopicpathsinYandthushavethesameendpoint.
showIninorderadditiontoobtainthattheforpfixedointyy∈∗(Yγ∙(andπ◦[γδ]))∈asπ1(X,definedx)aabwoellveisdefinedactuallymappingindepzenden→[tγ]of∙z,thewechoiceneedofto
thepathδconnectingyandz.Thisisprovedoverthecourseofthenextthreelemmas.
compGivenositionaπtop◦γ˜ologicaldefinesspacealoopX,inaXcovbasederingπat:xY=→π(yX).andInafact,looptheγ˜:follo[0,win1]g→Yholds:basedaty∈Y,the
LemmaA.11.(cf.proposition13.1of[20])Letπ:Y→XbeacoveringofatopologicalspaceXwith
π(y)=x.Thenπinducesagrouphomomorphism
π∗:π1(Y,y)→π1(X,x),[γ˜]→π∗([γ˜]):=[π◦γ˜](A.4.2)
betweenthefundamentalgroupsπ1(Y,y)andπ1(X,x).Furthermore,π∗isinjective.
Proof.Firstweshowthatthemapping˜π∗iswelldefined.Tothisend,let[γ˜1]=[γ˜2],i.e.γ˜1∼γ˜2fortwo
loopsγ˜1,γ˜˜2basedaty.DenotingbyH:[0,1]×[0,1]→Ythe˜homotopyfromγ˜1toγ˜2,weprovethat
H:=mappingπ◦H[0,1]defines×[0,a1]→homotopX.Fyfromurthermore,π◦γ˜1wtoeπha◦vγ˜e2:H(Ast,π0)=and(πH◦γ˜are1)(tcon)tinandHuous,(t,H1)=(definesπ◦γ˜a2)(cont)tinforuousall
159

tThis∈[0,pro1],vesas(wπell◦γ˜as)H∼(0(,πs)◦=γ˜π)(γ˜1and(0))th=usππ(γ˜([2γ˜(0))])=andπH([γ˜(1,]),s)=i.e.ππ(γ˜1is(1))well=π(γ˜2defined.(1))forNext,allswe∈[0pro,v1].e
12∗1∗2∗
thatπ∗definesagrouphomomorphism:Theneutralelements(withrespecttothecorrespondinggroup
operations)ofthefundamentalgroupsinvolvedaregivenbytheequivalenceclassesoftheconstantloops
ybasedatyandxbasedatx,respectively.Forthese,wehaveπ∗([y])=[π◦y]=[π(y)]=[x].
Furthermore,for[γ˜1]and[γ˜2]inπ1(Y,y),wehaveπ∗([γ˜1]∙[γ˜2])=π∗([γ˜1∙γ˜2])=[π◦(γ˜1∙γ˜2)]=
[(π◦γ˜1)∙(π◦γ˜2)]=[π◦γ˜1]∙[π◦γ˜2]=π∗([γ˜1])∙π∗([γ˜2]).Together,thisprovesthatπ∗definesagroup
homomorphism.Itremainstoshowthatπ∗isinjective.Tothisend,let[γ˜]∈π1(Y,y)withπ∗([γ˜])=[x],
whereconstantxloopdenotesbasedtheaty∈constanY.tByloopassumbasedption,atxwe∈haX.veW(πe◦γ˜need)∼tox.proveDenote[γ˜]=the[y],correspwhereondingydenoteshomotopthey
byHandconsiderthe(bylemmaA.10)uniquehomotopyliftH˜,whichinturndefinesahomotopyfrom
γ˜(theuniqueliftofπ◦γ˜basedaty)toy(theuniqueliftofxbasedaty).Butthisimpliesγ˜∼yand
therefore[γ˜]=[y],whichprovestheclaimandcompletestheproofofthelemma.
Themappingπ∗providesuswithsomeusefulresultsconcerningpathsinthecoveringspaceYofa
:XspaceologicaltopLemmaA.12.LetXbea(connected,locallypath-connectedandsemilocallysimplyconnected)topological
spaceandπ:Y→XtheuniversalcoveringofX.
1.Letγbealoopbasedatx∈Xandγ˜theliftofγstartingaty∈Y.Then:
γ˜endsaty⇐⇒[γ]∈π∗(π1(Y,y)).(A.4.3)
2.Letx,x∈Xandγ1,γ2betwopathsinXfromxtox.Furthermore,lety∈Yandγ˜1,γ˜2bethe
uniqueliftsofγ1,γ2,respectively,topathsinYstartingaty.Notingthatγ2∙γ1−1definesaloop
basedatx,thefollowingholds:
γ˜1,γ˜2havethesameendpoint⇐⇒[γ2∙γ1−1]∈π∗(π1(Y,y)).(A.4.4)
Proof.Westartwiththeproofof(A.4.3).Therearetwodirectionstoshow.Ontheonehand,ifγ˜
[γends]∈atπy(,πw(eY,hayv)),e[γ˜]there∈π1(existsY,ya)loandopthγ˜us[γbased]=at[πy◦γ˜]=satisfyingπ∗([γ˜[])π∈◦γ˜π]∗(=π1[(γY],y=)).π([Onγ˜])the=[πother◦γ˜].hand,Thisif
1∗∗impliestheexistenceofahomotopyHfromπ◦γ˜toπ◦γ˜,whichcanbeliftedtoahomotopyH˜fromγ˜
γ˜toisaliftofhomotopicπ◦γ˜toγ˜startingandaty,thereforewhich,nebycessarilytheauniquenessloopbasedpropatertyy,ofi.epathendinglifts,athasy.tobeAltogether,γ˜itself.weThhavus,e
(A.4.3).relationedvproApplying(A.4.3)toγ=γ2∙γ1−1,theclaimedrelation(A.4.4)reducesto
γ˜1,γ˜2havethesameendpoint⇐⇒γ˜endsaty,(A.4.5)
whereγ˜nowdenotesthe(unique)liftofγ2∙γ1−1startingaty.Weproof(A.4.5)intwosteps.First,
supposeγ˜1andγ˜2havethesameendpoint,sayy.Inthiscase,the(unique)liftγ˜ofγ2∙−1γ1−1starting
yaty(i.e.isγ˜giv−1en).bSo,yγ˜the=γ˜path∙γ˜−pro1,ductwhicofhthemeansliftofthatγ2γ˜endsstartingatyat.yThis(i.e.proγ˜2v)esandthethefirstliftofdirectionγ1ofstarting(A.4.5).at
2Now,supp1osethatthe(uni1que)liftγ˜ofγ2∙γ1−1startingatyalsoendsthere.Furthermore,denotethe
γ˜2endp)andointthofeγ˜2liftbyofyγ.1−1Notestartingthatatγ˜ycanandbe(bywrittenasassumption)thepathendingproatducty.oftheConseliftqueofnγtly2,thestartinginvaterseypath(i.e.
ofthissecondliftisaliftofγ1startingaty(andendingaty).Actually,bytheuniquenesspropertyof
pathlifts,thishastobeγ˜1,whichinturnnecessarilyhastoendaty.Thus,γ˜1andγ˜2havethesame
endpoint,whichprovestheseconddirectionof(A.4.5)andfinishestheproofofthelemma.
Finally,wecanprovethefollowingresult.
LemmaA.13.LetXbea(connected,locallypath-connectedandsemilocallysimplyconnected)topological
spaceandπ:Y→XtheuniversalcoveringofX.Furthermorelet[γ]∈π1(X,x),x∈Xandy∈Ywith
π(y)=x.Then,givenapointz∈Y,foranytwopathsδ1andδ2connectingyandzinYwehave
y∗(γ∙(π◦δ1))=y∗(γ∙(π◦δ2)).(A.4.6)

160

Promakingof.WuseeofneedtolemmashowA.12thatandthekliftseepingofinthemindpathsthγat∙(Yπ◦isδ1)simplyandγc∙on(πnec◦δ2ted)hav(andethethussameπ1(Y,endpy)oin={ts.[yB]}y,
whereydenotestheconstantloopbasedaty),thisisequivalenttoshowing
[(γ∙(π◦δ2))∙(γ∙(π◦δ1))−1]∈{[x]}.(A.4.7)
Asδ2∙δ1−1definesaloopbasedaty,which–asπ1(Y,y)={[y]}–ishomotopictotheconstantloopy
basedaty,wehave
[(γ∙(π◦δ2))∙(γ∙(π◦δ1))−1]=[(γ∙(π◦δ2))∙(π◦δ1)−1∙γ−1]=[γ∙(π◦(δ2∙δ1−1))∙γ−1]
=[γ]∙π∗([δ2∙δ1−1])∙[γ−1]=[γ]∙π∗([y])∙[γ−1]=[γ]∙[x]∙[γ−1]=[x]∈{[x]},(A.4.8)
wherewemadeuseofthefactthatthepathγ∙x∙γ−1ishomotopictotheconstantpathxbasedatx.
Sotheclaimisproved.
andWtheenowautomorphismresumeourgroupconsiderationsAut(Y/X)ofconcerningtheunivtheersalcorelationveringbetπw:Yeen→theXofXfundamen.Astalexplicgroupatedπ1(Xearlier,,x)
ponceointwze∈haYvethechposoinentyy∈∗(Yγ∙(withπ◦δπ()),y)=whicx,hdaneypendselemenonty[,γ]z,∈[γ]π1(andX,ax)pathalloδwsfromforyassotociz.atingHowevwither,anbyy
lemmaA.13,thechoiceofδwillactuallynotaffecttheresultingpoint.Thus,keepingy∈Yfixed,we
canrelatetoany[γ]∈π1(X,x)themappingY→Y,z→[γ]∙z:=y∗(γ∙(π◦δ)),onlydependingon
[γ].Moreprecisely,onecanprovethefollowingtheorem:
TheoremA.14.(Corollary13.15of[20])LetXbeaconnected,locallypath-connectedandsemilocally
simplyconnectedtopologicalspaceandπ:Y→XtheuniversalcoveringofX.Then,forafixedy∈Y
withπ(y)=xandany[γ]∈π1(X,x)themappingY→Y,z→[γ]∙z:=y∗(γ∙(π◦δ)),whereδdenotes
anarbitrarypathinYfromytoz,definesacoveringtransformationonY.Furthermore,themapping
π1(X,x)→Aut(Y/X),[γ]→(z→[γ]∙z),(A.4.9)
definesanisomorphismbetweenthefundamentalgroupπ1(X,x)andtheautomorphismgroupAut(Y/X)
i.e.,πofπ1(X,x)=∼Aut(Y/X).(A.4.10)
RemarkA.15.AssumingthepremisesoftheoremA.14,byevaluatingthecorrespondencebetween
π1(X,x)andAut(Y/X),itiseasytoseethattheautomorphismgroupAut(Y/X)ofπactstransitively
onπ−1(x).Thus,π:Y→X,definesasocalledG-covering,i.e.thespaceXcanbeidentifiedwiththe
setY/GoforbitsunderagivenactionofagroupGonY,whereby,inourcase,G=Aut(Y/X).In
liespimthisparticular,X=∼Y/Aut(Y/X)=∼Y/π1(X,x).(A.4.11)
Inviewofthis,wethinkofYasofthedisjointunionofsocalledsheetsFσ˜,whichare“indexed”bythe
elementsσ˜oftheautomorphismgroupofY:
Y=Fσ˜.(A.4.12)
σ˜∈Aut(Y/X)
WebijectiondefineFthe→X,“starting”andsettingsheetFFid:=byFc.hoAllosingotherasheetssubsetFareofthenYcongiventainingbythey,imsuchagesthatofFπ|Funderdefinesthea
elementsofAut(Y/X),idid
Fσ˜=σ˜(Fid).(A.4.13)
Furthermore,wenotethateachsheetFσ˜canbeidentifiedbijectivelywiththecoveredspaceXviathe
intosrestrictionheetsofasinthetrounivducedersalabcoovveiseringπreferredtothetoasaparticulartesselationsheet,(ofπ|FYσ˜).:Fσ˜→X.ThedecompositionofY
TheproofoftheoremA.14isbasedonthefollowinglemma(cf.section13bof[20]),whichadditionally
π1(impliesX,x)thatofXtheonY.mappingThisπ1(actionX,x)is×cYalle→dYthe,([γ]mono,z)dr→omy[γ]∙actziondefines(ofπan1(X,actionx)onofYthe).fundamentalgroup
LemmaA.16.LetXbeaconnected,locallypath-connectedandsemilocallysimplyconnectedtopological
spaceandπ:Y→XtheuniversalcoveringofX.ForfixedyinYand[γ]∈π1(X,x)definethemapping
z→[γ]∙zasin(A.4.1).

161

1.Let[γ1],[γ2]∈π1(X,x).Thenforallz∈Ywehave
([γ1]∙[γ2])∙z=[γ1]∙([γ2]∙z).(A.4.14)
2.Consider[x]∈π1(X,x),wherexdenotestheconstantloopbasedatxinX.Forallz∈Ywe
have[x]∙z=z.(A.4.15)
Proof.Westartwiththeproofofequation(A.4.14).For[γ1],[γ2]∈π1(X,x)andzinYwehaveby
definition([γ1]∙[γ2])∙z=([γ1∙γ2])∙z=y∗((γ1∙γ2)∙(π◦δ)),(A.4.16)
whereδdenotesapathinYconnectingyandz.As(γ1∙γ2)∙(π◦δ)ishomotopictoγ1∙(γ2∙(π◦δ)),
thecorrespondingliftsstartingatyhavethesameendpoints,andhenceweinferthat
y∗((γ1∙γ2)∙(π◦δ))=y∗(γ1∙(γ2∙(π◦δ))),(A.4.17)
sieimplhwhic([γ1]∙[γ2])∙z=y∗(γ1∙(γ2∙(π◦δ))).(A.4.18)
Ontheotherhandwehave
[γ1]∙([γ2]∙z)=y∗(γ1∙(π◦δ))(A.4.19)
foranarbitrarypathδinYconnectingyand[γ2]∙z.Choosingforδtheliftofγ2∙(π◦δ)startingat
obtainew,yy∗(γ1∙(π◦δ))=y∗(γ1∙(γ2∙(π◦δ)))(A.4.20)
usthand[γ1]∙([γ2]∙z)=y∗(γ1∙(γ2∙(π◦δ))).(A.4.21)
Altogether,equation(A.4.14)isproved.
Equation(A.4.15)canbeprovedbydirectcomputation:WithδagaindenotingapathinYfromy
tozwehave
[x]∙z=y∗(x∙(π◦δ))=y∗(π◦δ)=δ(1)=z,(A.4.22)
wherewemadeuseofthefactthatthepathsx∙(π◦δ)andπ◦δarehomotopicandthusinducelifts
withthesameendpoint.Thisfinishestheproofofthelemma.
ProofoftransformationtheoremonYA.14.,weTohavproevtoetheshowfactthatthatitistheaconmappingtinuousz→[γbijection]∙z:=ywith∗(cγ∙on(πtin◦δuous))invdefineserseacovmappingering
satisfyingπ([γ]∙z)=π(z).Westartwiththeproofofthebijectionproperty:Usingtheprecedinglemma
weobservethatthemappingz→[γ−1]∙zassociatedwiththehomotopyclassoftheinversepathγ−1of
γdefinestheinversemappingofthestudiedmappingz→[γ]∙z:
[γ]∙([γ−1]∙z)=([γ]∙[γ−1])∙z=[γ∙γ−1]∙z=[x]∙z=z,(A.4.23)
[γ−1]∙([γ]∙z)=([γ−1]∙[γ])∙z=[γ−1∙γ]∙z=[x]∙z=z.(A.4.24)
Thisalreadyprovesthatz→[γ]∙zisabijection.Next,weprovefor[γ]∈π1(X,x)thecontinuityofthe
aneighmappingbzorho→o[dγ]U∙z.ofTzowhicthishisend,mappletzed∈bYyuand→U[γ]b∙euaintoneighUb.orhoLetoNdbofezan:=ev[γ]enly∙zcoinvY.eredWeneighneedbtoorhoofindd
Vofπand(z)V=πb(ez)theinX.(disjoinAst)XopisenlosetscallyinYconpath-connectetainingd,zweandcanz,assumerespectivthately,Niswhicharepath-connected.mappedbyLetπ
homeomorphicallyontoN.DefiningtheopensetW:=U∩VinY,thepath-connectedneighborhood
N˜:=π(W)ofπ(z)˜inXandtheopensetW:=V∩π−1(N˜)inY,therestrictedmappingsπ|W:W→N˜
[γand]∙uπ|∈WU:.WTo→thisNend,arechoagainoseapathδhomeomorphisms.inN˜fromNoπ(w,z)settoπU(u:=)Wandanddenoteletuits∈liftU.Weconnectingneedztoandprovue
inUbyδ˜.Then,wehave
[γ]∙u=y∗(γ∙(π◦(δ˜∙δ˜))),(A.4.25)
whereδ˜denotesapathconnectingyandzinY.(Notethatthusδ˜∙δ˜definesapathconnectingyand
uinY.)Thiscanbetransformedinto
[γ]∙u=y∗(γ∙(π◦δ˜)∙(π◦δ˜))=y∗((γ∙(π◦δ˜))∙δ)=([γ]∙z)∗δ=z∗δ,(A.4.26)

162

whichmeansthat[γ]∙uistheendpointoftheliftofδtoapathstartingatz∈W.Asδisapath
inN˜,whichishomeomorphictoW,thisliftandinparticularitsendpoint[γ]∙ulieinW.Therefore,
[γ]∙u∈W⊆U,whichcompletestheproofofthecontinuityofthemappingz→[γ]∙z.Note−1that,as
hathevealsoargumentsalreadyinvproolvvededholdtheforcontinuitarbitraryyofthehomotopinveyrseclassesmappinginπ1z(X→,[xγ),−1in]∙z.particularFinally,alsokeepingfor[γin],mindwe
thatforanyzinYthepathγ∙(π◦δ)runsfromxtoπ(z)andthusthepointy∗(γ∙(π◦δ))isin
π−1(π(z)),wehave
π([γ]∙z)=π(y∗(γ∙(π◦δ)))=π(z),(A.4.27)
whichprovesthelastdesiredproperty.Altogether,themappingz→[γ]∙z:=y∗(γ∙(π◦δ))definesa
coveringtransformationonY.
Itremainstoprovethatthemappingπ1(X,x)→Aut(Y/X),[γ]→(z→[γ]∙z)definesanisomor-
phismbetweenπ1(X,x)andAut(Y/X).Thefactthatitdefinesahomomorphismbetweenthegroups
involvedisprovedbytheidentityA.4.14,soweonlyneedtoshowthatthemappingisbijective.Tothis
end,letσ˜:Y→Ybeacoveringtransformationofπ.Denoteσ˜(y)byy.Choosingapathγ˜0inYfrom
ytoyanddefiningγ0:=π◦γ˜,wehave
[γ0]∙y=y∗(γ0∙(π◦y))=y∗(γ0∙x)=y∗γ0=y,(A.4.28)
whereyandxdenotetheconstantloopsinY(resp.X)basedaty(resp.x).Thismeansthatthe
mappingz→[γ0]∙zalsomapsthepointytothepointy.BytheoremA.8thisimpliesthatthemappings
πz(→X,[xγ)0]∙→zandAut(Yσ˜/X)coincide,issurjectivi.e.σ˜(e.z)T=o[γpro0]v∙ezthatforallitiszinalsoY,injectivwhiche,prosuvppesosethat[γthe]∈π(X,homomorphismx)with
111[γ1]∙z=zforallzinY,inparticular[γ1]∙y=y,whichmeans
y=[γ1]∙y=y∗(γ1∙(π◦y))=y∗(γ1∙x)=y∗γ1.(A.4.29)
Butthisimpliesthattheliftγ˜1ofγ1toapathstartingatyalsoendsthere,whichbylemmaA.12
isequivalentto[γ1]∈π∗(π1(Y,y)).AsYissimplyconnected,wehaveπ∗(π1(Y,y))={[x]}and
therefore[γ]=[],whichcompletestheproofoftheinjectivity.Altogether,themappingπ(X,x)→
Aut(Y/X),1[γ]→x(z→[γ]∙z)definesanisomorphismbetweenthefundamentalgroupπ1(X,x1)andthe
automorphismgroupAut(Y/X)ofπ.
RemarkA.17.Inordertoexpressthecorrespondenceofthecoveringtransformationz→[γ]∙ztothe
loopγinMinvolved,w˜edenotethistransformationbyγ˜.Notethatwealsouse“γ˜”todenotethelift
oftheloopγinMtoM.However,itshouldbeclearfromtheparticularcontextwhichnotionweare
referringtowhenwriting“γ˜”.
Theconstructionofthecoveringtransformationγ˜fromagivenelement[γ]ofthefundamentalgroup
π1(X,x)involvesthechoiceofapointy∈Ywithπ(y)=x.Asthischoiceisnotunique,weareinterested
intheeffectofreplacingybyanotherpointinY,whichismappedbyπontox.
CorollaryA.18.LetXbeaconnected,locallypath-connectedandsemilocallysimplyconnectedtopo-
logicalspaceandπ:Y→XtheuniversalcoveringofX.Lety∈Ywithπ(y)=xandconsiderfor
any[γ]∈π1(X,x)thecoveringtransformationγ˜:Y→Y,γ˜(z):=[γ]∙zgivenbytheoremA.14.Then,
rtreplacingansformationy∈YoninYthe,cweobtainonstructionanotherofthecoveringmappingtrγ˜byaansformationpointσ˜γ˜(ynew)∈:Yz,→[wherγ]e∙σ˜newzdenotesonY,acwhichoveringis
relatedtoγ˜asfollows:
˜γnew(z)=(σ˜◦γ˜◦σ˜−1)(z).(A.4.30)
Proof.BytheoremA.14,thereexists[σ]∈π1(X,x),suchthat
σ˜(z)=[σ]∙z:=y∗(σ∙(π◦δ)),(A.4.31)
whereδdenotesanarbitrarypathinYfromytoz.Furthermore,denotingbyσ˜alsothe(unique)liftof
theloopσtoapathinYstartingaty(cf.remarkA.17),thepathproductσ˜−1∙δdefinesapathinY
startingaty∗σ=σ˜(y)andendingatz.Puttingtheseresultstogether,weobtain
γ˜new(z)=[γ]∙newz=(σ˜(y))∗(γ∙(π◦(σ˜−1∙δ)))=(y∗σ)∗(γ∙σ−1∙(π◦δ))=y∗(σ∙γ∙σ−1∙(π◦δ))
=[σ∙γ∙σ−1]∙z=[σ]∙([γ]∙([σ−1]∙z))=σ˜(γ˜(σ˜−1(z)))=(σ˜◦γ˜◦σ˜−1)(z),(A.4.32)
whichprovestheclaim.

163

(B.1)

BAppendix:Thefunctionµj=XjXj
Inthisappendixwestudytheλ-dependentfunction
µj=XjXj(B.1)
asintroducedinsection3.5inmoredetail.
RecallthatXjandXjaredefinedby
Xj=sjλ−1+tjλ,Xj=sjλ+tjλ−1(B.2)
forallλ∈C∗.Moreover,theparameterssj,tjsatisfysj∈[41,21)andsj+tj=21.Usingsj2+t2j=41−2sjtj
settingandwj=sjtj,(B.3)
wecanrewriteµjas
11
µj=XjXj=sj2+tj2+sjtj(λ−2+λ2)=4+wj(λ−2−2+λ2)=4+wj(λ−λ−1)2,(B.4)
(3.5.24).inearlierstatedasWenowturntothequestionforwhichλ∈C∗themappingµjiswelldefined.Tothisend,wefirst
needtodefinethecomplexsquarerootinvolvedinthedefinitionofµjexplicitly.Here,wedistinguish
betweenthecasethats=tandthecasethats=t=1.
Inthecasesj=tj,jwejwillusethefollowingjjholomorphic4extensionoftheusualrealsquarerootto
thecutcomplexplaneC>0=C\{x∈R;x≤0}:
√:C>0→C∗,λ=reiθ→√λ:=√rei2θ,(B.5)
wherewewriteλ∈C∗intheformλ=reiθwithr∈R+andθ∈(−π,π),and√rdefinesthevalueof
thepositiverealsquarerootofr.
Naturally,forθ=0,weobtaintheusualrealsquarerootR+→R+.Moreover,notethatθ∈
(−2π,2π)andthus√actuallymapsC>0to{z∈C∗;(z)>0}.Weextend√continuously(but2not
holomorphically)toC≥0=C>0∪{0}bysetting
√0:=0.(B.6)
Inviewof(B.5)and(B.6),thefunctionµjisinthecasesj=tjdefinedandcontinuousforallλ∈C∗,
expressionthehwhicforχj(λ)=41+wj(λ−λ−1)2(B.7)
takesvaluesinC≥0.Moreover,sinceχisholomorphicinλ∈C∗,µjisholomorphicinλforallλ∈C∗,
forwhichχjtakesvaluesinC>0(asacompositionofholomorphicfunctions).
Letλ∈C∗.Usingwj=sjtjand41−2wj=sj2+tj2,wecompute
χj(λ)=0⇐⇒wjλ4+(41−2wj)λ2+wj=0
2−(41−2wj)±(41−2wj)2−4wj2−sj2−tj2±|sj2−tj2|
⇐⇒λ=2wj=2sjtj
⇐⇒λ∈{±itj,±isj}.(B.8)
tsjjInparticular,thisimplies
µj(λ)=0⇐⇒λ∈{±istjj,±itsjj}.(B.9)
Moreover,writingλ∈C∗asλ=reiθwithr∈R+andθ∈[−π,π)andrecallingsj≥tj,standard
analysisofχj(λ)yields
χj(λ)∈C>0⇐⇒λ∈C∗\W1,j,(B.10)
164

whereW1,j={z∈C;(z)=0and(z)∈(−∞,−sj]∪[−tj,tj]∪[sj,+∞)}.(B.11)
tjsjsjtj
Wenowturntothecasesj=tj=41.Here,sincewj=sjtj=161,wehave
µj=41+161(λ−λ−1)2=161(λ+λ−1)2,(B.12)
whichallowsforresolvingtheoccurringcomplexsquareexplicitlyas:
µj=1(λ+λ−1).(B.13)
4Wewillusethisdefinitionofµjinthespecialcase∗thatsj=tj=41.Obviously,inthiscase,µjdefines
acontinuousandholomorphicfunctioninλ∈C.Wenotethat
µj(λ)=0⇐⇒λ∈{±i}.(B.14)
Basedonourconsiderationsabove,weobtainthefollowingresult:
LemmaB.1.Considerthemappingµjgivenin(B.1).
1.Inthecasesj=tj,µjdefinesacontinuousmapping
µj:C∗\W˜1,j→C,(B.15)
ewherW˜1,j=W1,j\{±itj,±isj}(B.16)
tsjjandW1,jisgivenin(B.11).Moreover,therestrictionofµjtoC∗\W1,jisholomorphic.
2.Inthecasesj=tj=41,µjdefinesacontinuousmapping
µj:C∗→C.(B.17)
Moreover,µjisholomorphicinλ∈C∗.
RemarkB.2.Considerthecasesj=tjinlemmaB.1.ExcludingthesetW1,j(resp.W˜1,j)fromthe
λ-domaincorrespondsto“cutting”thepuncturedλ-planeC∗alongtheimaginaryaxisthrice:oncefrom
−i∞to−itsjj,oncefrom−istjjtoistjjandoncefromitsjjto+i∞.Thefiniteendpoints±istjj,
±itsjjofthecutsarealsoremovedinthecasethatweexcludeW1,jfromC∗,whiletheyareretainedin
˜∗inthethecasecasethatsjw=etj,excludetheWmapping1,jfromµjisC.conWtinewuououldsandliketopholomorphicointoutonthat(athissufficienimpliestlyinsmallneighparticularborhothat,od
of)theunitcircleS1.(Thisisofcoursealsotrueforthemappingµjinthecasesj=tj=41.)
ThefollowinglemmasummarizessomebasicpropertiesofthemappingsXj,Xjandµj.
LemmaB.3.Considerthemappingµjgivenin(B.1).
1.Inthecasesj=tjwehave:
Xj=µjforallλ∈C∗\W1,j(B.18)
XµjjXj(λ=1)=Xj(λ=1)=1,(B.19)
21µj(λ=1)=2,(∂λµj)(λ=1)=0.(B.20)

165

(B.18)(B.19)(B.20)

2.Inthecasesj=tj=41,wehave:
Xj=1=µjforallλ∈C∗\{±i}(B.21)
Xµjj1Xj(λ=1)=Xj(λ=1)=2,(B.22)
1µj(λ=1)=2,(∂λµj)(λ=1)=0.(B.23)
Proof.Letfirstsj=tj.Forλ∈C∗\W1,jwehaveµj(λ)=0andXj(λ)=0,whichallowsforwriting
XjXjXjµj2µj
µj=Xjµj=Xjµj=Xj.(B.24)
Moreover,bydirectcomputationswehave
1Xj(λ=1)=sj+tj=2=sj+tj=Xj(λ=1),(B.25)
and

(B.28)

µj(λ=1)=1+wj(1−1)2=1,(B.26)
24−1−2
(∂λµj)(λ=1)=2wj(λ−λ)(1−λ)=0.(B.27)
241+wj(λ−λ−1)2λ=1
Letnowsj=tj=41.Then,wehaveforallλ∈C∗\{±i}therelationµj(λ)=Xj(λ)=Xj(λ)=0,
whichimpliesthefirstclaim.Theotherclaimsfollowbydirectcomputations.
Next,weturntotheparameterwj=sjtj:
LemmaB.4.Interpretingwj=sjtj=sj(21−sj)asafunctionofsj,
wj:[1,1)→R,(B.28)
24hasthefollowingproperties:
111wj([4,2))=(0,16],(B.29)
1111111
wj(sj)∈(0,18)⇐⇒sj∈(3,2),wj(sj)∈(18,16)⇐⇒sj∈(4,3),(B.30)
1111wj(sj)=18⇐⇒sj=3,wj(sj)=16⇐⇒sj=4.(B.31)
Proof.Elementaryanalysisofwj:[41,21)→R,sj→sj(21−sj).
BasedonlemmaB.4westudythebehaviourofµjforλ∈S1:
LemmaB.5.Letthemappingµjbegivenin(B.1).
1.Inthecasesj=tj,µjtakespositiverealvaluesforλ∈S1.Moreprecisely,wehave
µj(S1)=[1−4wj,1].(B.32)
242.Inthecasesj=tj=41,µjtakesrealvaluesforλ∈S1.Moreprecisely,wehave
µj(S1)=[−1,1].(B.33)
22

166

(B.29)(B.30)(B.31)

(B.32)(B.33)

Proof.Letfirstsj=tj.Writingλ∈S1aseiθwithθ∈(−π,π],wecompute
µj(λ)=41+wj(eiθ−e−iθ)2=41+wj(2isin(θ))2=41−4wjsin2(θ).(B.34)
Usingwj<161from1lemmaB.4,weobtain41−4wjsin2(1θ)≥41−4wj>0.Consequently,wehave
µj(λ)>0forallλ∈S.Moreover,weinferthat,forλ∈S,
1114−4wj≤µj(λ)≤4=2.(B.35)
Sincewehaveµj(λ=1)=21andµj(λ=i)=41−4wj,weconcludebycontinuityargumentsthat
actuallyµj(S1)=[41−4wj,21].
Wenowturntothecasesj=tj=41.Writingagainλ∈S1aseiθwithθ∈(−π,π],weobtain
µj(λ)=1(eiθ+e−iθ)=1cos(θ),(B.36)
2411111onwhicS1h,wedirectlyinferthatimpliesµj−(S21)≤=µj[(−λ)1,≤1].2forallλ∈S.Sinceµj(1)=2,µj(−1)=−2andµjiscontinuous
22theWeframewcloseorkthisofapthispendixthesis,withthisthecasefolloonlywingoccurslemma,forsjdealing=tjforwithallthej∈c{ase0,1µ,0=∞}µ.)1=µ1.(Notethat,in
LemmaB.6.Let,forj=0,1,∞,µj=41+wj(λ−λ−1)2,wherewj=sjtjforsj∈(41,21)and
sj+tj=12.(Inparticular,thisimpliessj=tj.)Then,thefollowingholds:
1.µ0=µ1=µ∞⇐⇒w0=w1=w∞⇐⇒s0=s1=s∞(B.37)
2.Letµ:=µ0=µ1=µ∞,w:=w0=w1=w∞ands:=s0=s1=s∞.Then
µsatisfies(3.5.28)forallλ∈S1⇐⇒w∈(0,181]⇐⇒s∈[31,21).(B.38)
21−λPr−o1)of.2=wFirst,∞(λw−eλgiv−1e)2theforproallofλ∈ofC∗\(B.37):(W˜1,0∩AssumeW˜1,1∩µ0W˜1=,∞µ)1=andµth∞.uswThis0=wyields1=ww0∞(.λ−Asλωj)=s=j(w11−(λsj−)
2isinjectiveforsj∈[41,21),weinferthats0=s1=s∞.Theotherwayround,assumes0=s1=s∞.This
impliesdirectlyw0=w1=w∞andµ0=µ1=µ∞.
Wenowturntothesecondclaim.As
w∈(0,1]⇐⇒s∈[1,1)(B.39)
2318isadirectconsequenceoflemmaB.4,itremainsonlytoprove
µsatisfies(3.5.28)forallλ∈S1⇐⇒w∈(0,1].(B.40)
18Inviewoftheassumption,the“unitarizabilitycondition”(3.5.28)reads
cos2(πµ)
0≤sin2(2πµ)≤1(B.41)
forallλ∈S1.Thisisequivalentto
10≤4sin2(πµ)≤1(B.42)
forallλ∈S1,whichinturnholdsifandonlyif
11µ(λ)∈[6,2](B.43)
forallλ∈S1.(Recallthat,bylemmaB.5,µtakesforλ∈S1onlyvaluesin(0,21].)Since,againbylemma
B.5,µj(S1)=[41−4wj,21],(B.43)holdsforallλ∈S1ifandonlyif41−4wj≥61,or,equivalently,
w≤181.Inviewofthegeneralrelationw∈(0,161]fromlemmaB.4,thisfinishestheproof.
167

CAppendix:Proofoflemma3.37
Inthisappendix,wegivetheproofoflemma3.37,whichexplicitlystatestheconnectioncoefficientsαj,
βj,δj,jrelating(bylemma3.12)thelocalsolutionΦjto(3.8.9)aroundzjtothefundamentalsystem
yj1,yj2(aroundzj)giveninequations(3.7.23)to(3.7.26):
αjyj1+βjyj2αjyj1+βjyj2
Φj=ν.(C.1)
δjyj1ν+jyj2δjyj1+jyj2
Lemma3.37.Theconnectioncoefficientsαj,βj,δj,joccuringin(C.1)aregivenby
λXµ2jjαj=−βj=(i)jXj,(C.2)
λX2jδj=j=(i)j1.(C.3)
Proof.Asearlierinthiswork(cf.(2.6.21)),wewrite
Dj=0Xj=µjRjSσ3S−1Rj−1,(C.4)
0Xj√λXj
√µj01−λ−1
0whereRj=√λ−1XjandS=√12λ1.(Notethat(2.6.21)appliessinceweconsider
õjonlythecasesj=0,1andsinceinthesecases,byassumption,sj>tj.)From(C.4),weinferthat
eln(z)D0=R0Seln(z)µ0σ3S−1R−1=R0Sz00S−1R−1=1s0,+µ0s0,−,(C.5)
µX0
00z−µ002µX0s0,−s0,+
0wheres0,+=zµ0+z−µ0,s0,−=zµ0−z−µ0,(C.6)
,similarlyand,eln(1−z)D1=R1Seln(1−z)µ1σ3S−1R−1
1=R1S(1−z)0−µS−1R1−1=1s1,+µ1s1,−,(C.7)
µ1X1
0(1−z)12µX1s1,−s1,+
1wheres1,+=(1−z)µ1+(1−z)−µ1,s1,−=(1−z)µ1−(1−z)−µ1.(C.8)
Byequations(3.8.33)and(3.8.34),wehave
1−P0=eln(z)D0Φ0V+,0,(C.9)
1−P1=eln(1−z)D1Φ1V+,1.(C.10)
Thus,byourconsiderationsabove,weobtain
1sj,+−µjsj,−αjyj1ν+βjyj2αjyj1+βjyj2
Xj
Pj=V+,j
2−µXjjsj,−sj,+δjyj1ν+jyj2δjyj1+jyj2
1∗sj,+(αjyj1+βjyj2)−µXjjsj,−(δjyj1+jyj2)
=V+,j.(C.11)
2∗−µXjjsj,−(αjyj1+βjyj2)+sj,+(δjyj1+jyj2)
Inparticular,weobtainfortheupperrightandthelowerrightentryofPj
X1jPj,12=(V+,j)22sj,+(αjyj1+βjyj2)−sj,−(δjyj1+jyj2)
µ2j1XjXj
=(V+,j)22yj1(αjsj,+−δjsj,−)+yj2(βjsj,+−jsj,−),(C.12)
µµ2jj168

X1jPj,22=2(V+,j)22−µjsj,−(αjyj1+βjyj2)+sj,+(δjyj1+jyj2)
XX1=(V+,j)22yj1(−jαjsj,−+δjsj,+)+yj2(−jβjsj,−+jsj,+),(C.13)
µµ2jjwhere(V+,j)22denotesthelowerrightentryofV+,j.
Next,recallfrom(3.7.23)to(3.7.26)thefundamentalsystemsyj1,yj2aroundzj(j=0,1)solving
theFuchsianequation(3.7.1):
y01=zr0(1−z)r1F(α,β,γ;z),(C.14)
y02=zr0+1−γ(1−z)r1F(α−γ+1,β−γ+1,2−γ;z),(C.15)
y11=zr0(1−z)r1F(α,β,α+β−γ+1;1−z),(C.16)
y12=zr0(1−z)r1+γ−α−βF(γ−β,γ−α,γ−α−β+1;1−z).(C.17)
Observingthatr0+1−γ=21−µ0andr1+γ−α−β=21−µ1,wecanrewritetheseequationsas
1y01=z2+µ0h0,(C.18)
y02=z21−µ0h˜0,(C.19)

y11=(1−z)12+µ1h1,(C.20)
y12=(1−z)21−µ1h˜1(C.21)
withmappingshj,h˜j,whichareholomorphicaroundzj,givenby
1h0=(1−z)2+µ1F(α,β,γ;z),(C.22)
1h˜0=(1−z)2+µ1F(α−γ+1,β−γ+1,2−γ;z),(C.23)
1h1=z2+µ0F(α,β,α+β−γ+1;1−z),(C.24)
1h˜1=z2+µ0F(γ−β,γ−α,γ−α−β+1;1−z).(C.25)
Weproceedbyinserting(C.18)to(C.21)intoequations(C.12)and(C.13).First,weconsiderthecase
0:=jP0,12=z−2λX0z2+µ0h0(z)α0(zµ0+z−µ0)−0δ0(zµ0−z−µ0)
111X
µ20+z21−µ0h˜0(z)β0(zµ0+z−µ0)−X00(zµ0−z−µ0)
µ0=1λX0z2µ0α0h0(z)−X0δ0h0(z)+α0h0(z)+X0δ0h0(z)+β0h˜0(z)−X00h˜0(z)

2µ0µ0µ0
+z−2µ0β0h˜0(z)+X00h˜0(z),(C.26)
µ0P0,22=1z−21λX0z21+µ0h0(z)−X0α0(zµ0−z−µ0)+δ0(zµ0+z−µ0)
µ20X1+z2−µ0h˜0(z)−0β0(zµ0−z−µ0)+0(zµ0+z−µ0)
µ0=1λX0z2µ0−X0α0h0(z)+δ0h0(z)+X0α0h0(z)+δ0h0(z)−X0β0h˜0(z)+0h˜0(z)

2µ0µ0µ0
X0+z−2µ0β0h˜0(z)+0h˜0(z).(C.27)
µ0

169

AsP0shallbeholomorphicatz=z0,alsothematrixcoefficientP0,12shallbeholomorphicatz=z0.
Sinceµ0isnotanintegerexceptforsomeλfromadiscretesubsetofC∗,thisisonlypossibleifthe
coefficientsofz2µ0andz−2µ0vanish.FurthermoreP0(z0)=IandthusP0,12(z0)=0.Altogether,we
inferthefollowingconditionsforthecoefficientP0,12:
δ0=µ0α0,(C.28)
X00=−µ0β0,(C.29)
X0α0h0(0)+X0δ0h0(0)+β0h˜0(0)−X00h˜0(0)=0.(C.30)
µµ00Similarly,asP0,22shallbeholomorphicatz=z0andP0,22(z0)=1,weobtain:
δ0=X0α0,(C.31)
µ0X0=−0β0,(C.32)
µ0λX0α0h0(0)+δ0h0(0)−β0h˜0(0)+0h˜0(0)=1.(C.33)
1X0X0
µµ200Since,bylemmaB.3giveninappendixB,µX0=Xµ0,conditions(C.31)and(C.32)areequivalentto
00(C.28)and(C.29).Moreover,h0(0)=h˜0(0)=1(cf.(3.7.16)).Altogether,conditions(C.28)to(C.33)
totalenequivareδ0=µ0α0,(C.34)
X0µ00=−β0,(C.35)
X02α0+2β0=0,(C.36)
1λX02µ0α0−2µ0β0=1.(C.37)
XX200Byaneasycomputation,theseequationsyield
Xα0=−β0=√0,(C.38)
λXµ2001δ0=0=2√λX.(C.39)
0Weturntothecasej=1.
P1,12=−1i(1−z)−21λX1
2X1∙(1−z)2+µ1h1(z)α1((1−z)µ1+(1−z)−µ1)−1δ1((1−z)µ1−(1−z)−µ1)
µ1+(1−z)21−µ1h˜1(z)β1((1−z)µ1+(1−z)−µ1)−X11((1−z)µ1−(1−z)−µ1)
µ11X(C.40)
=−iλX1(1−z)2µ1α1h1(z)−1δ1h1(z)
µ21+α1h1(z)+X1δ1h1(z)+β1h˜1(z)−X11h˜1(z)
Xµ1µ1
+(1−z)−2µ1β1h˜1(z)+11h˜1(z),
µ1170

11P1,22=−i(1−z)−2λX1∙
2X1∙(1−z)2+µ1h1(z)−1α1((1−z)µ1−(1−z)−µ1)+δ1((1−z)µ1+(1−z)−µ1)
µ1X1+(1−z)2−µ1h˜1(z)−1β1((1−z)µ1−(1−z)−µ1)+1((1−z)µ1+(1−z)−µ1)
µ11X(C.41)
=−iλX1(1−z)2µ1−1α1h1(z)+δ1h1(z)
µ21XX+1α1h1(z)+δ1h1(z)−1β1h˜1(z)+1h˜1(z)
X1µ1µ1
+(1−z)−2µ1β1h˜1(z)+1h˜1(z).
µ1AsP1shallbeholomorphicatz=z1,alsothematrixcoefficientsP1,12andP1,22shallbeholomorphic
atz=z1.Sinceµ1isnotanintegerexceptforsomeλfromadiscretesubsetofC∗,thisisonlypossible
ifthecoefficientsof(1−z)2µ1and(1−z)−2µ1vanish.FurthermoreP1(z1)=IandthusP1,12(z1)=0,
P1,22(z1)=1.Altogether,weinferthefollowingconditionsforthecoefficientsP1,12andP1,22,whereby
µwehavealreadyincorporatedµX11=X11andh1(1)=h˜1(1)=1:
µ1δ1=α1,(C.42)
X1µ1=−1β1,(C.43)
X12α1+2β1=0(C.44)
−iλX121α1−21β1=1.(C.45)
1µµ
XX211yieldsThisX1α1=−β1=i√,(C.46)
λXµ2111δ1=1=i√.(C.47)
λX21

yieldsThis

171

DAppendix:OnthenecessityoftheunitarizingmatrixT
Insection3.9,wepresentamatrixTsimultaneouslyunitarizingthemonodromymatricesM0(λ)and
M1(λ)givenin(3.9.3)and(3.9.4),respectively,forallλ∈S1.Thisismotivatedbytheclaim,that
M1(λ)isingeneralnotalreadyunitaryforλ∈S1.Inthisappendix,weprovethementionedclaimby
constructingacounterexample,inwhichactuallyM1(λ0)isnotunitaryforanappropriateλ0∈S1.
WeconsiderthematrixM1(λ).Assumingboths0ands1donotequal41andusingequations(2.6.23)
and(3.8.56)aswellasdet(A)=1fromlemma3.39,wecompute
M1(λ)=−Ae2πiD1A−1
=µ1R0Sκ1102λ02κ12S−1R−1R1Se−20πiµS−1R−1R1Sκ1202−λ01κ12S−1R−1
01−1012πiµ102−101
µ0λκ11κ1210e11−λκ11κ110
µ1e2πiµ1κ1101κ1202−e−2πiµ1κ1201κ1102λ−1κ1101κ1201(e−2πiµ1−e2πiµ1)−1−1
=µR0Sλκ02κ02(e2πiµ1−e−2πiµ1)e−2πiµ1κ01κ02−e2πiµ1κ01κ02SR0
0111211121211
1µ1A−C−B+AµX00(A−C+B−A)
(D.1),=X2µ0µ00(A+C−B−A)A+C+B+A

where

A=A(λ)=e2πiµ1κ1101κ1202−e−2πiµ1κ1201κ1102,(D.2)
1211B=B(λ)=κ01κ01(e−2πiµ1−e2πiµ1),(D.3)
C=C(λ)=κ1102κ1202(e2πiµ1−e−2πiµ1).(D.4)
Now,aneasycalculationshowsthatM1isunitaryforallλ∈S1,i.e.oftheformuvforallλ∈S1
u¯v¯−ifandonlyifC(λ)=−B(λ)forallλ∈S1,i.e.ifandonlyif
−κ1101κ1201sin(2πµ1)=κ1102κ1202sin(2πµ1)(D.5)
forallλ∈S1.(Notethatµ1aswellastheconnectioncoefficientsκkijlarerealvaluedonS1,aslongas
theyaredefinedatall,cf.lemmaB.5andequations(3.7.19)to(3.7.22).)
Toexplicitlyconstructacounterexample,inwhichM1(λ0)isnotunitaryforanappropriateλ0∈S1,
wesets:=s0=s1=s∞:=81.BylemmaB.6,weobtainµ:=µ0=µ1=µ∞=41+w(λ−λ−1)2
withw:=w0=w1=w∞=643.Moreover,alsobylemmaB.6,µ0,µ1andµ∞satisfytheunitarizability
condition(3.5.28)andthereforegiverisetoatrinoidpotentialη.Thus,thegivenchoiceofs0,s1and
s∞isvalidforourconsiderations.
Inviewofequations(3.7.19)to(3.7.22)aswellasequations(3.7.8)to(3.7.10),(D.5)readsunderthe
givenpresumptionsas
−sin(2πµ)Γ(1+2µ)Γ(−2µ)Γ(1+2µ)Γ(2µ)=sin(2πµ)Γ(−2µ)Γ(1−2µ)Γ(2µ)Γ(1−2µ).
Γ(21−µ)Γ(21+µ)Γ(21+3µ)Γ(21+µ)Γ(21−3µ)Γ(21−µ)Γ(21+µ)Γ(21−µ)
(D.6)Consequently,M1(λ)isunitaryforallλ∈S1ifandonlyif(D.6)holdsforallλ∈S1.
However,settingλ0:=i∈S1,wehaveµ(λ0)=41andthusobtainforthelefthandsideof(D.6):
Γ(1+21)Γ(−21)Γ(1+21)Γ(21)21Γ(21)Γ(−21)21Γ(21)Γ(21)(Γ(21))3Γ(−21)
−=−=−16.
Γ(41)Γ(1−41)Γ(1+41)Γ(1−41)Γ(41)(−41)Γ(−41)41Γ(41)(−41)Γ(−41)(Γ(41))2(Γ(−41))2
(D.7)(Here,wehaveusedtherelationΓ(z+1)=zΓ(z),whichholdsforzinCexcludingthenon-positive
integers(cf.[33],chapter2,2.1).Similarly,wecomputetherighthandsideof(D.6):
Γ(−21)Γ(21)Γ(21)Γ(21)Γ(−21)Γ(21)Γ(12)Γ(21)(Γ(21))3Γ(−21)
==−4.(D.8)
Γ(−41)Γ(41)Γ(1−41)Γ(41)Γ(−41)Γ(41)(−41)Γ(−41)Γ(41)(Γ(41))2(Γ(−41))2
Comparing(D.7)and(D.8),weinferthat(D.6)doesnotholdforλ=λ0,andthusthatM1(λ0)isnot
.unitary

172

EAppendix:Amendmentstotheproofoftheorem3.53
Inthisappendix,weprovidetheoutstandingcomputationsexcludedfromtheproofoftheorem3.53.I.e.,
weprovethatthematrixequation(3.9.56),reading
√µ1κ1101λ−1κ1201
−i√µ0λκ1102κ1202
=1ω0−1ω1λ−1(q0+q1−p0q1+p1q0)ω0−1ω1−1λ−1(−q0+q1−p0q1+p1q0),
2λ−1q0λ−1q1ω0ω1(−q0+q1+p0q1−p1q0)ω0ω1−1λ−1(q0+q1+p0q1−p1q0)
(E.1)isequivalenttothescalarequations
√02κ1201q0+q1+p0q1−p1q0
ω0=δκ12√−q0+q1−p0q1+p1q0,(E.2)
√κ1201q0+q1−p0q1+p1q0
ω1=δ˜κ1101√−q0+q1−p0q1+p1q0,(E.3)
p0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞),(E.4)
2sin(2πµ0)sin(2πµ1)
whereδ,δ˜∈{±1},suchthat
√√√δδ˜=√µ0q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0.(E.5)
µ1−2iλλ−1q0λ−1q1κ1101κ1202
Ofcourse,thisequivalenceholdsonlyforthoseλ∈S1,forwhichalloccurringtermsarewelldefined.
Inviewofremark3.43,wewilltacitlyassumethisinthefollowing.Inparticular,wewillignorethe
cases,inwhichcertainλ-dependenttermswedividebyshouldvanishforcertainisolatedvaluesofλ.
Throughoutthissection,wemakeuseofthefollowingrelations,holdingforj=0,1:
pj2+qjqj=1andpj=pj.(E.6)
Moreover,wewillusethefollowinglemma:
E.1.Lemma−4q0q1µ1κ1101κ1202=(q0+q1)2−(p0q1−p1q0)2
µ0⇐⇒p0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞)
2sin(2πµ0)sin(2πµ1)
µ⇐⇒−4q0q1µ1κ1102κ1201=(q0−q1)2−(p0q1−p1q0)2.(E.7)
0Proof.Recallequations(3.7.19)to(3.7.22)
01Γ(γ)Γ(γ−α−β)
κ11=Γ(γ−α)Γ(γ−β),(E.8)
κ1201=Γ(γ)Γ(α+β−γ),(E.9)
)β)Γ(αΓ(κ02=Γ(γ−α−β)Γ(2−γ),(E.10)
11Γ(1−α)Γ(1−β)
02Γ(α+β−γ)Γ(2−γ)
κ12=Γ(α−γ+1)Γ(β−γ+1),(E.11)
aswellasequations(3.7.8)to(3.7.10)
1α=2+µ0+µ1+µ∞,(E.12)
1β=+µ0+µ1−µ∞,(E.13)
2γ=1+2µ0.(E.14)
173

(E.15)(E.16)(E.17)

TheGammafunctionΓsatifiesthewellknownrelations(cf.[33],chapter2,2.1)
1π11Γ(+z)Γ(−z)=,z∈/{+w;w∈Z},(E.15)
22cos(πz)2
π−Γ(z)Γ(−z)=,z∈/Z,(E.16)
)zπsin(zzπΓ(1+z)Γ(1−z)=,z∈/Z.(E.17)
)zπsin(Usingtheserelations,weobtain
0102Γ(1+2µ0)Γ(−2µ1)Γ(2µ1)Γ(1−2µ0)
κ11κ12=1111
Γ(2+µ0−µ1−µ∞)Γ(2+µ0−µ1+µ∞)Γ(2−µ0+µ1+µ∞)Γ(2−µ0+µ1−µ∞)
2πµ0−π
sin(2πµ0)2µ1sin(2πµ1)µ0cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))
=ππ=−,(E.18)
cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))µ1sin(2πµ0)sin(2πµ1)
and0102Γ(1+2µ0)Γ(2µ1)Γ(−2µ1)Γ(1−2µ0)
κ12κ11=1111
Γ(2+µ0+µ1+µ∞)Γ(2+µ0+µ1−µ∞)Γ(2−µ0−µ1−µ∞)Γ(2−µ0−µ1+µ∞)
2πµ0−π
sin(2πµ0)2µ1sin(2πµ1)µ0cos(π(µ0+µ1+µ∞))cos(π(µ0+µ1−µ∞))
=ππ=−.(E.19)
cos(π(µ0+µ1+µ∞))cos(π(µ0+µ1−µ∞))µ1sin(2πµ0)sin(2πµ1)
yieldsThis

−4q0q1µµ1κ1101κ1202=4q0q1cos(π(µ0−µ1sin(2−µπµ∞)))cos(sin(2ππ(µµ0)−µ1+µ∞)),(E.20)
100−4q0q1µ1κ1102κ1201=4q0q1cos(π(µ0+µ1+µ∞))cos(π(µ0+µ1−µ∞)).(E.21)
µ0sin(2πµ0)sin(2πµ1)
Moreover,inviewof(E.6),wecompute
(q0±q1)2−(p0q1−p1q0)2=q02(1−p12)+q12(1−p02)+2q0q1(p0p1±1)
=q02q1q1+q12q0q0+2q0q1(p0p1±1)=2q0q1(q0q1+q0q1+p0p1±1).(E.22)
2Altogether,usingequations(E.20),(E.21)and(E.22)andthefactthat,ingeneral,q0,q1=0,theclaimed
equivalencecanberewrittenas
2cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))=q0q1+q0q1+p0p1+1
sin(2πµ0)sin(2πµ1)2
⇐⇒p0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞)
2sin(2πµ0)sin(2πµ1)
⇐⇒2cos(π(µ0+µ1+µ∞))cos(π(µ0+µ1−µ∞))=q0q1+q0q1+pp−1.(E.23)
sin(2πµ0)sin(2πµ1)201
Finally,usingtheformulascos(x)cos(y)=21(cos(x−y)+cos(x+y))andcos(x±y)=cos(x)cos(y)
sin(x)sin(y),thefollowingcomputationsfinishtheproof:
2cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))−1=cos(2πµ∞)+cos(2π(µ0−µ1))−1
sin(2πµ0)sin(2πµ1)sin(2πµ0)sin(2πµ1)
=cos(2πµ∞)+cos(2πµ0)cos(2π(µ1)+sin(2πµ0)sin(2π(µ1)−1
sin(2πµ0)sin(2πµ1)
=cos(2πµ∞)+cos(2πµ0)cos(2π(µ1),(E.24)
sin(2πµ0)sin(2πµ1)

174

2cos(π(µ0+µ1+µ∞))cos(π(µ0+µ1−µ∞))+1=cos(2πµ∞)+cos(2π(µ0+µ1))+1
sin(2πµ0)sin(2πµ1)sin(2πµ0)sin(2πµ1)
=cos(2πµ∞)+cos(2πµ0)cos(2π(µ1)−sin(2πµ0)sin(2π(µ1)+1
sin(2πµ0)sin(2πµ1)
=cos(2πµ∞)+cos(2πµ0)cos(2π(µ1).(E.25)
sin(2πµ0)sin(2πµ1)
Withthesepreparationsmade,wecanfinallyprovetheclaimedequivalenceofthematrixequation
(E.1)ontheonesideandthethreescalarequations(E.2),(E.3)and(E.4)ontheotherside.Since(E.1)
isequivalenttothefourscalarequations
√µ1011−1
10−iλ√µ0κ11=2λ−1qλ−1qω0ω1(q0+q1−p0q1+p1q0),(E.26)
√µ1011−1−1
10−iλ√µ0κ12=2λ−1qλ−1qω0ω1(−q0+q1−p0q1+p1q0),(E.27)
√1µ10210−iλ√µ0κ11=2λ−1qλ−1qω0ω1(−q0+q1+p0q1−p1q0),(E.28)
√µ1021−1
10−iλ√µ0κ12=2λ−1qλ−1qω0ω1(q0+q1+p0q1−p1q0),(E.29)
itremainstoshowthat
(E.27)(E.26)(E.2)
(E.4)(E.29)(E.28)⇐⇒(E.3)(E.30)
Asremarkedbefore,weexcludevaluesofλ∈S1fromourconsiderations,forwhichanyoftheterms
involvedinthefollowingcomputationsarenotwelldefined.
Proofof“⇒”in(E.30).Dividingequation(E.29)byequation(E.27),weobtain
κ12022q0+q1+p0q1−p1q0
κ1201=ω0−q0+q1−p0q1+p1q0,(E.31)
whichimplies√
0κ1201q0+q1+p0q1−p1q0
ω=δκ1202√−q0+q1−p0q1+p1q0,(E.32)
whereδ∈{±1}.Similarly,dividingequation(E.26)byequation(E.27)yields
κ11012q0+q1−p0q1+p1q0
κ1201=ω1−q0+q1−p0q1+p1q0,(E.33)
andthus01√
κ1201q0+q1−p0q1+p1q0
ω1=δ˜κ11√−q0+q1−p0q1+p1q0,(E.34)
whereδ˜∈{±1}.Bymultiplying(E.26)and(E.29),weinferthat
−µ1κ1101κ1202=11((q0+q1)2−(p0q1−p1q0)2),(E.35)
µ04q0q1
which,bylemmaE.1,implies(E.4).Asafurtherconsequenceof(E.35)wehave
√µ1√√
02λλq0λq1
(−i)√µ1κ1101κ1202=−1−1q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0,(E.36)
where√√√=√µ0q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0∈{±1}.(E.37)
µ1−2iλλ−1q0λ−1q1κ1101κ1202
175

(E.40)(E.41)

Thus,using(E.35)onceagain,weinferthat
˜κ1202κ1101−q0+q1−p0q1+p1q0
ω0ω1=δδ√√
κ1201q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
√√˜κ1202κ1101(−q0+q1−p0q1+p1q0)q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
δδ=µκ1201−4q0q1µ01κ1101κ1202
=δδ˜−q√0+q1−p0q1+p1q0∙.(E.38)
−2iλ√µµ1λ−1q0λ−1q1κ1201
0Inviewof(E.27),thisimpliesω0ω1=δδ˜ω0ω1,i.e.
δδ˜=,(E.39)
whichcompletestheproof.
Proofof“⇐”in(E.30).BylemmaE.1,wehavetherelations
µ1−4q0q1κ1101κ1202=(q0+q1)2−(p0q1−p1q0)2,(E.40)
µ0µ1−4q0q1κ1102κ1201=(q0−q1)2−(p0q1−p1q0)2.(E.41)
µ0Usingthesetogetherwiththeassumptions,weobtainbydirectcomputation
01κ√√02κ12ω0−1ω1(q0+q1−p0q1+p1q0)=δδ˜11q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
√√1110=√µ0(q0+q1+p0q1−p1q0)(q0+q1−p0q1+p1q0)=(E.40)−2iλ√µ1λ−1qλ−1qκ01,(E.42)
µ1−2iλλ−1q0λ−1q1κ1202µ0
−1−1κ1201√√
κκ1112ω0ω1(−q0+q1−p0q1+p1q0)=δδ˜0201q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
√01√1210√√=µ0(q0+q1+p0q1−p1q0)(q0+q1−p0q1+p1q0)κ12=(E.40)−2iλµ1λ−1qλ−1qκ01,(E.43)
µ1−2iλλ−1q0λ−1q1κ1202κ1101µ0
0102ωω(−q+q+pq−pq)=δδ˜κ12κ11(−√q0+q1−p0q1+p1q0)(√−q0+q1+p0q1−p1q0)
01010110
κ1201q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
√√1110√√=µ0(−q0+q1−p0q1+p1q0)(−q0+q1+p0q1−p1q0)=(E.41)−2iλµ1λ−1qλ−1qκ02(E.44)
µ1−2iλλ−1q0λ−1q1κ1201µ0
and−1κ1202√√
κ11ω0ω1(q0+q1+p0q1−p1q0)=δδ˜01q0+q1+p0q1−p1q0q0+q1−p0q1+p1q0
√√1210√√=µ0(q0+q1−p0q1+p1q0)(q0+q1+p0q1−p1q0)=(E.40)−2iλµ1λ−1qλ−1qκ02.(E.45)
µ1−2iλλ−1q0λ−1q1κ1101µ0
Theserelationsimplyequations(E.26)to(E.29).

176

FAppendix:Proofofremark3.55
Weprovethestatementofremark3.55:
LemmaF.1.Equation(3.9.51),i.e.
p0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞),(F.1)
2sin(2πµ0)sin(2πµ1)
issolvableforfunctionsp0,q0,p1,q1inλ∈S1satisfying(3.9.50),i.e.
pj2+qjqj=1andpj=pj,(F.2)
ifandonlyiftheeigenvaluesµjoftheDelaunaymatricesDjinducingthepotentialηmeettheunitariz-
abilitycondition(3.5.28),i.e.
0≤cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))≤1,(F.3)
sin(2πµ0)sin(2πµ1)
forallλ∈S1.
Proconditionof.Onthe(F.3)oneforλhand,∈S1.presumeThen,thebyeigentheoremvaluesµ3.14,joftheretheexistsDelaunaaymatrimatricesxTDjsimmeetultaneouslytheununitarizabilittarizingy
themonodromymatricesM0andM1.Bytheorem3.53,Tisoftheform(3.9.49)involvingfunctions
p0,q0,p1,q1inλ∈S1satisfying(F.2)and(F.1).I.e.,(F.1)issolvable.
Ontheotherhand,supposethereexistfunctionsp0,q0,p1,q1inλ∈S1satisfying(F.2)andsolving
(F.1).Decomposingqj=uj+ivjwithrealvaluedfunctionsujandvj,wenotethat
pj2=1−qjqj=1−uj2−vj2,(F.4)
q0q12+q0q1=21(u0u1+v0v1+i(v0u1−u0v1)+u0u1+v0v1−i(v0u1−u0v1))=u0u1+v0v1.(F.5)
Usingtheserelationsandapplyingelementaryestimates,weobtain
|p0p1+q0q1+q0q1|=|p0p1+u0u1+v0v1|
2≤|p0p1|+|u0u1|+|v0v1|≤21(p02+p12)+21(u02+u12)+21(v02+v12)=1.(F.6)
As,byassumption,p0,q0,p1,q1solve(F.1)forλ∈S1,weconcludethat,forλ∈S1,
−1≤cos(2πµ0)cos(2πµ1)+cos(2πµ∞)≤1.(F.7)
sin(2πµ0)sin(2πµ1)
Recallnowthat,byremark3.16,s0=t0ands1=t1.Thus,bylemmaB.5ofappendixB,wehave
0<µj≤21forj=0,1andforallλ∈S1andthereforesin(2πµ0)sin(2πµ1)≥0forallλ∈S1.
Consequently,(F.7)isequivalentto
−sin(2πµ0)sin(2πµ1)≤cos(2πµ0)cos(2πµ1)+cos(2πµ∞)≤sin(2πµ0)sin(2πµ1)(F.8)
and,byfurthertransformations,equivalentto
0≤cos(2πµ0)cos(2πµ1)+sin(2πµ0)sin(2πµ1)+cos(2πµ∞)≤2sin(2πµ0)sin(2πµ1)(F.9)
and0≤cos(2πµ0−2πµ1)+cos(2πµ∞)≤2sin(2πµ0)sin(2πµ1)(F.10)
and0≤21(cos(π(µ0−µ1−µ∞)+π(µ0−µ1+µ∞))+cos(π(µ0−µ1−µ∞)−π(µ0−µ1+µ∞)))
≤sin(2πµ0)sin(2πµ1).(F.11)
Thisyields,usingthetrigonometricidentity21(cos(x+y)+cos(x−y))=cos(x)cos(y),
0≤cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))≤sin(2πµ0)sin(2πµ1).(F.12)
Since,asstatedbefore,sin(2πµ0)sin(2πµ1)≥0forλ∈S1,weendupwith
0≤cos(π(µ0−µ1−µ∞))cos(π(µ0−µ1+µ∞))≤1.(F.13)
sin(2πµ0)sin(2πµ1)

177

GAppendix:Proofofremark3.56
Weprovethestatementofremark3.56:
LemmaG.1.Forj=0,1,∞,letpj,qjbethefunctionsoccurringintheunitarymonodromymatrixMˆj
asin(3.9.26),satisfying(3.9.27).Thefollowingholds:
p0,q0,p1,q1solvep0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞)
2sin(2πµ0)sin(2πµ1)
andp∞,q∞aregivenby(3.9.33)and(3.9.34)
⇐⇒p0,q0,p∞,q∞solvep0p∞+q0q∞+q0q∞=cos(2πµ0)cos(2πµ∞)+cos(2πµ1)
2sin(2πµ0)sin(2πµ∞)(G.1)
andp1,q1aregivenby(3.9.35)and(3.9.36)
⇐⇒p1,q1,p∞,q∞solvep1p∞+q1q∞+q1q∞=cos(2πµ1)cos(2πµ∞)+cos(2πµ0)
2sin(2πµ1)sin(2πµ∞)
andp0,q0aregivenby(3.9.37)and(3.9.38).
Proof.Recalltheidentity(3.9.32),Mˆ0Mˆ1Mˆ∞=I,whichinviewofremark3.48impliestherelations
(3.9.33),(3.9.34),(3.9.35),(3.9.36)(3.9.37)and(3.9.38).
Using(3.9.33)and(3.9.34)(and(3.9.27)),weprovetheimplication
p0,q0,p1,q1solvep0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞)
2sin(2πµ0)sin(2πµ1)(G.2)
=⇒p0,q0,p∞,q∞solvep0p∞+q0q∞2+q0q∞=cos(2πµ0sin(2)πcos(2µ)πµsin(2∞)π+µcos(2)πµ1).
0∞Tothisend,wecompute
isin(2πµ∞)(p0p∞+q0q∞+q0q∞)=p0(isin(2πµ∞)p∞)+q0(isin(2πµ∞)q∞)+q0(isin(2πµ∞)q∞)
222=p0[−cos(2πµ∞)−cos(2πµ0)cos(2πµ1)+icos(2πµ0)sin(2πµ1)p1+isin(2πµ0)cos(2πµ1)p0
+sin(2πµ0)sin(2πµ1)(p0p1+q0q1)]
+q20[icos(2πµ0)sin(2πµ1)q1+isin(2πµ0)cos(2πµ1)q0−sin(2πµ0)sin(2πµ1)(p0q1−p1q0)]
q0+2[icos(2πµ0)sin(2πµ1)q1+isin(2πµ0)cos(2πµ1)q0+sin(2πµ0)sin(2πµ1)(p0q1−p1q0)]
=p0(−cos(2πµ∞)−cos(2πµ0)cos(2πµ1))+p0p1(icos(2πµ0)sin(2πµ1))+p02(isin(2πµ0)cos(2πµ1))
+p02p1(sin(2πµ0)sin(2πµ1))+p0q0q1(sin(2πµ0)sin(2πµ1))
+q0q1(icos(2πµ0)sin(2πµ1))+q0q0(isin(2πµ0)cos(2πµ1))
22−p0q0q1(sin(2πµ0)sin(2πµ1))+p1q0q0(sin(2πµ0)sin(2πµ1))
22+q02q1(icos(2πµ0)sin(2πµ1))+q02q0(isin(2πµ0)cos(2πµ1))
+p0q0q1(sin(2πµ0)sin(2πµ1))−p1q0q0(sin(2πµ0)sin(2πµ1))
22=p0p1+q0q1+q0q1(icos(2πµ0)sin(2πµ1))+p02+q0q0(isin(2πµ0)cos(2πµ1))
2+pop0p1+q0q12+q0q1(sin(2πµ0)sin(2πµ1))+p0(−cos(2πµ∞)−cos(2πµ0)cos(2πµ1))
)µπcos(20=i(cos(2πµ0)cos(2πµ1)+cos(2πµ∞))sin(2πµ0)+isin(2πµ0)cos(2πµ1).

178

(G.3)

impliesThispp+q0q∞+q0q∞
0∞2=(cos(2πµ)cos(2πµ)+cos(2πµ))cos(2πµ0)+sin(2πµ0)cos(2πµ1)
01∞sin(2πµ0)sin(2πµ∞)sin(2πµ∞)
cos(2πµ0)cos(2πµ∞)+cos(2πµ1)
,=sin(2πµ0)sin(2πµinfty)

whichSimplyprovesshiftingtheclaimedindices,weimplication.provecompletelyanalogously
p0,q0,p∞,q∞solvep0p∞+q0q∞+q0q∞=cos(2πµ0)cos(2πµ∞)+cos(2πµ1)
2sin(2πµ0)sin(2πµ∞)
=⇒p1,q1,p∞,q∞solvep1p∞+q1q∞+q1q∞=cos(2πµ1)cos(2πµ∞)+cos(2πµ0)
2sin(2πµ1)sin(2πµ∞)
byusing(3.9.35)and(3.9.36),and,byusing(3.9.37)and(3.9.38)
p1,q1,p∞,q∞solvep1p∞+q1q∞+q1q∞=cos(2πµ1)cos(2πµ∞)+cos(2πµ0)
2sin(2πµ1)sin(2πµ∞)
=⇒p0,q0,p1,q1solvep0p1+q0q1+q0q1=cos(2πµ0)cos(2πµ1)+cos(2πµ∞).
2sin(2πµ0)sin(2πµ1)
Altogether,thestatementofthelemmafollows.

179

(G.4)

(G.5)

(G.6)

HAppendix:Proofoftheorem5.16
Inthisappendix,wegivetheproofof
Theorem5.16.Forallλ∈C∗thefollowingholds:
4sin2(πµ(λ))−1=4π2∞Ck(1−λ)(1+λ)(1−λ−1)(1+λ−1),(H.1)
k=−∞λkλkλkλk
where−λk2wfork=0
Ck:=λk2wfork=−1(H.2)
)(1+kk2λk2w131fork∈Z\{−1,0}
and,fork∈Z,
λk:=21wdk+dk2−4w2(H.3)
withdk:=(1+k)2−1+2w.(H.4)
46Recallfromlemma5.3that
µ(λ)=1+w(λ−λ−1)2,(H.5)
4wherew=s0t0=s1t1=s∞t∞andsj,tjdenotetheparameters1occurringintheDelaunaymatricesDj
definedTheinproof(3.5.7).oftheorem(Notethat5.16isremarkprepared3.16inimpliesthefollothatwingw=three16.)lemmas.
LemmaH.1.Forallk∈ZletIk:={±λk,±λk−1}asinlemma5.15.Then,forallλ∈C∗\k∈ZIk
holds:lowingfolthe2π4−4sin2(πµ(λ))−1=Γ(61−µ(λ))Γ(61+µ(λ))Γ(65−µ(λ))Γ(65+µ(λ)),(H.6)
whereΓdenotestheGammafunctionΓ(z)=0∞e−ttz−1dt.
RemarkH.2.Writingsin(πµ)initspowerseriesrepresentation,weobservethattheexpressionsin2(πµ)
onlyinvolves2evenpowersofµ.Since,byremark∗3.13,µ2definesaholomorphicfunctiononC∗,we
interpretsin(πµ)asaholomorphicfunctionofλ∈C.
ProofoflemmaH.1.WeusethefollowingwellknownformulafortheGammafunction:
πΓ(1−z)Γ(z)=sin(πz),z∈/Z.(H.7)
WeconsidertheproductofGammafunctions
Γ(61−µ(λ))Γ(61+µ(λ))Γ(65−µ(λ))Γ(65+µ(λ)).(H.8)
∗1This±µ(proλ),duct5±isµ(wλ)elltakesdefinedanon-p(andositivnon-zeerino)tegerforvallalue.λ∈CI.e.,,fortheprowhichductnoneiswofelltheodefinedccurring(andargumennon-zero)ts
6forallλ∈6C∗,suchthatµ(λ)=±(61+k)forallk∈Z,or,equivalentlyduetolemma5.15,suchthat
λ∈/k∈ZIk.Thus,using(H.7),wecancomputeforallλ∈C∗\k∈ZIk:
1Γ(61−µ(λ))Γ(61+µ(λ))Γ(65−µ(λ))Γ(65+µ(λ))
1sin(π(61−µ(λ)))sin(π(61+µ(λ)))
=Γ(61−µ(λ))Γ(1−(61−µ(λ)))Γ(61+µ(λ)))Γ(1−(61+µ(λ)))=ππ
=[sin(6π)cos(−πµ(λ))+cos(6π)sin(−πµ(λ))][sin(6π)cos(πµ(λ))+cos(6π)sin(πµ(λ))]
2π=sin2(6π)cos2(πµ(λ))−cos2(6π)sin2(πµ(λ))=cos2(πµ(λ))−3sin2(πµ(λ))=1−4sin2(πµ(λ)).
π24π24π2
(H.9)claim.theimpliesThis

180

Lemma−1H.3.LetΓdenotetheGammafunctionΓ(z)=0∞e−ttz−1dtandforallk∈ZletIk:=
{±λk,±λk}asinlemma5.15.Moreover,let
˜1fork∈{−1,0}
Ck=k2(1+1k1)31fork∈Z\{−1,0}(H.10)
Then,forallλ∈C∗\k∈ZIkwehave
1.1∞1
Γ(61−µ(λ))Γ(61+µ(λ))=C˜k(6+k)2−(µ(λ))2,(H.11)
=0k2.−1
Γ(5−µ(λ1))Γ(5+µ(λ))=C˜k(61+k)2−(µ(λ))2.(H.12)
66=k−∞Proof.Weapplythefollowingformula,whichallowstorepresentΓ(z)asaninfiniteproduct.Forall
z∈C,exceptingthenon-positiveintegers,wehave(cf.[33],chapter2,2.2)
Γ(z)=1∞(1+k1)z.(H.13)
zk=11+kz
expressionsThe11Γ(61−µ(λ))Γ(61+µ(λ))andΓ(65−µ(λ))Γ(65+µ(λ))(H.14)
lemma.Thus,applying(H.13),weobtainforallλ∈C\k∈ZIk
arewelldefinedforallλ∈C∗\k∈ZIk.Thishas∗alreadybeenexplainedintheproofoftheprevious
111=(1−µ(λ))(1+µ(λ))∞(k+61−µ(λ))(k+161+µ(λ))
Γ(6−µ(λ))Γ(6+µ(λ))66k=1k2(1+k1)3
=∞C˜k(1+k)2−(µ(λ))2,(H.15)
6=0kwhereC˜isdefinedin(H.10).
Inviewkoftheformula(k=−1)
(−k−1)2(1+1)35=(k+1)2(k)35=(k+1)31k35=(1+1)31k2(H.16)
−k−1k+1k
wefurthermorecomputeforallλ∈C∗\k∈ZIk
155∞(l+5−µ(λ))(l+5+µ(λ))
Γ(65−µ(λ))Γ(65+µ(λ))=(6−µ(λ))(6+µ(λ))l=16l2(1+l1)356
=(5)2−(µ(λ))2∞(65+l)2−(µ5(λ))2=(1−1)2−(µ(λ))2−2(61+k)2−(µ(λ))25
6l=1l2(1+l1)36k=−∞(−k−1)2(1+−k1−1)3
=(61−1)2−(µ(λ))2(6+k)1−31(µ(2λ))=C˜k(61+k)2−(µ(λ))2,(H.17)
−2122−1
k=−∞(1+k)kk=−∞
wherewehavesubstitutedk=−l−1andC˜kisdefinedin(H.10).
RemarkH.4.Notethattheinfiniteproduct(H.13)representstheGammafunctionΓandthustakes
profiniteductvofaluestwinoC(or,onmorethecomplgenerallyex,ofplanefinitelyexcludingmany)thesuchnon-pinfiniteositiveproinductegers.tsoftheConsequenformtly(H.13),alsoiswtheell
definedonthecomplexplaneexcludingthenon-positiveintegers.Thisjustifiesthecalculationsinvolving
infiniteproductsoccurringintheproofaboveaswellasintheproofoftheorem5.16below.
181

(H.18)

LemmaH.5.Forallλ∈C∗andallk∈Zthefollowingholds:
1λλλ−1λ−1
(6+k)2−(µ(λ))2=λk2w(1−λk)(1+λk)(1−λk)(1+λk)(H.18)
whereλkisdefinedby(5.5.4)inlemma5.15.
Proof.Recallingthatµ(λ)=41+w(λ−λ−1)2,thisisprovedbyadirectcomputation.Inviewof
lemma5.15wehave
λ0=1[d0+d02−4w2]andλ0−1=−1[d0−d02−4w2],(H.19)
112w2w
λk=2w[dk+dk2−4w2]andλk−1=2w[dk−dk2−4w2]fork=0,(H.20)
whichinanycase(k∈Z)implies
λk2+λk−2=dk.(H.21)
wHence,wecomputeforallλ∈C∗
(61+k)2−(µ(λ))2=dk−2w−w(λ−λ−1)2=w(λk2+λk−2)−w(λ2+λ−2)=λk2w(1+λk−4−λ2λk−2−λ−2λk−2)
11−−=λ2kw(1−λk−2λ2)(1−λk−2λ−2)=λk2w(1−λ)(1+λ)(1−λ)(1+λ),(H.22)
λkλkλkλk
whichprovestheclaim.
Proofoftheorem5.16.InviewofthelemmasH.1,H.3andH.5,weobtainforallλ∈C∗\k∈ZIk,where
Ik:={±λk,±λk−1},that
2π4−24sin(πµ(λ))−1=Γ(61−µ(λ))Γ(61+µ(λ))Γ(65−µ(λ))Γ(65+µ(λ))
=−4π2C˜k(1+k)2−(µ(λ))2=4π2Ck(1−λ)(1+λ)(1−λ)(1+λ).(H.23)
∞∞−1−1
k=−∞6k=−∞λkλkλkλk
Here,C˜kisgivenin(H.10).
thatinfereW11∞−−4sin2(πµ(λ))−1=4π2Ck(1−λ)(1+λ)(1−λ)(1+λ)(H.24)
k=−∞λkλkλkλk
equationarezeroforλ∈k∈ZIk(cf.lemma5.15).Altogether,theclaimfollows.
forallλ∈C∗\k∈ZIk.But,naturally,thisequationalsoholdsforallλ∈k∈ZIk,asbothsidesofthe

182

IAppendix:Proofsoflemma5.21andlemma5.24
First,wegivetheproofof
Lemma5.21.Forallk∈Z,letλkandCkbegivenby(5.5.4)and(5.5.32),respectively.Moreover,let
theλ-dependentfunctionspk(ν)(λ),ν∈{1,2,3,4}bedefinedby(5.5.44),(5.5.45),(5.5.46)and(5.5.47),
respectively.Then,wehave:
1.Theinfiniteproductk∞=−∞√Ckconverges.
2.Theinfiniteproductk∞=−∞pk(1)isnormallyconvergentonC∗.
3.Theinfiniteproductk∞=−∞pk(2)isnormallyconvergentonC∗.
4.Theinfiniteproductk∞=−∞pk(3)isdivergentonC∗.
5.Theinfiniteproductk∞=−∞pk(4)isdivergentonC∗.
Fortheconvenienceofthereader,werecallthedefinitionsofλk,ofCkandofthefunctionsp(kν)(λ),
ν∈{1,2,3,4}:Fork∈Zwehave
1λk=2wdk+dk2−4w2,(I.1)
wheredk:=(61+k)2−41+2w.(I.2)
Moreover,alsofork∈Z,
−λk2wfork=0
Ck:=λk2w2fork=−1(I.3)
k2λ(1+kwk1)31fork∈Z\{−1,0}
Finally,fork∈Z,
λλ(1)pk(λ)=(1−λk)(1+λk)(I.4)
(2)λ−1λ−1
pk(λ)=(1−λk)(1+λk)(I.5)
1−λλpk(3)(λ)=(1(1−−λλ0)(1)(1+−λλ−01))forforkk∈=Z0\{0}(I.6)
λλkk(4)(1+λλ0)(1+λλ−01)fork=0
pk(λ)=(1+λλk)(1−λλ−k1)fork∈Z\{0}(I.7)
Proofoflemma5.21.Westartwiththeproofofthefirstclaim:Theinfinite∞√productk∞∞=−∞√Ckcon-
vverge.erges.TothisReferringend,toweremarkapply5.19,lemmawep5.20,rovei.e.wthateprothevetheinfiniteconvproergenceductsofn=0theCserieskand∞n(=1√CkC−−k1)con-and
=0k∞k=1W(ewillC−kuse−the1),wrespellknoectivwnely.formula(“binomialtheorem”)
∞(x+y)α=nαxnyα−n,(I.8)
=0nwhichisvalidforx,y∈Csatisfying|yx|<1andα∈C.Here,thegeneralizedbinomialcoefficientis
definedby
α=1,α=α∙(α−1)∙∙∙(α−n+1)forn∈N.(I.9)
!nn0183

Consider(fork∈Z)
2w6464
λk=1(1+k)2−1+2w+((1+k)2−1+2w)2−4w2
1122w122w4w2
=k2w1+3k−9k2+k2+(1+3k−9k2+k2)2−k2
=k2w1+3k−9k2+k2+1+3k−3k2+k2−27k3+3k3+81k4−9k4.(I.10)
1122w214w44w48w
Byadirectcomputationoneverifiesthattheexpression
214w44w48w
xk:=3k−3k2+k2−27k3+3k3+81k4−9k4(I.11)
satisfies−1<xk<11foratleastallkgreaterorequalsomek0∈N.Thus,wecanapply(I.8)(with
x=xk,y=1andα=2)forallk≥k0toobtain
1122w√
λk=k2w1+3k−9k2+k2+1+xk

=k21w1+31k+O(k12)+1+x2k+O(k12)=kw11+31k+O(k12),(I.12)
k→where∞)webhehaavveesasusedthethefunctinotationonf:OZ(f→)Rfor,ka→f(functionk).ZAgain,→Rweofcank∈applyZ,whic(I.8)hforallkasymptotically≥k0(i.e.(adjustingfor
k0ifnecessary),tocontinueourcomputation:
11kλk=√w(1+6k+O(k2)).(I.13)
Inparticular,thisshowsthatλkisoftheformO(k).
Basedon(I.13),wecompute(oncemoreusing(I.8))
1√Ck−1=wλk−k(11+k1)6=k(1+61k+O(k12))−k(1+61k+O(k12))=O(1),(I.14)
k(1+k1)6k(1+61k+O(k12))k2
√1theseriesk=0(Ck−1)andk=1(C−k−1)converge,whichimpliesthefirstclaimoflemma5.21.
whichmeans∞that√Ck−1asymptotically∞behaveslikek2.ThesameholdsforC−k−1.Consequently,
O(k),Theweinferremainingthatλ−claims1isofofthelemmaform5.21O(1are).nowConseproqvuenedtly,quitetheeasily:seriesSince,by(I.13),λkisoftheform
kk∞λ2∞λ2∞λ−2∞λ−2
k=0−λk2,k=1−λ2−k,k=0−λk2,k=1−λ2−k(I.15)
convergenormallyonC∗,whiletheseries
∞λλ−11∞λλ−11∞λλ−11∞λλ−11
k=1−λk+λk−λk2,k=1−λ−k+λ−k−λ2−k,k=1λk−λk−λk2,k=1λ−k−λ−k−λ2−k(I.16)
divergeonC∗.Bydefinition5.18(inconjunctionwithremark5.19),weconcludethattheinfiniteproducts
∞∞∞∞pk(1)(λ)=1−λ22,pk(2)(λ)=1−λ−22(I.17)
k=−∞k=−∞λkk=−∞k=−∞λk
arenormallyconvergentonC∗,whiletheinfiniteproducts
∞∞pk(3)(λ),pk(4)(λ)(I.18)
k=−∞k=−∞
aredivergentonC∗.

184

Next,weprove
Lemma5.24.Theinfiniteproducts(offunctions)
gj,1(λ)andgj,2(λ),(I.19)
j∈Nj∈N
C∗\j∈N{±λj}andC∗\j∈N{±λj−1},respectively.
where,forallj∈N,gj,1isgivenin(5.5.94)andgj,2isgivenin(5.5.95),arenormallyconvergenton
Theexpressionsoccurringinlemma5.24aregivenby
(1−λ−1)(1+λ−1)
λλjjgj,1(λ)=(1−λ)(1+λ),(I.20)
λλjj(1−λλj)(1+λλj)
gj,2(λ)=λ−1λ−1,(I.21)
(1−λj)(1+λj)
whereλj:=1dj+dj2−4w2(I.22)
w2with11dj:=(+j)2−+2w.(I.23)
42Proofoflemma5.24.Analogouslyasintheproofoflemma5.21forλk(givenin(I.1)),oneshowsforλj
that(I.22)inengiv11jλj=√w(1+2j+O(j2)).(I.24)
ThisshowsinparticularthatλjisoftheformO(j).Consequently,theseries
2−2∞λ2−λ−2∞λλj2−λλj2
22=λ2,(I.25)
j=1λj−λj=11−λj2
22−2=j−2j(I.26)
∞λ−2−λ2∞λλ2−λλ2
j=1λj−λ−2j=11−λλj2
convergenormallyonC∗\j∈N{±λj}andC∗\j∈N{±λj−1},respectively.Bydefinition5.18,weconclude
thattheinfiniteproducts
λ−2λ2λ−2
gj,1(λ)=λ2=(1+λ2),(I.27)
∞∞1−λj2∞λj2−λj2
j=1j=11−λj2j=11−λj2
22−2gj,2(λ)==λ−j2=(1+jλ−2j)(I.28)
∞∞1−λλ2∞λλ2−λλ2
j=1j=11−λj2j=11−λj2
1−arenormallyconvergentonC∗\j∈N{±λj}andC∗\j∈N{±λj},respectively.
185

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