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Singular behaviour of rotationally symmetric surfaces of codimension two evolving under mean curvature flow [Elektronische Ressource] / Maren Stroot

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MarenSingularGradesb?eharer.viourMeppoferroNatationallyssertationsymmetricorensurfacessiofHannocoErlangungdimensionDoktorstwissenschaftenwgenehmigteoonevotolving12.10.1981under2010meantcurvtaturevozurwdesDereinesFderakult?tturf?rDr.Mathematiknat.undDiPhvysikDipl.-Math.derStroGottfriedgebWilhelmamLeibnizinUnivenerderReferenH.-Ch.t:2010Prof.TDr.04.K.Dr.SmoGrunauczykagKPromotion:orreferenJunit:Prof.41 3c :S !0 >0 3 3 1 4:= y2 ;y > 0>01 1 4F :S S ! M0 01 4y y 1 1 2 2 1 3 1 1 1 2 1 2 1M := c x cosx ;c x ;c x ;c x sinx ;x ;x 2S :0FtF03F c = M \t t t >0@c =H + ;@t3c (; 0) =c ; c ;0 t >0H [0;T )T <1cM 4 1 4y2 ;y = 0;y = 0 [0;T )j(p ;t )jn nlim = 0n!1jH(p ;t )jn n(p ;t )n n(p ;t )n n2 lim jj =1t!T?c = H 1 11vmsunhggest?rtensuBlossnRterHalbraumonimnfangswederKurv-uphlosseneengescallglatte,die(0.1a)neineerscRKr?mmseiKr?mmDazuolglictet.wiehalierungetrac,bamiliehenKr?mmFl?cenehedasstionssymmetriscdenRKurvrota-Singularit?timmergierte,dasRKr?mmdendurc(0.1b)istb.estimmobt.onHierbbeidiebtezeicDieshnenrecinessendenamilie(mittleren)enKr?mmeilfolgeungsvonektorergieren,derAlsKurvSingularit?tenetet,undSt?rtermf?rveinenwirdSt?rtermbersterergenzOrdnselbst?hnlicung,-EbderendurcSinne,herh?ltnisdieorsionRotationngenwindet:tstehet.VWisotropeiterderseiRungsuss,Kr?mmzummittleredesderessen,Fwirdhnet.eitexploArborsion,nicdasscmaximaleKr?mmZeitindasstervenall,RintlangdemBlo(0.1)olgeeineEin-PglattedieL?sungtlbeineresitzt.

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MarenSingularGradesb?eharer.viourMeppoferroNatationallyssertationsymmetricorensurfacessiofHannocoErlangungdimensionDoktorstwissenschaftenwgenehmigteoonevotolving12.10.1981under2010meantcurvtaturevozurwdesDereinesFderakult?tturf?rDr.Mathematiknat.undDiPhvysikDipl.-Math.derStroGottfriedgebWilhelmamLeibnizinUnivenerderReferenH.-Ch.t:2010Prof.TDr.04.K.Dr.SmoGrunauczykagKPromotion:orreferenJunit:Prof.4
1 3c :S !0 >0
3 3 1 4:= y2 ;y > 0>0
1 1 4F :S S ! M0 0
1 4y y

1 1 2 2 1 3 1 1 1 2 1 2 1
M := c x cosx ;c x ;c x ;c x sinx ;x ;x 2S :0
Ft
F0
3F c = M \t t t >0
@
c =H + ;
@t
3
c (; 0) =c ; c ;0 t >0
H
[0;T )
T <1
c
M

4 1 4y2 ;y = 0;y = 0
[0;T )
j(p ;t )jn nlim = 0n!1jH(p ;t )jn n
(p ;t )n n
(p ;t )n n
2
lim jj =1t!T
?c = H 1 11
vmsunhggest?rtensuBlossnRterHalbraumonimnfangswederKurv-uphlosseneengescallglatte,die(0.1a)neineerscRKr?mmseiKr?mmDazuolglictet.wiehalierungetrac,bamiliehenKr?mmFl?cenehedasstionssymmetriscdenRKurvrota-Singularit?timmergierte,dasRKr?mmdendurc(0.1b)istb.estimmobt.onHierbbeidiebtezeicDieshnenrecinessendenamilie(mittleren)enKr?mmeilfolgeungsvonektorergieren,derAlsKurvSingularit?tenetet,undSt?rtermf?rveinenwirdSt?rtermbersterergenzOrdnselbst?hnlicung,-EbderendurcSinne,herh?ltnisdieorsionRotationngenwindet:tstehet.VWisotropeiterderseiRungsuss,Kr?mmzummittleredesderessen,Fwirdhnet.eitexploArborsion,nicdasscmaximaleKr?mmZeitindasstervenall,RintlangdemBlo(0.1)olgeeineEin-PglattedieL?sungtlbeineresitzt.einerBeiener,dexereronAnalysedemdgeneseisingul?renerdenVdererhaltensetracderdiesemKurvdierten:dieserBilduntterZun?cdemMonotonieformelgest?rtenFlussFlussDiese(undondamitreskderzuerzeugtenL?sungFl?callenKr?m-henist:InebuniterdemdemdassmittlerenVK?mmvungsuss)TsindzuzwueivF?llehzuhuneintersconheiden.erhaltenZudasn?cist,hstungsusswmittlereerdenDaSingularit?tenImmersionuertnAtersucwheit,ungsussesdiemittlerenenL?sungentferneinettiellevw-uponolgederezeicRotationsebFenehZusammenfassungdiertmittlerenTgest?rteninRSingularit?tdemhtersounhnellRdie.ung.Rotationimpliziert,RnacdereinerKurvtspehendenineskReneneinertstehtiellenen.wInFdiesemvFarameter-FalldiebleibtsoderKurvSt?rtermenliefertaaufgdemTgesamzutenFZeitinebtervkallvonKurvvkerhaltenvVdiebmittlerenescungsusshr?nkt?gen.undzwdastedaswimdieallaufTRotationsebSingularit?tbderhwindieFungexploKurvderalsseihEsSt?rtermdasdieren.inSubjectariantion:un53A04,.Schlahstoreine:f?rKr?mmgest?rtenRotationssymmetrie,(0.1)eewiesen.oimplizierth?hereKovKr?mmderualiertenngsuss.enDaseinerhei?t,hendiedesKurvFlussesenRotationenwdererdeneneindersingul?reFVeinereyp-IrhinaltensounohlterKr?mmdemdergest?rteneFluaucssderistexplogleicAMShClassificadem53C44,singul?ren53A07.Vgwerhaltentevmittlereronungsuss,Kurvcurvenshorteningunw,terKdemdimensionmittleren4
1 3 3c : S ! :=0 >0 >0
3 1 4 1 1 4y2 ;y > 0 c F : S S !0 0
1 4M y y0

1 1 2 2 1 3 1 1 1 2 1 2 1
M := c x cosx ;c x ;c x ;c x sinx ;x ;x 2S :0
Ft
4F c :=M \0 t t
F ct t
@
c =H + ;
@t
3c (; 0) =c ; c ;0 t >0
H c t
T <1

4 1 4y2 ;y = 0;y = 0

j(p ;t )jn n
lim = 0n!1jH(p ;t )jn n
(p ;t )n n
(p ;t )n n
2
lim jj =t!T
1

?c =1
H 1 1
dobwytialthup,einbcurvehatheviourInofhtheageneratingalongcurvviouresydetermaturetotallysolutionsevcuolvingasunderofthew-uppshoerturbRedfamilymeanaturecurvofatuinreooypwdisformoferturbviouroaanishes,ehwbistheanisotropic,l(0.1a)tiisewForescalingatureRcurvsolutionsmeanutheex,Sinceunder.,RyandsurfacesdataaRthisinitialthewithprowyob(0.1b)thewheretermatureoisvthesolution(mean)wcurvtoatureandvatureectortheofbltheascurvcurvethecurvRmeanRandthenunderhisanalpceimmersedrtubrbationitvthatectoranofw-uprststudied.orderimmersedcomingvfromsubthetorotation.conFaurthermore,evolvingnevw.immersionsbdenotesathethemaximali.timerotationallyupreto,whicestigated.haaulasmoerturboth(0.1)solutioned.toccurring(0.1)ofexists.andWhentheanalyzingectortheesingularpbbloehamovicitiouimpliesrofofa(0.1)the(andthereforeththeisingularrsibnehatheviourrvofvthehence,surfaces)torsiononenothasotoupdfastistinguishthetaturewimageothatcases.sucFirst,immersionthedenessingularinofrotationnedRRfspaceAMSaClassifitheafor53C44,essen53A07.awbloandsequen:ecurvcurvoclosedrotationallyesurfaces,.eurthermore,eningisw,wnhafterralongdimensionessenfamilybloone-parametersequencetheLetRiseinbtheLetcon-plane:ergetheainseqrotationsenceallaunderoftvarianplanvr,esi.olvinge.meaifcurvinoremainsSecondlybtheouehanded.inIsingularitnonthisplanecase,rotation,curve.essymmetricevofolvingoundertuthecurvpiserturbvedInocase,wmonotonicitbformehaforvpeedlikweiscurvvesIfunderomeansingularitcisutrve-Iatureifoothwcurvinvtheofsingularitcurvy:anththeeerturbationcurvmeaneswbtheecomenplanartoninytheulasingulconaergencerirescaledttoyself-similarinofthpeedsenseothatesis,thethisratioAbstractoftheb.ehaSubjectviourcistion:studied53A04,ifKeytheordssingularitphrasesymeanoatureccursw,asymmetricwcurvashort-yofromhigtheeplanecoofrotationeuclideanContents.1..Introvdu.cimilartolutioni.on449on2..Mean59curvature.oertieswlintheeuclidean.spaceand16.2.1.moEv.olution.equationswofA.1.the.deriv.ativspaceesCylindricalof.theerturbsecondolutionsfundamenRotationallytal.form..in16I2.2..Dilation-inNotesvrotationarianicitt.estimates....5.1.1..p...a...........immersed...surfa.the...69.manifolds...B.3..space.o.a.84....19.3.BehaMeanypcurvatureypo.w.fo.r50rotationallysingulasymmetricplanesurfaces5.1.23o3.1.formEv.olution.equations.for.space.curv.es.evonolvingtounderedthe.p.ertuEvolutionrbsed.o.w...25.3.2..Some.geometric.prop.ertiesA.2.forforcoindimension.tB.wrotationallyo69surfacesordinates.curv.......Ev.equations.the.o..26.4.urtherFforoermationerturbof75singulasurfacesritiesrya80w.a.y.from.the.plane.of.rotation4.3.2.29viour4.1.tDilation-ine-Ivtariane-Itsingularitiesestimates.for.the.p.erturb.ed.o5.won.rities.the.of.54.A.n.ton.y.ula......32.4.1.1..Con.trolling.the.dissipation.of.the.curv.ature.and54torsionNotes.self-s.solutions.the.erturb.o35.4.2..The.to.talA.curvEquaturetion.64.Notations...................................64.Ev.equations.an.manifold.euclidean.....65.Prop.of4symmetric0ces4.3.B.1.Fcoormingandsingularitiesmeanbatureecomeectorplanar..........B.2..o.ution.for.under.p.ed.w.......72.F.ev.equations.a42curv4.3.1.underTheplimitedofwrescaledC.solutionssymmetric.in.rbitra.dimensional.space.Bibliography..nM (N;h)
nF :U !F (U)M F :M !N0 0 0 t t
@
F =H;
@t
F (; 0) =F ;0
H M Mt t
M [0;T )0
M Mt 0
M0
mF :U!F (U)M 0 0 0
2jFj
@ 2 2jFj = jFj 2n
@t
2jFj + 2nt
t2 [0;T ) T =1 M
M r
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M0
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n 2nF :M !N N !;J;g0 0
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