The contact Yamabe flow [Elektronische Ressource] / von Yongbing Zhang

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The Contact Yamabe FlowVon der Fakult˜at fur˜ Mathematik und Physikder Universit˜at Hannoverzur Erlangung des Grades einesDoktors der NaturwissenschaftenDr. rer. nat.genehmigte DissertationvonM. Sc. Yongbing Zhanggeboren am 19.11.1978 in Anhui, China2006Referent: Prof. Dr. K. SmoczykKoreferent: Prof. Dr. E. SchroheTag der Promotion: 09.02.2006iAcknowledgementsI am greatly indebted to my supervisor, Prof. Knut Smoczyk who has givenme many help and support during the past years from fall of 2002. His helpfulsuggestions made it possible that my dissertation appears in the present form.I am also grateful to Prof. Jiayu Li and Prof. Jurgen˜ Jost for their help andsupport.I am grateful to Prof. Guofang Wang, Prof. Xinan Ma and Prof. Xiux-iongChenfortheirconversationsinmathematicsduringmypreparationforthedissertation.IhavetogivemanythankstoDr.XianqingLi,Dr.WeiLi,Dr. QingyueLiu,Ye Li, Xiaoli Han, Liang Zhao, Prof. Chaofeng Zhu, Prof. Yihu Yang, Prof.QunChen,Prof.HuijunFan,Prof.ChunqinZhou,Dr.KanghaiTan,Dr. BoSu,Dr. Ursula Ludwig, Konrad Groh, Melanie Schunert and many other friendswith them I had learned a lot in mathematics and enjoyed a pleasant period inChina and Germany. I am also grateful to the referee, Prof. Schrohe, for thediscussion. He carefully reads this thesis and gives helpful suggestions.I am not only grateful to my family and many other friends for their manyyears’ favor.

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The Contact Yamabe Flow
Von der Fakult˜at fur˜ Mathematik und Physik
der Universit˜at Hannover
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
von
M. Sc. Yongbing Zhang
geboren am 19.11.1978 in Anhui, China
2006Referent: Prof. Dr. K. Smoczyk
Koreferent: Prof. Dr. E. Schrohe
Tag der Promotion: 09.02.2006i
Acknowledgements
I am greatly indebted to my supervisor, Prof. Knut Smoczyk who has given
me many help and support during the past years from fall of 2002. His helpful
suggestions made it possible that my dissertation appears in the present form.
I am also grateful to Prof. Jiayu Li and Prof. Jurgen˜ Jost for their help and
support.
I am grateful to Prof. Guofang Wang, Prof. Xinan Ma and Prof. Xiux-
iongChenfortheirconversationsinmathematicsduringmypreparationforthe
dissertation.
IhavetogivemanythankstoDr.XianqingLi,Dr.WeiLi,Dr. QingyueLiu,
Ye Li, Xiaoli Han, Liang Zhao, Prof. Chaofeng Zhu, Prof. Yihu Yang, Prof.
QunChen,Prof.HuijunFan,Prof.ChunqinZhou,Dr.KanghaiTan,Dr. BoSu,
Dr. Ursula Ludwig, Konrad Groh, Melanie Schunert and many other friends
with them I had learned a lot in mathematics and enjoyed a pleasant period in
China and Germany. I am also grateful to the referee, Prof. Schrohe, for the
discussion. He carefully reads this thesis and gives helpful suggestions.
I am not only grateful to my family and many other friends for their many
years’ favor.iii
Zusammenfassung
Das Yamabe-Problem ist eine klassische Fragestellung aus der Difierential-
geometrie. Es lautet: Ist eine gegebene kompakte und zusammenh˜angende
Mannigfaltigkeit konform ˜aquivalent zu einer Mannigfaltigkeit mit konstanter
Skalarkrumm˜ ung? DieseFragewurde1960vonYamabeformuliert. N.Trudinger
and T. Aubin erzielten erste Resultate. Das Yamabe-Problem wurde von R.
SchoenmitHilfedesPositiven-Masse-TheoremsimJahr1984vollst˜andiggel˜ost.
Das CR-Yambabe Problem wurde 1987 von D. Jerison und J. M. Lee for-
muliert. Man fragt: Existiert fur˜ eine gegebene kompakte, strikt pseudokon-
vexe CR-Mannigfaltigkeit eine pseudohermitische Struktur, welche konstante
Webster-Skalarkrumm˜ ung besitzt? Eine bejahende L˜osung dieser Frage wurde
von D. Jerison, J. M. Lee, N. Gamara und R. Yacoub im Jahr 2001 gefunden.
Das Yamabe-Problem wurde auch mit Hilfe von geometrischen Flu…en˜ von
R. Hamilton, R. Ye, H. Schwetlick, M. Struwe, und S. Brendle untersucht. Das
Verhalten des Flu…es ist auch fur˜ sich genommen interessant. In dieser Arbeit
benutzenwirdensogenanntenKontakt-Yamabe-Flu…umL˜osungendesKontakt-
Yamabe-Problems zu flnden. Das Kontakt-Yamabe-Problem ist eine naturlic˜ he
Verallgemeinerung des CR-Yamabe-Problems.
DerKontaktYamabe u…isteinedegeneriertesemilineareW˜armeleitungsgle-
ichung. W˜armeleitungsgleichungen dieses Typs wurden bisher nicht weiter un-
tersucht. Aus diesem Grund mussen˜ wir zun˜achst einige geometrische und ana-
lytischeBeobachtungenetablieren. DanachzeigenwirdieExistenzeinerL˜osung
fur˜ ein kleines Zeitintervall. Schlie…lich beweisen wir, da… der Yamabe u… fur˜
alle Zeiten existiert und gegen eine L˜osung des Yamabe-Problems konvergiert,
falls wir annehmen, da… entweder die Yamabe-Invariante negativ ist, oder das
Anfangsdatum K-Kontakt ist.
Schlusselw˜˜ orter: Yamabe-Problem, Kontakt-Mannigfaltigkeit,
Kontakt-Yamabe u… .iv
Abstract
TheYamabeproblemisaclassicproblemindifierentialgeometryconcerning
the question: whether a given compact and connected manifold is necessarily
conformallyequivalenttooneofconstantscalarcurvature? Itwasformulatedby
Yamabein1960. Yamabe,N.TrudingerandT.Aubinmadecontributiontothis
problem,anditwascompletelysolvedbyR.Schoenusingpositivemassthoerem
in1984. LateronD. JerisonandJ.M. Lee introducedtheCRYamabeproblem
in1987. Thatis, foragivencompact, strictlypseudoconvexCRmanifold, ifit’s
possible to flnd a choice of pseudohermitian structure with constant Webster
scalar curvature? This problem was solved in a–rmative due to D. Jerison,
J. M. Lee, N. Gamara and R. Yacoub in 2001.
A owapproachwasalsoappliedtotheclassicYamabeproblembyR.Hamil-
ton, R. Ye, H. Schwetlick, M. Struwe and S. Brendle. The ow behavior has
also its own interests. Here we use the contact Yamabe ow to flnd solutions
of the contact Yamabe problem. The contact Yamabe problem is a natural
generalization of the CR Yamabe problem.
The contact Yamabe ow corresponds to a degenerate semilinear heat equa-
tion. However the analytic theory regarding such heat equation has not been
well studied up to now. For this reason we have to resort to some geometrical
and also analytic observations. After we obtain the local existence in general,
we prove the contact Yamabe ow exists for all time and tends to a solution
of the contact Yamabe problem when the Yamabe invariant is negative or the
initial data is K-contact.
Key words: Yamabe problem, contact manifold, contact Yamabe ow.Contents
1 Introduction 1
1.1 The main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Open questions and remarks . . . . . . . . . . . . . . . . . . . . . 5
2 The Riemannian Yamabe problem 7
2.1 The elliptic approach . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 History and motivations . . . . . . . . . . . . . . . . . . . 7
2.1.2 Basic materials . . . . . . . . . . . . . . . . . . . . . . . . 9
n2.1.3 The solution when ‚(M;g)<‚(S ;g) . . . . . . . . . . . 12
2.1.4 The solutions on the standard sphere. . . . . . . . . . . . 14
2.1.5 Aubin’s results . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.6 Schoen’s work and positive mass theorem . . . . . . . . . 19
2.2 The Yamabe ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Ye’s approach by using the heat equation . . . . . . . . . 25
2.2.2 Some recent works . . . . . . . . . . . . . . . . . . . . . . 31
3 The geometry of contact manifolds 35
3.1 Contact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Contact metric manifolds . . . . . . . . . . . . . . . . . . . . . . 37
3.3 CR manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 The Webster scalar curvature . . . . . . . . . . . . . . . . . . . . 44
3.5 The generalized Webster scalar curvature . . . . . . . . . . . . . 46
4 The CR Yamabe problem 49
4.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Analytic aspect on CR manifolds . . . . . . . . . . . . . . . . . . 55
4.3 The solution of the CR Yamabe problem . . . . . . . . . . . . . . 57
4.4 The CR Yamabe solutions on the sphere . . . . . . . . . . . . . . 59
5 The Contact Yamabe ow 61
5.1 Standard results for the contact Yamabe ow . . . . . . . . . . . 62
5.1.1 Basic materials . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.2 The short time existence . . . . . . . . . . . . . . . . . . . 65
vvi CONTENTS
5.2 The contact Yamabe ow with ‚(M;? )<0 . . . . . . . . . . . . 700
5.2.1 The long-time existence . . . . . . . . . . . . . . . . . . . 71
5.2.2 The asymptotic behavior . . . . . . . . . . . . . . . . . . 74
5.2.3 Regularity of the limit solution . . . . . . . . . . . . . . . 76
5.3 The contact Yamabe ow on K-contact manifolds . . . . . . . . . 77
5.3.1 Basic material on K-contact . . . . . . . . . . . 77
5.3.2 The long-time existence and convergence . . . . . . . . . 81Chapter 1
Introduction
In this chapter let us brie y overview what we are going to do in this thesis.
Wewillstateourmaintheorems,explainhowweorganizethisthesisanddiscuss
some open questions. In particular, we assume that the reader is familiar with
certain aspects in conformal geometry and in contact geometry. The analytic
andgeometricdetailsofmythesiswillbeexplainedintheforthcomingchapters.
1.1 The main theorems
In this thesis we focus our interests on a Yamabe type ow on contact metric
manifolds, i.e. we will use heat equations to solve Yamabe type problems on
contact metric manifolds.
Let (M;? ;J;g ) be a connected and compact contact metric manifold of0 0
dimension 2n+1, where as usual ? ;J denote the underlying contact form and0
the almost complex structure on the contact distribution given by ker(? ). The0
Riemannian metric g is associated with d and compatible with J (for details0 0
see chapter 3). The background contact form ? deflnes a conformal class0
1[? ]:=f?2› (M;R)j?=f? ;f >0g:0 0
Toeachelement?intheconformalclass [? ]onecanassignaconnection, called0
the generalized Tanaka connection (see [Tan89] and in addition section 3.5 in
thisthesis). The(generalized)Websterscalarcurvature W thenisthefulltrace
of the curvature tensor associated to the Tanaka connection.
Any contact manifold (M;? ) admits a pair (J;g ) consisting of an almost0 0
complex structure and an associated metric (not necessarily unique). With any
such choice (J;g ), (M;? ;J;g ) is called a contact metric manifold. A Yamabe0 0 0
type problem on contact metric manifolds is to flnd a 1-form ?2[? ] such that0
the Webster scalar curvature w.r.t. ? is constant, i.e.
W(x)¡W =0; 8x2M; (1.1)
12 CHAPTER 1. INTRODUCTION
R
nW(x;t)?^d MRwhere W := . To distinguish this Yamabe type problem on
n?^d M
contact metric manifolds form the Riemannian Yamabe problem, we call it the
contact Yamabe problem.
The semilinear, subelliptic equation (1.1) can be attacked also by considering
the subparabolic analogue given by the contact Yamabe ow

@?(x;t) =(W(t)¡W(x;t))?(x;t)@t (1.2)0?(x;0)=? 2[? ]:0
Aswewilloutlineinchapters4and5,thecontactYamabeproblemoncontact
metric manifolds is a natural generalization of the CR Yamabe problem on CR
manifolds that was initiated by D. Jerison and J. M. Lee in [JL87] and was
completely solved by D. Jerison, J. M. Lee, N. Gamara and R. Yacoub (see
[JL87], [JL89], [GY01] and [Gam01]).
In this thesis we use the ow approach to prove the following main theorems:
Theorem 1.1 Let (M;? ;J;g ) be a connected, compact contact metric mani-0 0
fold of dimension 2n+1.
(a) The contact Yamabe ow (1.2) admits a smooth solution on a maximal
time interval [0;T);0<T •1.
(b) If the contact Yamabe invariant ‚(M;[? ]) is negative, then there exists0
a contact metric structure (M;? ;J;g ) with negative constant Webster1 1
0 0scalarcurvature. Inparticular,foranychoice? 2[? ]satisfyingW(? )<0
0 the solution ?(t) of (1.2) exists for all time and as t!1 the Webster
scalar curvature approaches some negative constant exponentially.
Theorem 1.2 Let (M;? ;J;g ) be a K-contact metric manifold. Then the con-0 0
0tact Yamabe ow (1.2) with initial data ? =? exists for all time and converges0
smoothly to a smooth limit ? with constant Webster scalar curvature.1
In Theorem 1.1(b) we can only prove ?(t) of the contact Yamabe ow (1.2)
converges continuously to a limit ?(1) , and the limit is actually smooth by
(1.1). Better regularity can be proved in Theorem 1.2 where we prove all deriv-
atives of the solution to the ow (1.2) are bounded uniformly in space and time
which implies the smooth convergence. Theorem 1.2 flnds a solution of the
contact Yamabe problem on any K-contact metric manifold (for the deflnition
see chapter 5). K-contact metric manifolds are a special class of contact metric
manifolds. Any Sasakian manifold is a K-contact metric manifold and a K-
contact metric manifold is not necessarily a CR manifold. For the deflnition of
K-contact metric see deflnition 5.10.