Table Of ContentRecent Progress in
Conformal Geometry
Imperial College Press Advanced Texts in Mathematics
Series Editor: Dennis Barden (Univ. of Cambridge, UK)
Vol. 1 Recent Progress in Conformal Geometry
by Abbas Bahri (Rutgers Univ. USA) & YongZhong Xu
(Courant Inst. for the Mathematical Sciences, USA)
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ICP Advanced Texts in Mathematics – Vol. 1
Recent Progress in
Conformal Geometry
Abbas Bahri
Rutgers University, USA
Yongzhong Xu
Courant Institute for the Mathematical Sciences, USA
Imperial College Press
ICP
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RECENT PROGRESS IN CONFORMAL GEOMETRY
Copyright © 2007 by Imperial College Press
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Preface
This book is divided into two parts. The first part is about Sign-Changing
Yamabe-typeproblems. AMorseLemmaatinfinity,underreasonablebasic
conjectures,isprovedforsuchproblems. This workisanattemptto define
anewareaofresearchfornonlinearanalysts. Wehavetriedinittoprovide
a family ofestimates and techniques with the helpof whichthe problemof
finding infinitely many solutions to these equations on domains of R3 can
be studied.
Our estimates and our work is a “cas d’ecole” in that we work on R3
or S3, a framework where solutions are known to exist, in fact in infinite
number; and we have chosen to study the asymptots generated by these
solutionsandtheircombinationsundertheactionoftheConformalGroup.
This work could also be useful for other variationalproblems such as Ein-
stein or Yang-Mills equations.
The second part of this book is about Contact Form Geometry via
Legendriancurves. Givenathree-dimensionalcompactorientablemanifold
M3 and a contact form α on it, we have assumed in earlier works the
existenceofa“dual”contactformβ,β =dα(v,·),witht(cid:1)hesameorientation
1
thanαandwehaveintroducedthevariationalproblem αx(x˙)dtonCβ =
0
{x ∈ H1(S1,M)|βx(x˙) ≡ 0}. We have defined a homology related to the
periodic orbits of the Reeb vector-field ξ of α on Cβ.
We prove in this framework two main results. First, we establish that
the hypothesisthatβ isacontactformwiththe sameorientationthanαis
notessential. The techniques involvedinorderto provesucha result(ona
typicalexample)havethe definite advantagethatthey arequantitative: as
we allow regions where β is no longer a contact form with the same orien-
tation than α, we track down the modifications of the variational problem
and we provide bounds on a key quantity (denoted τ) as we introduce a
v
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vi Recent Progress in Conformal Geometry
large amount of rotation for kerα along the orbits of v near these areas.
We then move to prove a compactness result about the flow-lines of
this variational problem which originate at a periodic orbit of ξ. This —
still slightly imperfect — compactness result indicates that all flow-lines
originatingatperiodic orbitsgotoperiodic orbits(atleastifthe difference
ofMorseindexes is 1),unless the numberofzerosofthe v-componentofx˙,
the tangent vector to the curves under deformation, drops.
Nocriticalpointatinfinity(asymptot)interfereswiththishomology. We
strongly suspect that this homology is, in the case of the standard contact
structure of S3, the homology of PC∞. We expect that we will be able to
compute this homology in some easy cases at least.
We had been searching for a long time for such a result. This work
entitled “Compactness” will be published independently by the first au-
thor and dedicated to his long time friend and collaborator Haim Brezis
for his sixtieth birthday. Both directions of researchi.e. Conformal Geom-
etry, Einstein equations, Yang-Mills equations on one hand, Contact Form
Geometry on the other hand, are also studied by other techniques due to
“hard-chore”Geometry and Symplectic Geometry.
In fact, Geometers have always been our “co-area researchers”. We
view these areas which we have contributed to define — for Contact Form
Geometry — in a different way and with different techniques.
This book is a book of collaboration and research. It also defines new
goals and presents a new understanding. It is not (yet) a textbook for
graduate students. It rather informs our collaborators about a definite
progress in the two above mentioned areas.
This research has been long; and at times hard and difficult. It has
been a strain on our friends and companions. Thanks are due: Abbas
Bahri wishes to thank Haim Brezis, his long-time collaborator and friend,
not only for his obvious support but more so for his friendship. Having a
friend — and of such a quality — is a rare blessing in life.
Abbas Bahri wishes also to thank Diana Nunziante, his wife, for her
patience, her understanding and her love as this book was being written.
Lines and equations are written, but only with the overwhelming intel-
ligence and love of those closest to us.
Both of us extend our warmest thanks to Barbara Mastrian for her
wonderful work as well as her wit and life. It has been a pleasure to work
with her all these years.
Finally, we would like to thank H. Brezis, S. Chanillo, R. Nussbaum
and Z. Han for giving up so much of their time and patiently listen to our
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Preface vii
arguments as they developed.
We also thank them, as well as our friends and colleagues of the Math-
ematics Department at Rutgers, for their thoughtful remarks and observa-
tions.
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TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
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Contents
Preface v
A. Bahri and Y. Xu
1. Sign-Changing Yamabe-Type Problems 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Results and Conditions. . . . . . . . . . . . . . . . . . . . . 2
1.3 Conjecture 2 and Sketch of the Proof of Theorem 1; Outline 7
1.4 The Difference of Topology . . . . . . . . . . . . . . . . . . 11
1.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Understand the difference of topology . . . . . . . . 14
1.5.2 Non critical asymptots . . . . . . . . . . . . . . . . . 15
1.5.3 The exit set from infinity. . . . . . . . . . . . . . . . 15
1.5.4 EstablishingConjecture2andcontinuousformsofthe
discrete inequality . . . . . . . . . . . . . . . . . . . 16
1.5.5 The Morse Lemma at infinity, Part I, II, III . . . . . 16
1.5.6 Notations v¯,v¯i,¯hi . . . . . . . . . . . . . . . . . . . . 16
1.6 Preliminary Estimates and Expansions, the Principal Terms 17
1.7 Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . 18
1.7.1 The equation satisfied by v . . . . . . . . . . . . . . 19
1.7.2 First estimates on vi and hi . . . . . . . . . . . . . . 23
1.7.3 The matrix A . . . . . . . . . . . . . . . . . . . . . . 25
1.7.4 Towards an H01-estimate on vi and an L∞-estimate
on hi . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.5 The formal estimate on hi . . . . . . . . . . . . . . . 31
1.7.6 Remarks about the basic estimates . . . . . . . . . . 35
ix
