Table Of Content

Recent 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) ZhangJi - Recent Progress.pmd 2 1/12/2007, 5:09 PM 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 Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. RECENT PROGRESS IN CONFORMAL GEOMETRY Copyright © 2007 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13978-1-86094-772-8 ISBN-101-86094-772-7 Printed in Singapore. ZhangJi - Recent Progress.pmd 1 1/12/2007, 5:09 PM January17,2007 11:55 WSPC/BookTrimSizefor9inx6in finalBB 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 January17,2007 11:55 WSPC/BookTrimSizefor9inx6in finalBB 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 January17,2007 11:55 WSPC/BookTrimSizefor9inx6in finalBB 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. January17,2007 11:55 WSPC/BookTrimSizefor9inx6in finalBB TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January17,2007 11:55 WSPC/BookTrimSizefor9inx6in finalBB 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