Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture Qi S. Zhang CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-3459-6 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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QA295.Z43 2011 512.9’7--dc22 2010018868 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Preface First weprovidea treatment of Sobolevinequalities invarious settings: the Euclidean case, the Riemannian case and especially the Ricci flow case. Then, we discuss several applications and ramifications. These include heat kernel estimates, Perelman’s W entropies and Sobolev in- equality with surgeries, and the proof of Hamilton’s little loop conjec- ture with surgeries, i.e. strong noncollapsing property of 3 dimensional Ricci flow. Finally, using these tools, we present a unified approach to thePoincar´econjecture,whichseemstoclarifyandsimplifyPerelman’s original proof. The work is based on Perelman’s papers[P1], [P2], [P3], and the works Chow etc. [Cetc], Chow, Lu and Ni [CLN], Cao and Zhu [CZ], Kleiner and Lott [KL], Morgan and Tian [MT], Tao [Tao] and earlier work of Hamilton’s. The first half of the book is aimed at graduate students and the second half is intended for researchers. Acknowledgment This writing is derived from the lecture notes for a special summer course in Peking University in 2008 and another summer course in Nanjing University in 2009. I am deeply grateful to Professor Gang Tian and Professor Meiyue Jiang for the invitation to the School of Mathematics at Peking University and to Professor Gang Tian again for the invitation to Nanjing University. I feel fortunate to have the opportunity to enjoy the hospitality, generosity and excellent working condition in both universities and the cities where ancient tradition and modernity are displayed in splendor. Thanks also go to Professors Xiao Dong Cao, Bo Dai, Yu Guang Shi, Xing Wang Xu, Hui Cheng Yin, and Xiao Hua Zhu for their interests in the course and discussions on related mathematical problems. I am also indebted to the students who come from many parts of China to attend the classes, and to v vi Ms Wu and Ms Yu for their technical assistance. Special thanks go to Professors Bennett Chow and Lei Ni who invited me to a summer workshop on geometric analysis in 2005, which introduced me to Ricci flow, toProfessor Gang Tian whorecommended to publishtheChinese version of the lecture notes as a book by Science Press Beijing, and to Dr. Sunil Nair for inviting me to submit the English version to CRC Press. During the preparation of the book, I have also received helpful suggestions or encouragement from Professors Huaidong Cao, Jianguo Cao,XiuXiongChen,BoGuan,NicolaGarofalo,QingHan,Emmanuel Hebey, Zhen Lei, Junfang Li, John Lott, Peng Lu, Jie Qing, Yanir Rubinstein, Philippe Souplet, Bun Wong, Sumio Yamada, Zhong-Xin Zhao, Yu Zheng and Xiping Zhu. Materials from Chapter 2 and 4 were also used for a graduate course at University of California, Riverside. I thank Jennifer Burke and Shilong Kuang for taking notes, parts of which were incorporated in the book. I am also indebted to Professors Xiao Yong Fu and Murugiah Muraleetharan for reading and checking through the whole book. Professor Xiao Yong Fu with the assistance of Jun Bin Li also translated Chapters 2–6 into Chinese and made numerous corrections. Finally I have benefited amply from studying the works Chow etc. [Cetc], Chow-Lu-Ni, [CLN], Cao-Zhu [CZ], Kleiner-Lott [KL], Morgan- Tian[MT],Tao[Tao]andPerelman[P1],[P2].Iwishtousetheoccasion to thank them all. I dedicate this book to my family members: Wei, Ray, Weiwei, Misha, and to my parents. vii Contents 1 Introduction 1 2 Sobolev inequalities in the Euclidean space 7 2.1 Weak derivatives and Sobolev space Wk,p(D), D Rn . 7 1,p ⊂ 2.2 Main imbedding theorem for W (D) . . . . . . . . . . 10 0 2.3 Poincar´e inequality and log Sobolev inequality . . . . . 23 2.4 Best constants and extremals of Sobolev inequalities . . 25 3 Basics of Riemann geometry 27 3.1 Riemann manifolds, connections, Riemann metric . . . . 27 3.2 Second covariant derivatives, curvatures . . . . . . . . . 44 3.3 Common differential operators on manifolds . . . . . . . 52 3.4 Geodesics, exponential maps, injectivity radius etc. . . . 56 3.5 Integration and volume comparison . . . . . . . . . . . . 80 3.6 Conjugate points, cut-locus and injectivity radius . . . . 90 3.7 Bochner-Weitzenbock type formulas . . . . . . . . . . . 98 4 Sobolev inequalities on manifolds 103 4.1 A basic Sobolev inequality . . . . . . . . . . . . . . . . . 103 4.2 Sobolev, log Sobolev inequalities, heat kernel . . . . . . 108 4.3 Sobolev inequalities and isoperimetric inequalities . . . . 127 4.4 Parabolic Harnack inequality . . . . . . . . . . . . . . . 133 4.5 Maximum principle for parabolic equations . . . . . . . 151 4.6 Gradient estimates for the heat equation . . . . . . . . 155 5 Basics of Ricci flow 167 5.1 Local existence, uniqueness and basic identities . . . . . 167 5.2 Maximum principles under Ricci flow . . . . . . . . . . . 187 5.3 Qualitative properties of Ricci flow . . . . . . . . . . . . 199 5.4 Solitons, ancient solutions, singularity models . . . . . . 209 ix