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Sobolev Inequalities, Heat Kernels Under Ricci Flow, and The i \bookQiCRC-FM" | 2010/5/28 | 9:46 | page 2 | i i i Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture i i i i i \bookQiCRC-FM" | 2010/5/28 | 9:46 | page 3 | i i i i i i i i \bookQiCRC-FM" | 2010/5/28 | 9:46 | page 4 | i i i Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture Qi S. Zhang i i i i CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140515 International Standard Book Number-13: 978-1-4398-3460-2 (eBook - PDF) 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. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com i i “bookQiCRC” — 2010/6/2 — 10:03 — page v — i i Preface First we provide a treatment of Sobolev inequalities in various 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 the Poincar´econjecture, which seems to clarify and simplify Perelman’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 i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page vi — i i 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, to Professor Gang Tian who recommended to publish the Chinese 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, Xiu Xiong Chen, Bo Guan, Nicola Garofalo, Qing Han, 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] and Perelman [P1], [P2]. I wish to use the occasion to thank them all. i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page vii — i i I dedicate this book to my family members: Wei, Ray, Weiwei, Misha, and to my parents. vii i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page viii — i i i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page ix — i i Contents 1 Introduction 1 2 Sobolev inequalities in the Euclidean space 7 2.1 Weak derivatives and Sobolev space W k,p(D), D Rn . 7 1,p ⊂ 2.2 Main imbedding theorem for W0 (D) .......... 10 2.3 Poincar´einequality 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 i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page x — i i x Contents 6 Perelman’s entropies and Sobolev inequality 225 6.1 Perelman’s entropies and their monotonicity . 225 6.2 (Log) Sobolev inequality under Ricci flow . 238 6.3 Critical and local Sobolev inequality . 248 6.4 Harnack inequality for the conjugate heat equation . 272 6.5 Fundamental solutions of heat type equations . 281 7 Ancient κ solutions and singularity analysis 291 7.1 Preliminaries ........................ 291 7.2 Heat kernel and κ solutions . .. .. .. .. .. 297 7.3 Backward limits of κ solutions. 308 7.4 Qualitative properties of κ solutions . 316 7.5 Singularity analysis of 3-dimensional Ricci flow . 331 8 Sobolev inequality with surgeries 341 8.1 A brief description of the surgery process . 341 8.2 Sobolev inequality, little loop conjecture with surgeries . 354 9 Applications to the Poincar´econjecture 381 9.1 Evolution of regions near surgery caps . 382 9.2 Canonical neighborhood property with surgeries . 394 9.3 Summaryandconclusion . 405 Bibliography 409 Index 421 i i i i i i “bookQiCRC” — 2010/6/2 — 10:03 — page 1 — i i Chapter 1 Introduction The book is centered around Sobolev inequalities and their applications to analysis on manifolds, and in particular to Ricci flow. There are two objectives. One is to serve as an introduction to the field of analysis on Riemann manifolds. The other is to use the tools of Sobolev imbed- ding and heat kernel estimates to study Ricci flows, especially in the case with surgeries, a research field that has attracted much attention. Rather than making a comprehensive presentation, the aim is to present key ideas, to explain the hard proofs and most important applications in a succinct, accessible and unified manner. Roughly speaking, a Sobolev inequality states that if the derivative of a function is integrable in certain sense (Lp, etc.), then the func- tion itself has better integrability. It lies in the foundation of modern analysis. For example, Sobolev imbedding is an essential tool in study- ing partial differential equations since the goal of solving a differential equation is to integrate out the derivatives to recover the unknown func- tion.
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