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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

© 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

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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 inequality with surgeries, and the proof of Hamilton’s little loop conjecture with surgeries, i.e. strong noncollapsing property of 3 dimensional Ricci flow. Finally, using these tools, we present a unified approach to the Poincar´e conjecture, 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

vvi 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], MorganTian [MT], Tao [Tao] and Perelman [P1], [P2]. I wish to use the occasion 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

  • 7
  • 2.1 Weak derivatives and Sobolev space Wk,p(D), D ⊂ Rn .

2.2 Main imbedding theorem for W01,p(D) . . . . . . . . . . 10 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 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´e conjecture
  • 381

9.1 Evolution of regions near surgery caps . . . . . . . . . . 382 9.2 Canonical neighborhood property with surgeries . . . . 394 9.3 Summary and conclusion . . . . . . . . . . . . . . . . . 405

Bibliography Index
409 421

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 imbedding 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 function itself has better integrability. It lies in the foundation of modern analysis. For example, Sobolev imbedding is an essential tool in studying partial differential equations since the goal of solving a differential equation is to integrate out the derivatives to recover the unknown function. On the other hand, a Sobolev inequality will also yield interesting partial differential equations via minimizing the Sobolev constants. It can also reveal useful information on the underlying space or manifold. This last property is the focus of this book.
The book is divided into three parts. Chapter 2 is the first part, where we will present basic materials on Sobolev inequalities in the Euclidean case. These include the standard W1,p(Rn) imbedding into Lnp/(n−p), Orlicz and Cα spaces, when p ∈ [1, n), p = n and p > n respectively. We will also present the Poincar´e inequality and Log Sobolev inequality. All these materials can be found in standard books, such as [GT], [Maz], [Ad], [LL]. The prerequisite for this part is graduate Real Analysis.

1
2

Chapter 1. Introduction

The second part consists of Chapters 3–4. Here we discuss Sobolev imbedding on compact or noncompact Riemann manifolds with fixed metrics. The main theme is to prove several close relatives of the Sobolev imbedding. These include log Sobolev inequality, certain heat kernel estimates, Poincar´e inequality and doubling condition, Harnack inequalities, etc. We will also show that the validity of certain Sobolev inequalities imply such geometric properties as volume noncollapsing, isoperimetric inequalities. Much of the material in this part is taken from [Heb2] and [Sal]. The reader needs some basic knowledge of Riemann geometry. In Chapter 3, a very brief summary of the most relevant results in basic Riemann geometry is provided.
The third part starts from Chapter 5, where we outline a few basic results of R. Hamilton’s Ricci flow. Starting from Chapter 6, we turn to some recent research on the Poincar´e conjecture.
From Perelman’s original papers [P1], [P2], [P3] and the works by
Cao and Zhu [CZ], Kleiner and Lott [KL] and Morgan and Tian [MT], and Tao [Tao2], [Tao], it is clear that the bulk of the proof of the Poincar´e conjecture is consisted of two items. One is the proof of local noncollapsing with or without surgeries, and the other is the classification of backward limits of ancient κ solutions. After these are done, by a clever blow-up argument, also due to Perelman, one can show that regions where the Ricci flow is close to forming singularity have simple topological structure, i.e. canonical neighborhoods. Then one proceeds to prove that the singular region can be removed by finite number of surgeries in finite time. When the initial manifold is simply connected, the Ricci flow becomes extinct in finite time [P3] (see also [CM]). Thus the manifold is diffeomorphic to S3, as conjectured by Poincar´e.
Besides the results and techniques by R. Hamilton, the main new tools Perelman used in carrying out the proof are several monotone quantities along Ricci flow. These include the W entropy, reduced volume and the associated reduced distance. In [P1], Perelman first used his W entropy to prove local noncollapsing for smooth Ricci flows, i.e. the little loop conjecture by Hamilton. This result is a major breakthrough for the theory of Ricci flow. However he then turned completely to the reduced volume (distance) to prove the classification and a weaker form of noncollapsing with surgeries. The W entropy is not used anymore. The reduced distance is a distance in space time, which is suitably weighted by the scalar curvature. Even though it is not smooth or positive in general, Perelman shows that the reduced distance miraculously satisfies certain differential equalities and inequal-

Introduction

3ities in the weak sense. But a rigorous proof of these is lengthy and intensive.
The main aim for the third part of the book is to show that the W entropy and its monotonicity actually imply certain uniform Sobolev inequalities along Ricci flows, which have many ramifications. One of them is the aforementioned local noncollapsing result with or without surgeries. Another one is the classification of the backward limits of ancient κ solutions. This enables one to give a proof of the Poincar´e conjecture using techniques unified around W entropy, Sobolev imbedding and related heat kernel estimate. This method allows one to bypass reduced distance and volume which are central to Perelman’s original proof. The reduced distance, being neither smooth nor positive in general, is one reason for the complexity of the original proof. The proof presented here, within Perelman’s framework, seems more accessible to wider audience since the techniques involving Sobolev inequalities and heat kernel estimates are familiar to many mathematicians. Much of the highly intensive analysis involving reduced distance and volume is now replaced by the study of the W entropy and the related uniform Sobolev inequalities and heat kernel estimates. These results have independent interest since they are instrumental to analysis on manifolds. Besides, due to the relative simplicity and versatility, we believe the current technique can lead to better understanding of other problems for Ricci flow. One such result is the proof of Hamilton’s little loop conjecture with surgeries in Chapter 8. Some applications are also found by several authors for K¨ahler Ricci flow, as indicated in Section 6.2.
We should mention that the reduced distance and volume are still needed so far for the proof of the geometrization conjecture. Actually, they are needed, but only in the proof of Perelman’s no collapsing Theorem II with surgeries, i.e. Proposition 6.3 (a) in [P2]. A short proof of the nonsurgery version is given in Section 6.3.
Let us briefly describe the content of Chapters 6–9. In Chapter 6, we show that Perelman’s W entropy is just the formula in a log Sobolev inequality and the monotonicity of a family of W entropies implies certain uniform Sobolev inequalities along a smooth Ricci flow. By earlier result of Carron [Ca] and Akutagawa [Ak], local noncollapsing of Ricci flow follows as a corollary. In Section 6.3, we also report two new results which are not directly related to the proof of Poincar´e conjecture. One is a uniform Sobolev inequality with critical exponents. The other is a localized uniform Sobolev inequality which implies the smooth version of Perelman’s no local collapsing theorem II [P1] as a special case. These
4

Chapter 1. Introduction

results are not published elsewhere. The content in Sections 6.4 and 6.5 are taken from the papers [KZ] and [Z1] respectively. The former is a differential Harnack inequality for all positive solutions of the conjugate heat equation. The later is certain upper bound for the fundamental solution of the conjugate heat equations.
In Chapter 7, we present Perelman’s classification of backward limits of ancient κ solutions and the canonical neighborhood property for 3 dimensional Ricci flow.
The main goal of Section 7.2 is to establish certain Gaussian type upper bound for the heat kernel (fundamental solutions) of the conjugate heat equation associated with 3 dimensional ancient κ solutions to the Ricci flow. As an application, in Section 7.3, using the W entropy associated with the heat kernel, we give a different and shorter proof of Perelman’s classification of backward limits of these ancient solutions. The question of whether or not the classification can also be done in this way has been raised in [Tao] e.g.
In Chapter 8, using the idea of Sobolev imbedding developed in
Chapter 6 and being inspired by the last section of [P2] and [KL], we prove a uniform Sobolev inequality which is independent of the number of surgeries. As one application, a strong finite time κ noncollapsing result for Ricci flow with surgeries is proven. This gives the first proof of Hamilton’s little loop conjecture with surgeries in 3 dimension case. We mention that the smooth version of the little loop conjecture is proven by Perelman [P1] using the monotonicity of his W entropy. The main work in the surgery case involves the analysis of eigenvalues of the minimizer equation of the W entropy over a horn like manifold. As a result it is shown that the best constant of the associated log Sobolev inequalities differ at most by the change in volume if the underlying manifold undergoes surgeries (cut and paste). The proof, without using reduced distance or volume, is short and seems more accessible. Its main advantage over the weaker κ noncollapsing proved by Perelman is that the relevant geometric information concentrates only on one time slice, thus avoiding the complication associated with surgeries which can happen shortly before this time slice. This chapter is based on [Z4].
In Chapter 9, with the help of strong κ noncollapsing, we will give a detailed proof of Perelman’s existence theorem of Ricci flow with surgeries. Something new about this chapter is the proof of Lemma 9.1.1, which describes the evolution of regions near surgery caps. This lemma is a key step in proving there are finitely many surgeries in finite time. Here we provide a proof which is considerably different from the

Introduction

5one outlined in [P2], Lemma 4.5. The reason is that we are unable to follow the original proof by Perelman, which seems to require a little more justification. The proof also allows one to bypass sophisticated uniqueness theorems for noncompact Ricci flows.
This result, together with the finite time extinction theorem in [P3] or [CM] imply the Poincar´e conjecture.
As described above, some of the materials in the book are not directly related to the proof of the Poincar´e conjecture. As an experienced worker solely interested in the Poincar´e conjecture, one can start with Section 5.4, then go to Sections 6.1 and 6.2, and then read Chapters 7, 8 and 9.
Due to limited space and time, we will only provide in detail those results or proofs that are different from Perelman’s original one. These include the proof of strong κ noncollapsing with surgeries, backward limit of κ solutions and the proof of the canonical neighborhood property with surgeries. Those parts which are more or less the same as the original ones are only sketched or referred to other sources.
Owing to the vastness and depth of the topic, some necessary selections of the material is necessary. This selection only reflects the current personal preference and limited knowledge and is in no way a snub to the materials that are left out. At the beginning of each section, some basic background material will be sketched with little or no proof. However standard references will be given. Coming to the proofs, we will strive to present the main ideas. Sometimes this is done at the price of sacrificing generality. The reader is led to references for further development and generalizations. Due to my limited knowledge and time constraint, the book contains many imperfections and omission. It is hoped that improvements will be made constantly. The author also welcomes all constructive suggestions and corrections, which can be sent to [email protected].
To close the introduction, we list a number of notations to be used throughout the book. More notations will be introduced in each chapter or section.
R or Rn: Euclidean space of dimension 1 or n; M or M: a Riemann manifold; g or gij: the Riemann metric; Ric or Rij: the Ricci curvature; Rm: the full curvature tensor; R: the scalar curvature.
∇: covariant derivative or gradient; ∆: Laplace operator; Hess: Hes-

  • 2
  • 2

sian; ∇ , or ∇i,j, or ∇i∇j: second covariant derivative. dx, dg, dµ or dµ(g), dg(x, t) etc.: volume element; d(x, y): distance; d(x, y, t), d(x, y, g(t)): distance with respect to metric g(t); B(x, r):
6

Chapter 1. Introduction

geodesic ball of radius r centered at x; B(x, r, t) or B(x, r, g(t)): geodesic ball of radius r centered at x, with respect to metric g(t); |B(x, r)|: volume of the ball B(x, r) under a given metric; |B(x, r, g(t))|h: volume of the ball under the metric h.

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    Yau, Shing-Tong I 1233 government manipulation of evidence in the Japanese article by the eminent Chinese American mathemati­ internmentcases, offeredto bring a challenge to Yasui cian Shiing-Shen Chern. In 1966, Yau entered the and his fellow wartime defendants' convictions by Chinese University of Hong Kong to study mathemat­ means of a coram nobis petition. Yasui consented, ics but moved three years later to the University of and a legal team headed by Oregon attorney Peggy Californiaat Berkeley to pursue graduate studies under Nagae took up his case. Unlike in the case of Fred Chern. At Berkeley, besides working with Chern in Korematsu, however, Yasui's petition failed to bring differential geometry, Yau also studied differential about a reconsideration of the official malfeasance equations with other professors, believing that cross­ involved in his prosecution. In 1984, district judge fe1tilizationwas key to the future of mathematics. In­ Robert C. Belloni issued an order vacating Yasui's depth knowledge of both fields indeed proved to be conviction, in accordance with a motion by Justice crucial to his success as it helped lay the foundation Department officials anxious to dispose of the case, for Yau's research in integrating thetwo. Yau received but declined to either grant Yasui's coram nobis peti­ his PhD in 1971, after spending less than two years at tion or to make findings of fact regarding the record Berkeley. of official misconduct. Yasui and his lawyers appealed After graduation from Berkeley, Yau went to the the ruling, but he died in ovember 1986, thereby Institute for Advanced Study at Princeton where he mooting the case before the appeal could be decided.
  • List of Publications

    List of Publications

    References [1] Yau, Shing-Tung, On the fundamental group of manifolds of non-positive curvature, Proc. Nat. Acad. Sci., (1970) Vol. 67, No. 2, page 509. [2] Yau, Shing-Tung, On the fundamental group of compact manifolds of non-positive curvature, Ann. Math., 93 (1971), pages 579-585. [3] Yau, Shing-Tung, Compact flat Riemannian manifolds, J. Diff. Geom., 6 (1972), pages 395-402. [4] Lawson, Jr. H. Blaine and Yau, Shing-Tung, Compact manifolds of nonpositive curvature, J. Diff. Geom., 7 (1972), pages 211-228. [5] Yau, Shing-Tung, Remarks on conformal transformations, J. Diff Geom., 8 (1973), pages 369-381. [6] Yau, Shing-Tung, Some Global Theorems on non-complete surfaces, Comment. Math., Helv. 48 (1973), pages 177-187. [7] J.-P. Bourguignon and Yau, Shing-Tung, Sur les metriques riemanniennes a courbure de Ricci nulle sur le quotient d’une surface K3, C.R. Acad. Sci. Paris Ser. A-B, 277 (1973), A1175-A1177. [8] J.-P. Bourguignon and Yau, Shing-Tung, Geometrie differentielle, C.R. Acad. Sc., Paris, 277 (1973), pages 1175-1177. [9] Yau, Shing-Tung, On the curvature of compact Hermitian manifolds, Inv. Math., 25 (1974), pages 213-239. [10] H.B. Lawson and Yau, Shing-Tung, Scalar curvature, non-Abelian group actions, and the degree of symmetry of exotic spheres, Comm. Math. ,Helv., 49 (1974), pages 232-244. [11] Yau, Shing-Tung, Submanifolds with constant mean curvature, I, Amer. J. Math., 98 (1974), pages 346-366. [12] Yau, Shing-Tung, Curvature preserving diffeomorphisms, Ann. of Math., 100 (1974), pages 121-130. [13] Yau, Shing-Tung, Non-existence of continuous convex functions on certain Riemannian manifolds, Math.
  • Plasma Physics Laboratory Expenditures by Amoutn and Percentage

    Plasma Physics Laboratory Expenditures by Amoutn and Percentage

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  • A Mathematical Journal

    A Mathematical Journal

    ISSN 0716-7776 VOLUME 15, No 01 2 0 1 3 Cubo A Mathematical Journal "Dedicated to Professor Gaston M. N'Guérékata on the occasion of his 60th birthday". G u e s t E d i t o r s Claudio Cuevas Bruno de Andrade Jin Liang Universidad de La Frontera Universidade Federal de Pernambuco Faculdad de Ingeniería, Ciencias y Administración Centro de Ciências Exatas e da Natureza Departamento de Matemática y Estadística Departamento de Matemática Temuco - Chile Recife - Brazil CUBO A Mathematical Journal Bruno de Andrade (bruno0*luis@g+ail.co+) Editor in Chief (UFPE-Brazil) Claudio Cuevas ([email protected]) Emeritus Editor (UFPE-BR)/ !) C´esar Burgue˜no ([email protected]) Managing Editor (UFRO-CH !E) "os´e!abrin (#[email protected]) $ec%nical &u''ort EDITORIAL BOARD Agarwal R. P. Ladas Gerry Ambrosetti Antonio Li Peter Anastassiou George A. Moslehian M. S. Arad Tzvi Nagel Rainer Avramov Luchezar Nirenberg Louis Benguria Ra ael Pet!ov "esselin Bollob#as B#ela Pinto Manuel Burton T.A. Ramm Ale$ander G. %ardoso &ernando Rebolledo Rolando %arlsson Gunnar Robert 'idier %uevas %laudio S#aBarreto Antonio (c!mann )ean Pierre Shub Michael (laydi Saber Simis Aron (snault *#el+ene S,-ostrand )ohannes &omin Sergey Stanley Richard .alai Gil Tian Gang .ohn )ose/h ). 0hlmann Gunther .urylev 1aroslav "ainsencher 2srael The Department of Mathematics of the Universidad de La Frontera Temuco-Chile and the De- partment of Mathematics of the Universidade Federal de Pernambuco, Recife-Brazil, edit CUBO, A Mathematical Journal. CUBO appears in three issues per ear and is indexed in ZentralBlatt Math., Mathematical Reviews, MathSciNet and Latin 'ndex# CUBO publishes original research papers, preferably not longer than )* printed pages, that contain substantial mathematical results in all branches of pure and applied mathematics.
  • Heat Kernel and Curvature Bounds in Ricci Flows with Bounded

    Heat Kernel and Curvature Bounds in Ricci Flows with Bounded

    HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE RICHARD H. BAMLER AND QI S. ZHANG Abstract. In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform L2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger ε-regularity theorem for Ricci flows. This result is particularly useful in the study of K¨ahler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results. Contents 1. Introduction 1 2. Preliminaries 8 3. Heat kernel bounds, distance distortion estimates and the construction of a cutoff function 10 4. Mean value inequalities for the heat and conjugate heat equation 18 5. Bounds on the heat kernel and its gradient 26 6. Backward pseudolocality 28 7. L2 curvature bound in dimension 4 34 Acknowledgements 43 References 43 arXiv:1501.01291v3 [math.DG] 18 Nov 2015 1. Introduction In this paper we analyze Ricci flows on which the scalar curvature is locally or globally bounded by a time-dependent or time-independent constant.

  • The Variations of Yang-Mills Lagrangian

    The Variations of Yang-Mills Lagrangian. Tristan Rivi`ere∗ Dedicated to Gang Tian on his sixtieth Birthday. I Introduction Yang-Mills theory is growing at the interface between high energy physics and mathemat- ics It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential topology, Hitchin Kobayashi conjecture relating the existence of Hermitian-Einstein metric on holomorphic bundles over K¨ahlermanifolds and Mumford stability in complex geometry or also Gromov Witten invariants in symplectic geometry...etc. While the influence of Gauge theory in geometry is quite notorious, one tends sometimes to forget that Yang-Mills theory has been also at the heart of fundamental progresses in the non-linear analysis of Partial Differential Equations in the last decades. The purpose of this survey is to present the variations of this important lagrangian. We shall raise analysis question such as existence and regu- larity of Yang-Mills minimizers or such as the compactification of the \moduli space" of critical points to Yang-Mills lagrangian in general. II The Plateau Problem. Before to move to the Yang-Mills minimization problem we will first recall some funda- mental facts regarding the minimization of the area in the parametric approach and some elements of the resolution of the so called Plateau problem. Let Γ be a simple closed Jordan Curve in R3 : there exists γ 2 C0(S1; R3) such that Γ = γ(S1). Plateau problem : Find a C1 immersion u of the two dimensional disc D2 which is continuous up to the boundary, whose restriction to @D2 is an homeomorphism and which minimizes the area Z Area(u) = j@x1 u × @x2 uj dx1 dx2 : D2 ∗Forschungsinstitut f¨urMathematik, ETH Zentrum, CH-8093 Z¨urich, Switzerland.