IAS/PARK CITY 22

Geometric

Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors

American Mathematical Society Institute for Advanced Study

https://doi.org/10.1090//pcms/022

IAS/PARK CITY MATHEMATICS SERIES Volume 22

Geometric Analysis

Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors

American Mathematical Society Institute for Advanced Study Hubert Lewis Bray, Gregory J. Galloway, Rafe Mazzeo, and Natasa Sesum, Volume Editors

IAS/Park City Mathematics Institute runs programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 53-06, 35-06, 83-06.

Library of Congress Cataloging-in-Publication Data Geometric analysis / Hubert L. Bray, editor [and three others]. pages cm. — (IAS/Park City mathematics series ; volume 22) Includes bibliographical references. ISBN 978-1-4704-2313-1 (alk. paper) 1. Geometric analysis. 2. . I. Bray, Hubert L., editor. QA360.G455 2015 515.1—dc23 2015031562

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c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 22212019181716 Contents

Preface xiii

Gerhard Huisken Heat Diffusion in 1

Heat Diffusion in Geometry 3 1. Heat diffusion 3 2. Curve shortening 5 3. flow 8 4. Ricci flow 10 5. Towards surgery 12 Bibliography 13

Peter Topping Applications of Hamilton’s Compactness for 15

Overview 17

Lecture 1. Ricci flow basics – existence and singularities 19 1.1. Initial PDE remarks 19 1.2. Basic Ricci flow theory 20

Lecture 2. Cheeger-Gromov convergence and Hamilton’s compactness theorem 23 2.1. Convergence and compactness of 23 2.2. Convergence and compactness of flows 25

Lecture 3. Applications to Singularity Analysis 27 3.1. The rescaled flows 27 3.2. Perelman’s no local collapsing theorem 28

Lecture 4. The case of compact surfaces – an alternative approach to the results of Hamilton and Chow 31

Lecture 5. The 2D case in general – Instantaneously complete Ricci flows 35 5.1. How to pose the Ricci flow in general 35 5.2. The existence and uniqueness theory 36 5.3. Asymptotics 38 5.4. Singularities not modelled on shrinking 39

v vi CONTENTS

Lecture 6. Contracting Cusp Ricci flows 41

Lecture 7. Subtleties of Hamilton’s compactness theorem 45 7.1. Intuition behind the construction 46 7.2. Fixing proofs requiring completeness in the extended form of Hamilton’s compactness theorem 47 Bibliography 49

Ben Weinkove The K¨ahler-Ricci flow on compact K¨ahler manifolds 51

Preface 53

Lecture 1. An Introduction to K¨ahler geometry 55 1.1. Complex manifolds 55 1.2. Vector fields, 1-forms, Hermitian metrics and 56 1.3. K¨ahler metrics and covariant differentiation 59 1.4. Curvature 61

Lecture 2. The K¨ahler-Ricci flow and the K¨ahler cone 65 2.1. The K¨ahler-Ricci flow and simple examples 65 2.2. The K¨ahler cone and the first Chern class 66 2.3. Maximal existence time for the K¨ahler-Ricci flow 68

Lecture 3. The parabolic complex Monge-Amp`ere 71 3.1. Reduction to the complex Monge-Amp`ere equation 71 3.2. Estimates on ϕ andϕ ˙ 74 3.3. Estimate on the 76 3.4. Higher order estimates 78

Lecture 4. Convergence results 81 4.1. Negative first Chern class 81 4.2. Zero first Chern class 85

Lecture 5. The K¨ahler-Ricci flow on K¨ahler surfaces, and beyond 89 5.1. Riemann surfaces 89 5.2. K¨ahler surfaces, blowing up and Kodaira dimension 89 5.3. Behavior of the K¨ahler-Ricci flow on K¨ahler surfaces 91 5.4. Non-K¨ahler surfaces and the Chern-Ricci flow 95

Appendix A. Solutions to exercises 99

Bibliography 105

Steve Zelditch Park City lectures on Eigenfunctions 109 Section 1. Introduction 111 1.1. The eigenvalue problem on a compact Riemannian 112 1.2. Nodal and critical point sets 114 CONTENTS vii

1.3. Motivation 115 Section 2. Results 116 2,1. Nodal hypersurface volumes for C∞ metrics 116 2.2. Nodal hypersurface volumes for real analytic (M,g) 117 2.3. Number of intersections of nodal sets with geodesics and number of nodal domains 117 2.4. Dynamics of the geodesic or billiard flow 119 2.5. Quantum ergodic restriction and nodal intersections 120 2.6. Complexification of M and Grauert tubes 121 2.7. Equidistribution of nodal sets in the complex domain 122 2.8. Intersection of nodal sets and real analytic curves on surfaces 122 2.9. Lp norms of eigenfunctions 124 2.10. Quasi-modes 126 2.11. Format of these lectures and references to the literature 127 Section 3. Foundational results on nodal sets 128 3.1. Vanishing order and scaling near zeros 128 3.2. Proof of Proposition 1 129 3.3. A second proof 130 3.4. Rectifiability of the nodal 130 m−1 ∞ Section 4. Lower bounds for H (Nλ)forC metrics 130 4.1. Proof of Lemma 4.4 132 4.2. Modifications 135 4.3. Lower bounds on L1 norms of eigenfunctions 135 4.4. Dong’s upper bound 135 4.5. Other level sets 136 4.6. Examples 137 Section 5. Quantum ergodic restriction theorem for Dirichlet or Neumann data 138 5.1. Quantum ergodic restriction theorems for Dirichlet data 138 5.2. Quantum ergodic restriction theorems for Cauchy data 139 Section 6. Counting intersections of nodal sets and geodesics 141 6.1. Kuznecov sum formula on surfaces 141 Section 7. Counting nodal domains 142 Section 8. Analytic continuation of eigenfuntions to the complex domain 144 8.1. Grauert tubes 144 8.2. Weak * problem for Husimi measures in the complex domain 145 8.3. Background on currents and PSH functions 146 8.4. Poincar´e-Lelong formula 146 8.5. Pluri-subharmonic functions and compactness 146 8.6. A general weak* limit problem for of Husimi functions 147 Section 9. Poisson operator and Szeg¨o operators on Grauert tubes 147 9.1. Poisson operator and analytic Continuation of eigenfunctions 147 9.2. Analytic continuation of the Poisson 148 9.3. Complexified spectral projections 149 9.4. Poisson operator as a complex Fourier operator 149 viii CONTENTS

9.5. Toeplitz dynamical construction of the wave group 151 Section 10. Equidistribution of complex nodal sets of real ergodic eigenfunctions 151 10.1. Sketch of the proof 152 10.2. Growth properties of complexified eigenfunctions 154 10.3. Proof of Lemma 10.6 and Theorem 10.3 155 10.4. Proof of Lemma 10.8 156 10.5. Proof of Lemma 10.7 156 Section 11. Intersections of nodal sets and analytic curves on real analytic surfaces 157 11.1. Counting nodal lines which touch the boundary in analytic plane domains 158 11.2. Application to Pleijel’s conjecture 162 11.3. Equidistribution of intersections of nodal lines and geodesics on surfaces 163 11.4. Real zeros and 166 Section 12. Lp norms of eigenfuncions 166 12.1. Generic upper bounds on Lp norms 166 12.2. Lower bounds on L1 norms 167 12.3. Riemannian manifolds with maximal eigenfunction growth 168 12.4. Theorem 9 169 12.5. Sketch of proof of Theorem 9 170 12.6. Size of the remainder at a self-focal point 172 12.7. Decomposition of the remainder into almost loop directions and far from loop directions 173 12.8. Points in M\TL 173 12.9. of the remainder 173 12.10. Conclusions 174 Section 13. Appendix on the phase and the geodesic flow 174 Section 14. Appendix: Wave equation and Hadamard parametrix 176 14.1. Hormander parametrix 177 14.2. Wave group: r2 − t2 178 14.3. Exact formula in spaces of constant curvature 179 14.4. Sn 180 14.5. Analytic continuation into the complex 181 Section 15. Appendix: Lagrangian distributions, quasi-modes and Fourier integral operators 181 15.1. Semi-classical Lagrangian distributions and Fourier integral operators 181 15.2. Homogeneous Fourier integral operators 183 15.3. Quasi-modes 184 Section 16. Appendix on Spherical Harmonics 185 16.1. Highest weight spherical harmonics 187 16.2. Spherical harmonics as quasi-modes 188 Bibliography 189 CONTENTS ix

Jeff A. Viaclovsky Critical Metrics for Riemannian Curvature Functionals 195

Introduction 197

Lecture 1. The Einstein-Hilbert 199 1. Notation and conventions 199 2. First variation 200 3. Normalized functional 201 4. Second variation 202 5. Transverse-traceless variations 203

Lecture 2. Conformal geometry 205 1. Conformal variations 205 2. Global conformal minimization 206 3. Green’s metric and mass 207 4. The 208 5. Generalizations of the Yamabe Problem 209

Lecture 3. Diffeomorphisms and gauging 211 1. Splitting 211 2. Second variation as a bilinear form 212 3. Ebin-Palais slice theorem (infinitesimal version) 213 4. Saddle point structure and the smooth Yamabe invariant 215

Lecture 4. The moduli space of Einstein metrics 217 1. Moduli space of Einstein metrics 217 2. The nonlinear 218 3. Structure of nonlinear terms 219 4. Existence of the Kuranishi map 220 5. Rigidity of Einstein metrics 221

Lecture 5. Quadratic curvature functionals 225 1. Quadratic curvature functionals 225 2. Curvature in dimension four 227 3. Einstein metrics in dimension four 228 4. Optimal metrics 229 5. Anti-self-dual or self-dual metrics 230

Lecture 6. Anti-self-dual metrics 233 1. Deformation theory of anti-self-dual metrics 233 2. Weitzenb¨ock formulas 234 3. Calabi-Yau metric on K3 surface 235 4. Twistor methods 237 5. Gluing theorems for anti-self-dual metrics 237

Lecture 7. Rigidity and stability for quadratic functionals 239 1. Strict local minimization 239 xCONTENTS

2. Local description of the moduli space 241 3. Some rigidity results 243 4. Other dimensions 244

Lecture 8. ALE metrics and orbifold limits 245 1. Ricci-flat ALE metrics 245 2. Non-collapsed limits of Einstein metrics 247 3. Bt-flat metrics 249 4. Non-collapsed limits of Bt-flat metrics 251

Lecture 9. Regularity and volume growth 253 1. Local regularity 253 2. Volume growth estimate 254 3. ALE order and removable singularity theorems 255 4. Chen-LeBrun-Weber metric 257

Lecture 10. A gluing theorem for Bt-flat metrics 259 1. Existence of critical metrics 259 2. Lyapunov-Schmidt reduction 260 3. The building blocks 261 4. Remarks on the proof 262

Bibliography 267

Fernando C. Marques and Andr´e Neves Min-max theory and a proof of the Willmore conjecture 275 Introduction 277 Part 1. Canonical Family and Degree Calculation 281 Part 2. Min-Max Theory and Ruling Out Great Spheres 287 Part 3. Proof of the Main Theorems and the Energy of Links 291 Bibliography 299

Tristan Rivi´ere Weak immersions of surfaces with L2-bounded 301

Introduction 303

Lecture 1. Notations and fundamental results on the differential geometry of surfaces 307 1. Notations 307 2. Immersions and their geometry 308 3. Conformal invariance of the Willmore Energy 314 4. Two-dimensional geometry in isothermal charts 316 5. Existence of isothermal coordinates and the Chern moving frame method 321 6. Some facts on Riemann surfaces 324 CONTENTS xi

Lecture 2. The space of weak immersion with L2-bounded second fundamental form 329 1. Definition 329 2. Fundamental results on integrability by compensation 330 3. Existence of isothermal coordinates in the weak framework 335 4. H´elein’s energy controlled lifting theorem 336 5. Construction of local Coulomb frames with controlled W 1,2-energy 336 6. The Chern moving frame method in the weak framework 337

Lecture 3. of weak immersions 341 1. Compactness question 341 2. Control of the conformal factor 346 3. The monotonicity formula and consequences 350 4. Proof of the almost-weak theorem 352 5. Weak branched immersions 359 6. Weak sequentially closedness of FΣ 363

Lecture 4. The Willmore surface equation 365

Lecture 5. Conservation laws for weak Willmore immersions 371 1. The regularity of weak Willmore immersions 376 2. A minimization procedure for the Willmore energy among weak branched immersions 380

Bibliography 383

Brian White Introduction to Theory 385

Introduction 387

Lecture 1. The First Variation Formula and Consequences 389 Monotonicity 392 Density at infinity 393 Extended monotonicity 394 The isoperimetric inequality 398

Lecture 2. Two-Dimensional Minimal Surfaces 401 Relation to harmonic maps 401 Conformality of the Gauss map 402 Total curvature 403 The Weierstrass Representation 406 The geometric meaning of the Weierstrass data 408 Rigidity and Flexibility 409

Lecture 3. Curvature Estimates and Compactness Theorems 411 The 4π curvature estimate 413 A general principle about curvature estimates 414 xii CONTENTS

An easy version of Allard’s Regularity Theorem 415 Bounded total 415 Stability 418

Lecture 4. Existence and Regularity of Least- Surfaces 423 Boundary regularity 428 Branch points 428 The theorems of Gulliver and Osserman 429 Higher genus surfaces 430 What happens as the genus increases? 431 Embeddedness: The Meeks-Yau Theorem 432

Bibliography 435 Preface

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Foundation. In mid 1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and ed- ucation in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, gradu- ate students, undergraduate students, high school teachers, undergraduate faculty, and researchers in mathematics education. One of PCMI’s main goals is to make all of the participants aware of the total spectrum of activities that occur in math- ematics education and research. We wish to involve professional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that end, the summer institute features general sessions designed to encourage interaction among the various groups. In-year activities at the sites around the country form an integral part of the High School Teachers Program. Each summer a different topic is chosen as the focus of the Research Program and Graduate Summer School. Activities in the Undergraduate Summer School deal with this topic as well. Lecture notes from the Graduate Summer School are being published each year in this series. The first twenty one volumes are:

• Volume 1: Geometry and Quantum Theory (1991) • Volume 2: Nonlinear Partial Differential in Differential Geom- etry (1992) • Volume 3: Complex (1993) • Volume 4: and the of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: of Lie Groups (1998) • Volume 9: Algebraic Geometry (1999) • Volume 10: Computational Complexity Theory (2000) • Volume 11: , , and Enumerative Geometry (2001) • Volume 12: Automorphic Forms and their Applications (2002) • Volume 13: Geometric (2004) • Volume 14: Mathematical (2005) • Volume 15: Low Dimensional Topology (2006) • Volume 16: Statistical Mechanics (2007)

xiii xiv PREFACE

• Volume 17: Analytic and Algebraic Geometry: Common Problems, Dif- ferent Methods (2008) • Volume 18: Arithmetic of L-functions (2009) • Volume 19: Mathematics in Image Processing (2010) • Volume 20: Moduli Spaces of Riemann Surfaces (2011) • Volume 21: Geometric (2012) • Volume 22: Geometric Analysis (2013) Volumes are in preparation for subsequent years. Some material from the Undergraduate Summer School is published as part of the Student Mathematical Library series of the American Mathematical Society. We hope to publish material from other parts of the IAS/PCMI in the future. This will include material from the High School Teachers Program and publications documenting the interactive activities that are a primary focus of the PCMI. At the summer institute late afternoons are devoted to seminars of common interest to all participants. Many deal with current issues in education: others treat mathematical topics at a level which encourages broad participation. The PCMI has also spawned interactions between universities and high schools at a local level. We hope to share these activities with a wider audience in future volumes.

Rafe Mazzeo Director, PCMI January, 2016 Introduction

Amongst the many great advances in mathematics in the last part of the twenti- eth century, the successes in geometric analysis rank very highly. The most spectac- ular are the resolutions of the Poincar´e conjecture and Thurston’s geometrization conjecture, through the work of Perelman and the many people who clarified and extended his ideas. These were proved using Hamilton’s Ricci flow, a powerful tool in the subject later used by Brendle and Schoen to prove the differentiable theorem in higher dimensions. Beyond these, we mention the many dra- matic advances in the study of minimal submanifolds, harmonic mappings and related variational problems, the deeper understanding of using tools from PDE, Riemannian and Lorentzian geometry, the use of gauge theory to detect subtle new topological invariants, and the relationship between the spec- tral behavior of the Laplace-Beltrami operator on a and the dynamical properties of the underlying geodesic flow. It is not easy to give a comprehensive definition of this subject, and the name ‘geometric analysis’ has only been in common currency for the last 25 years or so. Loosely speaking, this field involves the many interlocking relationships between geometry and partial differential equations. These interconnections go both ways. For example, a ‘purely geometric’ problem, such as finding the optimal shape of a manifold, can be translated into an equivalent problem which involves solving a PDE. If a solution of that equation can be found, this can then be translated back into a solution of the original geometric problem. A classic instance is the uniformization theorem, where one seeks optimal (constant curvature) metrics on surfaces. There are several different analytic approaches, the earliest involving complex analysis and later ones involving semilinear elliptic PDE’s. In the other direction, new perspectives in the field of PDE and many new techniques to solve various classes of equations have been inspired by the geometry underlying these equations. Among the many examples here, deep advances in fully nonlinear ellip- tic equations originated in the fundamental breakthroughs by Yau and others on Monge-Ampere equations arising in geometry. In a different direction, the entire modern theory of linear partial differential using microlocal analysis, pioneered by H¨ormander, Kohn, Nirenberg and others, relies on a new way of viewing linear PDE through a geometric lens and exploiting the deep connections with symplectic geometry. The research area highlighted in the 2013 session of the Park City Mathematics Institute was geometric analysis. The program of this summer school included lec- tures by: Michael Eichmair, Fernando Coda Marques, Tristan Riviere, Igor Rodni- anski, Peter Topping, Jeff Viaclovsky, Ben Weinkove, Brian White, Steve Zelditch, and the Clay Scholars Gerhard Huisken and . All were chosen both for the excellence of their mathematical work as well as their expository talents.

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This volume collects papers contributed by Huisken, Marques, Riviere, Topping, Viaclovsky, Weinkove, White and Zelditch. The topics covered include general rela- tivity, the proof of the long-standing Willmore conjecture as well as analytic aspects of the Willmore equation, Ricci and K¨ahler-Ricci flow, critical metrics, minimal sur- faces and spectral theory. The papers here represent the lectures for the Graduate Summer School at PCMI, presented to an audience of 80 graduate students and 60 researchers. We also include a paper by the Clay Senior Scholar Gerhard Huisken, loosely based on his public lecture during the summer session. The organizers are grateful to the Clay Mathematics Institute for their sponsorship and of two Clay Senior Scholars during the 2013 PCMI session: Gerhard Huisken and Richard Schoen. As with every PCMI volume, this collection of papers is meant to serve as a high level introduction to many of the most important topics in geometric analysis by some of the great experts in the field, and is intended for graduate students or anyone else wishing an entr´ee into the field. The 2013 session was marked by recollections by the more seasoned researchers of their participation in the previous PCMI session on geometric analysis in 1992, and the lasting influence that workshop had on their careers. We can only hope that the 2013 session of PCMI will have a similarly long-lasting and far-reaching effect in this wonderful field.

Hugh Bray, Greg Galloway, Rafe Mazzeo, Natasa Sesum Published Titles in This Series

22 Hubert L. Bray, Greg Galloway, Rafe Mazzeo, and Natasa Sesum, Editors, Geometric Analysis, 2016 21 Mladen Bestvina, Michah Sageev, and Karen Vogtmann, Editors, , 2014 20 Benson Farb, Richard Hain, and Eduard Looijenga, Editors, Moduli Spaces of Riemann Surfaces, 2013 19 Hongkai Zhao, Editor, Mathematics in Image Processing, 2013 18 Cristian Popescu, Karl Rubin, and Alice Silverberg, Editors, Arithmetic of L-functions, 2011 17 Jeffery McNeal and Mircea Mustat¸˘a, Editors, Analytic and Algebraic Geometry, 2010 16 Scott Sheffield and Thomas Spencer, Editors, Statistical Mechanics, 2009 15 Tomasz S. Mrowka and Peter S. Ozsv´ath, Editors, Low Dimensional Topology, 2009 14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini, Editors, Mathematical Biology, 2009 13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, Geometric Combinatorics, 2007 12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications, 2007 11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum Field Theory, Supersymmetry, and Enumerative Geometry, 2006 10 Steven Rudich and Avi Wigderson, Editors, Computational Complexity Theory, 2004 9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001 8 Jeffrey Adams and David Vogan, Editors, Representation Theory of Lie Groups, 2000 7 Yakov Eliashberg and Lisa Traynor, Editors, Symplectic Geometry and Topology, 1999 6 EltonP.HsuandS.R.S.Varadhan,Editors, Probability Theory and Applications, 1999 5 Luis Caffarelli and Weinan E, Editors, Hyperbolic Equations and Frequency Interactions, 1999 4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology of Four-Manifolds, 1998 3 J´anos Koll´ar, Editor, Complex Algebraic Geometry, 1997 2 Robert Hardt and Michael Wolf, Editors, Nonlinear partial differential equations in differential geometry, 1996 1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum Field Theory, 1995 This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kähler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduc- tion to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in R 3, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace– Beltrami operators.

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