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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 Geometric Analysis 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 mathematics education 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. Mathematical analysis. I. Bray, Hubert L., editor. QA360.G455 2015 515.1—dc23 2015031562 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. 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 Geometry 1 Heat Diffusion in Geometry 3 1. Heat diffusion 3 2. Curve shortening 5 3. Mean curvature flow 8 4. Ricci flow 10 5. Towards surgery 12 Bibliography 13 Peter Topping Applications of Hamilton’s Compactness Theorem for Ricci Flow 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 manifolds 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 spheres 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 tensors 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 equation 71 3.1. Reduction to the complex Monge-Amp`ere equation 71 3.2. Estimates on ϕ andϕ ˙ 74 3.3. Estimate on the metric 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 manifold 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 theorems 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 set 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 * limit 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 logarithms 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 wave group 148 9.3. Complexified spectral projections 149 9.4. Poisson operator as a complex Fourier integral 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 complex analysis 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. Perturbation theory of the remainder 173 12.10. Conclusions 174 Section 13. Appendix on the phase space 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.
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