570 Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors 570 Geometric Analysis: Partial Differential Equations and Surfaces UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain Joaquín Pérez José A. Gálvez Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor George Andrews Abel Klein Martin J. Strauss Editorial Committee of the Real Sociedad Matem´atica Espa˜nola Pedro J. Pa´ul, Director Luis Al´ıas Emilio Carrizosa Bernardo Cascales Javier Duoandikoetxea Alberto Elduque Rosa Maria Mir´o Pablo Pedregal Juan Soler 2010 Mathematics Subject Classification. Primary 53A10; Secondary 35B33, 35B40, 35J20, 35J25, 35J60, 35J96, 49Q05, 53C42, 53C45. Library of Congress Cataloging-in-Publication Data UIMP-RSME Santal´o Summer School (2010 : University of Granada) Geometric analysis : partial differential equations and surfaces : UIMP-RSME Santal´o Summer School geometric analysis, June 28–July 2, 2010, University of Granada, Granada, Spain / Joaqu´ın P´erez, Jos´eA.G´alvez, editors p. cm. — (Contemporary Mathematics ; v. 570) Includes bibliographical references. ISBN 978-0-8218-4992-7 (alk. paper) 1. Minimal surfaces–Congresses. 2. Geometry, Differential–Congresses. 3. Differential equations, Partial–Asymptotic theory–Congresses. I. P´erez, Joaqu´ın, 1966– II. G´alvez, Jos´e A., 1972– III. Title. QA644.U36 2010 516.362–dc23 2012004897 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. 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 171615141312 Contents Preface vii Geometric PDEs in the presence of isolated singularities JoseA.G´ alvez´ and Pablo Mira 1 Constant mean curvature surfaces in metric Lie groups William H. Meeks III and Joaqu´ın Perez´ 25 Stochastic methods for minimal surfaces Robert W. Neel 111 The role of minimal surfaces in the study of the Allen-Cahn equation Frank Pacard 137 On curvature estimates for constant mean curvature surfaces Giuseppe Tinaglia 165 v Preface Llu´ıs Antoni Santal´o Sors (1911–2001) was a Spanish mathematician that stud- ied at Hamburg under the differential geometer Wilhelm Blaschke. In 1936, Pro- fessor Blaschke initiated two young students, Santal´o and S.S. Chern, to integral geometry. Unlike Chern, who spent long periods of his career in different places as France, China and USA, Santal´o moved definitely to Argentina after a short pe- riod in Spain, because of the Spanish Civil War; nevertheless he become recognized worldwide by his studies in Integral Geometry, Stereology, Geometric Probability and Projective Geometry (see the recent volume Luis Antonio Santal´oSelected Works edited by A. Naveira and A. Revent´os under Springer Verlag, for a selection of Santal´o’s best papers and bibliography). In his honor, the Royal Mathematical Society of Spain (RSME) organizes since 2002 a yearly advanced summer School about various aspects in Mathematics. Except for the 2010 occasion, the RSME Santal´o School had been developed within the framework of the Summer Courses of the International University Men´endez y Pelayo in Santander, Spain. The 2010 event was held at the Uni- versity of Granada, and the chosen topic was that of Geometric Analysis. This is a rather vague term to refer to a part of mathematics whose width has been in- creasing in the last decades, and whose frontiers have trespassed areas that a priori, were not assumed to be reachable. In a very simplified manner, we can describe Geometric Analysis as the interface between Differential Geometry and Differential Equations. The primary example of this interaction could be that of the calculus of variations, where the object of study is the perturbation of a functional acting on geometric objects. The critical points of the functional are often characterized as solutions of a differential equation (the associated Euler-Lagrange equation). These critical points are tools for understanding the geometry of the manifold over which the functional is defined. Different problems coming from areas apparently far, have been solved by application of tools in Geometric Analysis: among others, we can mention the solution by R. Schoen and S. T. Yau of the positive mass conjecture, and the more recent positive answer by G. Perelman to the Poincar´e conjecture. The main objective of the 2010 Santal´o School was to carry out mini-courses and talks in which distinguished researchers in Geometric Analysis would explain from the basics to the some of the most up-to-date aspects of this area. The School was mainly intended for researchers in Mathematics and degree or PhD students, although everyone with mathematical interest, an inquisitive mind and strong geometrical intuition was invited to join it. This set of lecture notes was originated from the series of lectures given at the 2010 Santal´o School on Geometric Analysis, held at the University of Granada from June 28 to July 2, 2010. The organization of this volume is as follows. The vii viii PREFACE proceedings are opened by the article Geometric PDEs in the presence of isolated singularities, by Jos´eA.G´alvez and Pablo Mira, where they describe the conical singularities of geometric PDEs of Monge-Amp`ere type. Next we include a self- contained version of the mini-course Constant mean curvature surfaces in metric Lie groups, taught at the School by William H. Meeks III, about several aspects of minimal and constant mean curvature surfaces in homogeneous three-manifolds, with special emphasis in the almost unexplored theory of such surfaces in met- ric Lie groups (i.e., three-dimensional Lie groups equipped with a left invariant metric). Except for this longer article, the volume consists of survey articles with an expository character. The remaining articles in the proceedings are Stochastic Methods for Minimal Surfaces, by Robert W. Neel, where stochastic methods as Brownian motion and its relation to conformal structure are applied to obtain re- sults for minimal surfaces in Euclidean three-space; The role of minimal surfaces in the study of the Allen-Cahn equation, by Frank Pacard, where he explains the role of minimal and constant mean curvature surfaces in the construction of entire solutions of the Allen-Cahn equation in Rn and in the study of extremal domains for the first eigenvalue of the Laplacian; and On curvature estimates for constant mean curvature surfaces, by Giuseppe Tinaglia, where the significance of curvature estimates for constant mean curvature surfaces is discussed. Besides the above topics, the 2010 Santal´o School scheduled two other mini- courses, namely On a fully nonlinear version of the Yamabe problem, by YanYan Li, where he revisited the classical Yamabe problem and its solution in order to apply these techniques to a fully nonlinear version of the Yamabe problem; and Introduction to the work of Colding-Minicozzi on minimal surfaces in R3, by Harold Rosenberg, where we could learn about various aspects of the recent theory by Colding and Minicozzi to study limits of simply connected minimal surfaces in R3. We want to thank all of them for their talks and courses, without which the School would not have been possible. All papers in this volume went through a blind-refereed process. We also want to thank all those who provided manuscripts for publication in these proceedings, as well as to give a special thanks to the reviewers, whose effort and hard work reflected their commitment and dedication. This book is published in cooperation with the Royal Mathematical Society of Spain (RSME). Granada, January 2012. Joaqu´ın P´erez and Jos´eA.G´alvez Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11303 Geometric PDEs in the presence of isolated singularities Jos´eA.G´alvez and Pablo Mira Abstract. This is a short course on the behavior of solutions to some geo- metric elliptic PDEs of Monge-Amp`ere type in two variables, in the presence of non-removable isolated singularities. We will describe local classification theorems around such an isolated singularity, as well as global classification theorems for the case of finitely many isolated singularities.