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http://dx.doi.org/10.1090/pspum/054.3 Recent Titles in This Series 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. Browder, editor, Nonlinear functional analysis and its applications (University of California, Berkeley, July 1983) 44 William K. Allard and Frederick J. Almgren, Jr., editors, Geometric measure theory and the calculus of variations (Humboldt State University, Areata, California, July/August 1984) 43 Francois Treves, editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984) 42 Anil Nerode and Richard A. Shore, editors, Recursion theory (Cornell University, Ithaca, New York, June/July 1982) 41 Yum-Tong Siu, editor, Complex analysis of several variables (Madison, Wisconsin, April 1982) 40 Peter Orlik, editor, Singularities (Humboldt State University, Areata, California, July/ August 1981) 39 Felix E. Browder, editor, The mathematical heritage of Henri Poincare (Indiana University, Bloomington, April 1980) 38 Richard V. Kadison, editor, Operator algebras and applications (Queens University, Kingston, Ontario, July/August 1980) 37 Bruce Cooperstein and Geoffrey Mason, editors, The Santa Cruz conference on finite groups (University of California, Santa Cruz, June/July 1979) 36 Robert Osserman and Alan Weinstein, editors, Geometry of the Laplace operator (University of Hawaii, Honolulu, March 1979) 35 Guido Weiss and Stephen Wainger, editors, Harmonic analysis in Euclidean spaces (Williams College, Williamstown, Massachusetts, July 1978) 34 D. K. Ray-Chaudhuri, editor, Relations between combinatorics and other parts of mathematics (Ohio State University, Columbus, March 1978) 33 A Borel and W. Casselman, editors, Automorphic forms, representations and L-f unctions (Oregon State University, Corvallis, July/August 1977) 32 R. James Milgram, editor, Algebraic and geometric topology (Stanford University, Stanford, California, August 1976) 31 Joseph L. Doob, editor, Probability (University of Illinois at Urbana-Champaign, Urbana, March 1976) (Continued in the back of this publication) Differential Geometry: Riemannian Geometry Proceedings of Symposia in PURE MATHEMATICS Volume 54, Part 3 Differential Geometry: Riemannian Geometry Robert Greene S. T. Yau Editors z((l*0S^iw American Mathematical Society Sll u h n n rj £ Mm * 3\\ ILliilil In Providence, Rhode Island PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON DIFFERENTIAL GEOMETRY HELD AT THE UNIVERSITY OF CALIFORNIA, LOS ANGELES LOS ANGELES, CALIFORNIA JULY 8-28, 1990 with the support from the National Science Foundation Grant DMS-8913610. 1991 Mathematics Subject Classification. Primary 53A10, 49F20, 58E20, 58E12, 58G11 (Part 1) 81E13, 53C80, 53B50, 32C10, 53C15, 58F05, 83C75 (Part 2) 53C20, 53C40, 58C40, 58G25, 58F07, 58F11, 58F17 (Part 3). Library of Congress Cataloging-in-Publication Data Riemannian geometry/Robert Greene and S. T. Yau, editors. p. cm.—(Proceedings of symposia in pure mathematics; v. 54, pt. 3) (Differential ge• ometry; pt. 3) "Proceedings of a Summer Research Institute on Differential Geometry, held at the Uni• versity of California, Los Angeles, July 8-28, 1990"—T.p. verso. ISBN 0-8218-1494-X (Part 1) ISBN 0-8218-1495-8 (Part 2) ISBN 0-8218-1496-6 (Part 3) ISBN 0-8218-1493-1 (set) (alk. paper) 1. Geometry, Riemannian—Congresses. I. Greene, Robert Everist, 1943- . II. Yau, Shing- Tung, 1949- . III. Summer Research Institute on Differential Geometry (1990: University of California, Los Angeles) IV. Series. V. Series: Differential geometry; pt. 3. QA641.D3833 1993 pt. 3 516.3'6s—dc20 92-34056 [516.3'73] CIP COPYING AND REPRINTING. Individual readers of this publication, and nonprofit li• braries acting for them, are permitted to make fair use of the material, such as to copy an article 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 publi• cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright ©1993 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ This publication was typeset using AMS-TgX, the American Mathematical Society's TgX macro system. 10 9 8 7 6 5 4 3 2 1 98 97 96 95 94 93 Dedication. To Professor S.-S. Chern, in appreciation of his formative influence on modern differential geometry. Contents (* denotes one-hour survey lectures) Preface xxi PART 1 Open Problems in Geometry* SHING TUNG YAU 1 Questions and Answers about Area-Minimizing Surfaces and Geometric Measure Theory* FRED ALMGREN 29 On the Geometrical Origin and the Solutions of a Degenerate Monge-Ampere Equation DAVID BAO AND TUDOR RATIU 55 Differential Geometry and the Design of Gradient Algorithms* ROGER W. BROCKETT 69 Spectral Geometry of F-Manifolds and its Application to Harmonic Maps YUAN-JEN CHIANG 93 Constructing Harmonic Maps into the Hyperbolic Space HYEONG IN CHOI AND ANDREJS TREIBERGS 101 Spherical Minimal Immersions of Spherical Space Forms DENNIS DETURCK AND WOLFGANG ZILLER 111 Banach Manifolds of Solutions to Nonlinear Partial Differential Equa• tions, and Relations with Finite-Dimensional Manifolds JOSEF DORFMEISTER 121 Some New Harmonic Maps ROBERT HARDT 141 x CONTENTS Existence of Unstable Minimal Surfaces in Manifolds with Homology and Applications to Triply Periodic Minimal Surfaces JOEL HASS, JON T. PITTS, AND J. H. RUBINSTEIN 147 Closed Minimal Submanifolds in the Spheres* WU-YIHSIANG 163 Local and Global Behaviour of Hypersurfaces Moving by Mean Curvature* GERHARD HUISKEN 17 5 The Level-Set Flow on a Manifold TOM ILMANEN 193 Unstable Solutions of Two-Dimensional Geometric Variational Problems* JURGEN JOST 205 Harmonic Maps and Superrigidity JURGEN JOST AND SHING TUNG YAU 245 Harmonic Functions of Polynomial Growth on Complete Manifolds ATSUSHI KASUE 281 The Structure of Constant Mean Curvature Embeddings in Euclidean Three Space NICK KOREVAAR AND ROB KUSNER 291 Uniformization of Spherical CR Manifolds and the CR Yamabe Problem ZHONGYUAN LI 299 The Theory of Harmonic Functions and Its Relation to Geometry* PETER LI 307 Harmonic Maps with Prescribed Singularities YAN YAN LI AND GANG TIAN 317 Some Recent Results on Harmonic Maps to Spheres* FANG HUA LIN 327 The Geometry, Topology, and Existence of Periodic Minimal Surfaces* WILLIAM H. MEEKS III 333 CONTENTS xi Soap Films and Mathematics FRANK MORGAN 375 Uniform Boundary Regularity Estimates for Minima of Certain Quadratic Functional LIBINH. Mou 381 Self-similar Solutions and Asymptotic Behavior of Flows of Nonpara- metric Surfaces Driven by Gauss or Mean Curvature VLADIMIR OLIKER 389 A Report on Geometric Quantization P. L. ROBINSON 403 Motion of Curves by Crystalline Curvature, Including Triple Junctions and Boundary Points JEAN E. TAYLOR 417 Recent Progress in Submanifold Geometry* CHUU-LIAN TERNG 439 Constant Mean Curvature Surfaces in the Heisenberg Group PER TOMTER 485 Complete Immersions of Constant Mean Curvature* HENRY C. WENTE 497 Banach Manifolds of Minimal Surfaces in the 4-Sphere HONGYOU Wu 513 On the Isolatedness for the Solutions of Plateau's Problem STEPHEN ZHENG 541 PART 2 Invariants of Gauss Maps of Theta Divisors MALCOLM ADAMS, CLINT MCCRORY, TED SHIFRIN, AND ROBERT VARLEY 1 On Characteristics of Hypersurfaces in Symplectic Manifolds AUGUSTIN
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