Complex Differential Geometry and Nonlinear Differential Equations Proceedings of a Summer Research Conference held August12-18,1984

AMERICAn MATHEMATICAL SOCIETY UOLUME 49 http://dx.doi.org/10.1090/conm/049 COnTEMPORARY MATHEMATICS

Titles in This Series

Volume 1 Markov random fields and their 15 Advanced analytic number theory. applications, Ross Kindermann and Part 1: Ramification theoretic J. Laurie Snell methods. Carlos J. Moreno 2 Proceedings of the conference 16 Complex representations of on integration. topology, and GL(2. K) for finite fields K, geometry in linear spaces. llya Piatetski-Shapiro William H. Graves. Editor 17 Nonlinear partial differential 3 The closed graph and P-closed equations, Joel A. Smoller. Editor graph properties in general topology, T. R. Hamlett and 18 Fixed points and nonexpansive L. L. Herrington mappings, Robert C. Sine. Editor 4 Problems of elastic stability and 19 Proceedings of the Northwestern vibrations, Vadim Komkov. Editor homotopy theory conference. 5 Rational constructions of Haynes R. Miller and Stewart B. modules for simple Lie algebras. Priddy, Editors George B. Seligman 20 Low dimensional topology, 6 Umbral calculus and Hopf algebras, Samuel J. Lomonaco. Jr.. Editor Robert Morris. Editor 21 Topological methods in nonlinear 7 Complex contour integral functional analysis, S. P. Singh. representation of cardinal spline S. Thomeier. and B. Watson. Editors functions, Walter Schempp 22 Factorizations of b" 1, b 8 Ordered fields and real algebraic ± = 2, 3, 5, 6, 7, 10,11,12 up to high geometry, D. W. Dubois and powers. John Brillhart. D. H. Lehmer. T. Recio. Editors J. L. Selfridge. Bryant Tuckerman. and 9 Papers in algebra. analysis and S. S. Wagstaff. Jr. statistics. R. Lidl. Editor 23 Chapter 9 of Ramanujan's second 10 Operator algebras and K-theory, notebook-Infinite series identities, Ronald G. Douglas and Claude transformations, and evaluations. Schochet. Editors Bruce C. Berndt and Padmini T. Joshi 11 Plane ellipticity and related problems, Robert P. Gilbert. Editor 24 Central extensions. Galois groups, and ideal class groups of number 12 Symposium on algebraic topology in fields, A. Frohlich honor of Jose Adem, Samuel Gitler. Editor 25 Value distribution theory and its 13 Algebraists' homage: Papers in applications. Chung-Chun Yang, ring theory and related topics. Editor S. A. Amitsur. D. J. Saltman. and 26 Conference in modern analysis G. B. Seligman. Editors and probability, Richard Beals. 14 Lectures on Nielsen fixed point Anatole Beck. Alexandra Bellow. and theory, Boju Jiang Arshag Hajian. Editors Titles in This Series

Volume 27 Microlocal analysis. M. Salah 40 Integral bases for affine Lie algebras Baouendi. Richard Beals. and and their universal enveloping Linda Preiss Rothschild. Editors algebras. David Mitzman 28 Fluids and plasmas: geometry and 41 Particle systems. random dynamics, Jerrold E. Marsden. Editor media and large deviations. 29 Automated theorem proving, Richard Durrett. Editor W. W. Bledsoe and Donald Loveland. 42 Classical real analysis. Daniel Editors Waterman. Editor 30 Mathematical applications of 43 Group actions on rings. Susan category theory, J. W. Gray. Editor Montgomery. Editor 31 Axiomatic set theory, James E. 44 Combinatorial methods in Baumgartner. Donald A. Martin. and topology and algebraic geometry, Saharon Shelah. Editors John R. Harper and Richard 32 Proceedings of the conference Mandelbaum. Editors on Banach algebras and several 45 Finite groups-coming of age. complex variables. F. Greenleaf and John McKay. Editor D. Gulick. Editors 46 Structure of the standard modules 33 Contributions to group theory, for the affine Lie algebra Ap>. Kenneth I. Appel. John G. Ratcliffe. James Lepowsky and Mirko Prime and Paul E. Schupp, Editors 47 Linear algebra and its role in 34 Combinatorics and algebra. systems theory, Richard A. Brualdi. Curtis Greene. Editor David H. Carlson. Biswa Nath Datta. 35 Four-manifold theory, Cameron Charles R. Johnson. and Robert J. Gordon and Robion Kirby, Editors Plemmons. Editors 36 Group actions on manifolds. 48 Analytic functions of one complex Reinhard Schultz. Editor variable. Chung-chun Yang and 37 Conference on algebraic topology Chi-tai Chuang. Editors in honor of Peter Hilton. 49 Complex differential geometry and Renzo Piccinini and Denis Sjerve. nonlinear differential equations. Editors Yum-Tong Siu. Editor 38 Topics in complex analysis. 50 Random matrices and their Dorothy Browne Shaffer. Editor applications, Joel E. Cohen. Harry 39 Errett Bishop: Reflections Kesten. and Charles M. Newman. on him and his research. Editors Murray Rosenblatt. Editor Complex Differential Geometry and Nonlinear Differential Equations COnTEMPORARY MATHEMATICS Volume49

Complex Differential Geometry and Nonlinear Differential Equations Proceedings of the AMS-IMS-&IAM Joint Summer Research Conference held August 12-181 19841 with support from the National Science Foundation

Yum-Tong Siu, Editor

AMERICAn MATHEMATICAL SOCIETY Providence • RhOda Island EDITORIAL BOARD R. 0. Wells, Jr., Jan Mycielski managing editor Johannes C. C. Nitsche Adriano M. Garsia Carl M. Pearcy James I. lepowsky Irving Reiner Alan D. Weinstein

The AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Complex Differential Geometry and Nonlinear Differential Equations was held at Bowdoin College, Brunswick, Maine on August 12-18, 1984 with support from the National Science Foundation, Grant DMS-8218075.

1980 Mathematics Subject Classification. Primary 53C55; Secondary 32C1 0, 58G30.

Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference (1984: Bowdoin College) Complex differential geometry and nonlinear differential equations. (Contemporary mathematics, ISSN 0271-4232; v. 49) Bibliography: p. 1. Global differential geometry-Congresses. 2. Differential equations, Nonlinear-Congresses. I. Siu, Yum-Tong, 1943- . II. American Mathematical Society. Ill. Institute of Mathematical Statistics. IV. Society for Industrial and Applied Mathematics. V. Title. VI. Series: Contemporary mathematics (American Mathematical Society); v. 49. QA641.A598 1984 516.3'62 85-28656 ISBN 0-8218-5049..0

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Copyright© 1986 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. This volume was printed directly from author prepared copy. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. LIST OF ARTICLES

Preface ix Program Schedule xi List of Participants xiii The New Approach to the Local Embedding Theorem of CR-Structures for n ~ 4 (the Local Solvability for the Operator ab in the Abstract Sense) TAKAO AKAHORI 1 Remarks on Curvature Integrals and Minimal Varieties MICHAEL T. ANDERSON 11 On the Automorphism Group of Strictly Convex Domains in cn J. BLAND, T. DUCHAMP, and M. KALKA 19 Inequality Between Chern Numbers of Singular Kahler Surfaces and Characterization of Orbit Space of Discrete Group of SU(2,1) S. Y. CHENG and S. T. YAU 31 Laplacian on Manifolds and Analogous Difference Operators for Graphs JOZEF DODZIUK 45 On Isotropic Harmonic Maps to Real and Quaternionic Grassmannians J. F. GLAZEBROOK 51 Characterizing CPn by the Spectrum of the Laplacian SAMUEL I. GOLDBERG 63 Stable Minimal Surfaces in Flat Tori MARIO J. MICALLEF 73 Foliation Techniques and Vanishing Theorems NGAIMING MOK 79 Complex Finsler Metrics H. L. ROYDEN 119 Applications of Harmonic Maps to Kahler Geometry J. H. SAMPSON 125

vii viii LIST OF ARTICLES

On the Relation Between Chern and Pontrjagin Numbers S. M. WEBSTER 135 Harmonic Morphisms, Foliations and Gauss Maps J. C. WOOD 145 PREFACE

These proceedings contain the papers contributed by the speakers of the Summer Research Conference on Complex Differential Geometry and Nonlinear Differential Equations that took place in Bowdoin College, Brunswick, Maine from August 12, 1984 to August 18, 1984. Most of the papers are the actual talks given in the conference. Some papers were developed from the talks and have titles somewhat different from those of the talks to reflect the difference in content. Not every speaker contributed a paper, because some of the talks reported on papers that were already published elsewhere or will soon appear in other publications.

Yum-Tong Siu

ix Program Schedule

Monday, August l_~ 1984

E. Ca1abi Yang-Mills problems for K~hler manifolds

J. Spruck Nonlinear elliptic problems and some geometric applications

Y. Tong Special harmonic forms on locally Hermitian symmetric spaces

B. White Harmonic maps and homotopy classes

N. Mok K~hler-Einstein manifolds of semipositive curvature

Construction of bounded holomorphic functions on simply connected negatively curved K~hler manifolds

Tuesday, August l_~ 1984

R. Schoen Constant mean curvature surfaces in R3 and s 3

L. Sibner The removable point singularity problem for Yang-Mills fields

T. Akahori A new approach to the local embedding theorem of CR structure in the case of dimRM = 2n-1 > 7

R. Sibner A free boundary minimal surface problem

M. Kalka The automorphisms of a convex domain

J. Glazebrook Harmonic maps to quaternionic projective spaces

Wednesday, August~ 1984

S. Webster Characteristic numbers of real submanifolds of en

N. Stanton The heat equation for the a b equation

xi xii Program Schedule

Thursday, Augus_t: ~ 1984

S. Goldberg Characterizing CPn by the spectrum of the Laplacian

J. Wolfson Minimal 2-sphere in complex Grassman manifold

A. Friedman Blow up of solutions of semilinear parabolic equation

P. Aviles Conformal deformations of complete manifolds with negative curvature

M. Micallef Stable minimal surfaces in Euclidean space

Friday, August 17, 1984

J. Sampson Applications of harmonic maps to K~hler geometry

J. Wood Holomorphic and harmonic maps and morphisms

M. Anderson Compactification of minimal submanifolds in En

S. T. Yau K~hler-Einstein metrics with singularities

J. Dodziuk Laplacian on manifolds and analogous difference operators for graphs

Y. T. Siu Estimating the dimensions of cohomology groups by curvature ORGANIZING COMMITTEE

Luis A. Caffarelli Richard M. Schoen Yum-Tong Siu (chairman)

LIST OF PARTICIPANTS

Takao A. Akahori Max-Planck Institute, West Germany

Michael T. Anderson Califonia Institute of Technology

Paticio Aviles University of Illinois

John s. Bland Tulane University

Eugenio Calabi University of Pennsylvania

Koji Cho Harvard University

Hyeong I. Choi Mathematical Sciences Research Institue, Berkeley

Patick R. Coulton Eastern Illinois University

Jozef Dodziuk Queen's College

Franco Favilli Universita di Pisa, Italy

Alan D. Fekete Harvard University

Avner Friedman Northwestern University

James F. Glazebrook Centro de Investigacion y de Estudios Avanzados del Instituto Politecnio Nacional, Mexico

Samuel I. Goldberg University of Illinois

David A. Hoffman University of Massachusetts

~!orris Kalka Tulane University

Luc Lemaire Free University of Bruxelles

Mario J. Micallef University of Michigan

Naiming Mok Princeton University

Nirmala Parkash Harvard University

Enrique Ramirez de Centro de Investigacion y de Estudios Avanzados Arellano del Instituto Politecnio Nacional, Mexico

xiii XiV LIST OF PARTICIPANTS

Halsey L. Royden Stanford University

Joseph H. Sampson Johns Hopkins University

Richard M. Schoen University of California, Berkeley

Lesley M. Sibner Polytech Institute of New York

Robert Sibner City University of New York

Yum-Tong Siu Harvard University

Joel Spruck University of Massachusetts

Nancy K. Stanton University of Notre Dame

Duraiswamy Sundararaman Centro de Investigacion y de Estudios Avanzados del Institute Politecnio Nacional, Mexico

Yue Lin L. Tong Purdue University

Al Vitter Tulane University

Sidney M. Webster University of Minnesota

Brian c. White Stanford University

Jon G. Wolfson Rice University

Pit-Mann Wong University of Notre Dame

John c. Wood University of Leeds

Paul Yang University of Southern California

Shing-Tung Yau University of California, San Diego

Stephen Yau University of Illinois at Chicago