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 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 an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication. systematic copying. or multiple reproduction of any material in this pub- lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director. American Mathematical Society. P.O. Box 6248. Providence. Rhode Island 02940. The appearance of the code on the first page of an article in this volume 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 Copyright Clearance Center. Inc .. 21 Congress Street, Salem. Massachusetts 01970. This consent does not extend to other kinds of copying. such as copying for general distribution. for advertising or promotion purposes. for creating new collective works or for resale. 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