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Stratified Multivarifolds, and the Plateau Problem Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem BA0 TRONG THI A. T. FOMENKO TRANSLATIONS OF MATHEMATICAL MONOGRAPHS TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME84 Minimal Surfaces, Stratified M u ltiva rifo lds, and the Plateau Problem 8AO TRONG THI A. T. FOMENKO American Mathematical Society ProvidenceRhode island AAO 'IOHI' TXH A. T. 4)OMEHKO MHHHMAJIbHbIE IIOBEPXHOCTH H HPOEJIEMA HJIATO «HAYKA», MOCKBA, 1987 Translated from the Russian by E. J. F. Primrose Translation edited by Ben Silver 1980 Mathematics Subject Classification (1985 Revision). Primary49F10, 53A10; Secondary 58E12, 58E20. ABSTRACT. The book is an account of the current state of the theory of minimal surfaces and of one of the most important chapters of this theory-the Plateau problem, i.e. the problem of the existence of a minimal surface with boundary prescribed in advance. The authors exhibit deep connections of minimal surface theory with differential equations, Lie groups and Lie algebras, topology, and multidimensional variational calculus. The presentation is simplified to a large extent; the book is furnished with a wealth of illustrative material. Bibliography: 471 titles. 153 figures, 2 tables. Library of Congress Cataloging-ln-Publicatlon Data Dao, Trong Thi. [Minimal'nye poverkhnosti i problema Plato. English] Minimal surfaces, stratified multivarifolds, and the Plateau problem/Dio Trong Thi, A. T. Fomenko. p.cm. - (Translations of mathematical monographs; v. 84) Translation of: Minimal'nye poverkhnosti i problema Plato. Includes bibliographical references and index. ISBN 0-8218-4536-5 1. Surfaces, Minimal.2. Plateau's problem.I. Fomenko, A. T.II. Title.III. Series. QA644.D2813 1991 90-22932 516.3'62-dc2O CIP Copyright ©1991 by the American Mathematical Society. All rights reserved. Translation authorized by the All-Union Agency for Authors' Rights, Moscow The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. 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-TEJC, the American Mathematical Society's TEJC macro system. 10987654321 969594939291 Table of Contents Preface vii Introduction 1 CHAPTER I. Historical Survey and Introduction to the Classical Theory of Minimal Surfaces 21 §1. The sources of multidimensional calculus of variations 21 §2. The 19th century-the epoch of discovery of the main properties of minimal surfaces 30 §3. Topological and physical properties of two-dimensional mimimal surfaces 43 §4. Plateau's four experimental principles and their consequences for two-dimensional mimimal surfaces 62 §5. Two-dimensional minimal surfaces in Euclidean space and in a Riemannian manifold 68 CHAPTER II. Information about Some Topological Facts Used in the Modern Theory of Minimal Surfaces 95 § 1. Groups of singular and cellular homology 95 §2. Cohomology groups 97 CHAPTER III. The Modern State of the Theory of Minimal Surfaces 99 1. Minimal surfaces and homology 99 §2. Theory of currents and varifolds 129 §3. The theory of minimal cones and the equivariant Plateau problem 138 CHAPTER IV. The Multidimensional Plateau Problem in the Spectral Class of All Manifolds with a Fixed Boundary 167 iii iv TABLE OF CONTENTS 1. The solution of the multidimensional Plateau problem in the class of spectra of maps of smooth manifolds with a fixed boundary. An analog of the theorems of Douglas and Rado in the case of arbitrary Riemannian manifolds. The solution of Plateau's problem in an arbitrary class of spectra of closed bordant manifolds 167 12. Some versions of Plateau's problem require for their statement the concepts of generalized homology and cohomology 178 §3. In certain cases the Dirichlet problem for the equation of a minimal surface of large codimension does not have a solution 18.3 §4. Some new methods for an effective construction of globally minimal surfaces in Riemannian manifolds 186 CHAPTER V. Multidimensional Minimal Surfaces and Harmonic Maps 207 ,q1. The multidimensional Dirichlet functional and harmonic maps. The problem of minimizing the Dirichlet functional on the homotopy class of a given map 207 §2. Connections between the topology of manifolds and properties of harmonic maps 217 §3. Some unsolved problems 228 CHAPTER VI. Multidimensional Variational Problems and Multivarifolds. The Solution of Plateau's Problem in a Homotopy Class of a Map of a Multivarifold 233 § 1. Classical formulations 233 §2. Multidimensional variational problems 234 §3. The functional language of multivarifolds 239 §4. Statement of Problems B, B', and B" in the language of the theory of multivarifolds 249 CHAPTER VII. The Space of Multivarifolds 253 1. The topology of the space of multivarifolds 253 §2. Local characteristics of multivarifolds 260 §3. Induced maps 266 CHAPTER VIII. Parametrizations and Parametrized Multivarifolds 273 §1. Spaces of parametrizations and parametrized multivarifolds 273 §2. The structure of spaces of parametrizations and parametrized multivarifolds 280 TABLE OF CONTENTS v §3. Exact parametrizations 290 §4. Real and integral multivarifolds 295 CHAPTER IX. Problems of Minimizing Generalized Integrands in Classes of Parametrizations and Parametrized Multivarifolds. A Criterion for Global Minimality 299 §1. A theorem on deformation 299 §2. Isoperimetric inequalities 309 §3. Statement of variational problems in classes of parametrizations and parametrized multivarifolds 314 §4. Existence and properties of minimal parametrizations and parametrized multivarifolds 317 CHAPTER X. Criteria for Global Minimality 331 § 1. Statement of the problem in the functional language of currents 331 §2. Generalized forms and their properties 334 §3. Conditions for global minimality of currents 336 §4. Globally minimal currents in symmetric problems 342 §5. Specific examples of globally minimal currents and surfaces 350 CHAPTER XI. Globally Minimal Surfaces in Regular Orbits of the Adjoint Representation of the Classical Lie Groups359 § 1. Statement of the problem. Formulation of the main theorem 359 §2. Necessary information from the theory of representations of the compact Lie groups 361 §3. Topological structure of the space G/TG 365 §4. A brief outline of the proof of the main theorem 368 Appendix. Volumes of Closed Minimal Surfaces and the Connection with the Tensor Curvature of the Ambient Riemannian Space 377 Bibliography 381 Subject Index 401 Preface Plateau's problem is a scientific trend in modern mathematics that unites several different problems connected with the study of minimal surfaces, that is, surfaces of least area or volume. In the simplest version, we are con- cerned with the following problem: how to find a surface of least area that spans a given fixed wire contour in three-dimensional space. A physical model of such surfaces consists of soap films hanging on wire contours af- ter dipping them in a soap solution. From the mathematical point of view, such films are described as solutions of a second-order partial differential equation, so their behavior is quite complicated and has still not finally been studied. Soap films or, more generally, interfaces between physical me- dia in equilibrium, arise in many applied problems, in chemistry, physics, and also in nature. A well-known example is that of marine organisms, Radiolaria, whose skeletons enable us to see clearly the characteristic singu- larities of interfaces between media and soap films that span complicated boundary contours. In applications there arise not only two-dimensional but also multidimensional minimal surfaces that span fixed closed "con- tours" in some multidimensional Riemannian space (manifold). It is con- venient to regard such surfaces as extremals of the functional of mul- tidimensional volume, which enables us when studying them to employ powerful methods of modern analysis and topology. We should mention that an exact mathematical statement of the problem of finding a surface of least area (volume) requires a suitable definition of such fundamental concepts as a surface, its boundary, minimality of a surface, and so on. It turns out that there are several natural definitions of all these concepts, which enable us to study minimal surfaces by different methods, which complement one another. In the framework of a comparatively small book it is practically impos- sible to cover all aspects of the modern problem of Plateau, to which a vast literature has been devoted. The authors have therefore tried to construct the book in accordance with the following principle: a maximum of clarity VII viii PREFACE and a minimum of formalization. Of course, it is possible to satisfy such a requirement only approximately, so in certain cases (this applies mainly to the last chapters of the book) we dwell on certain nontrivial mathematical constructions that are necessary for a concrete investigation of minimal surfaces. The book can be conventionally split into three parts: (a) Chapter 1, which contains historical information about Plateau's problem, referring to the period preceding the 1930s, and a description of its connections with the natural sciences; (b) Chapters 2-5, which give a fairly complete survey of various modem trends in Plateau's problem; (c) Chapters 6-11, in which we give a detailed exposition of one of these trends (the homotopic version of Plateau's
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