The Plateau Problem, Part I: Historical Survey (Studies in the Development of Modern Mathematics)

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Studies in the Development of Modern Mathematics A Series of Books and Monographs on the development of mathematical concepts within their scientific and historical context. Edited by Yu. I. Manin, Steklov Institute of Mathematics, Moscow, USSR.

Volume 1 The Plateau Problem I Historical Survey II Present State of the Theory A.T. Fomenko

Additional Volumes in Preparation: Kronecker's Jugendtraum and Modular Functions by S.G. Vladuts Space, Points of Which are Lines by S. G. Gindikin Solitons by Yu. A. Danilov and V.I. Perviashvili

This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

THE PLATEAU PROBLEM Historical survey

A. T. F omenko Faculty of Mechanics and Mathematics Moscow State University USSR

Part I

GORDON AND BREACH SCIENCE PUBLISHERS New York. Philadelphia. London. Paris. Montreux • Tokyo. Melbourne

©1990 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers Post Office Box 786 Cooper Station New York, New York 10276 United States of America

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Pages 123-141 of Part I: Copyright Springer-Verlag, 1933. Reproduced with permission. Library of Congress Cataloging-in-Publication Data Fomenko, A.T. The Plateau problem/A.T. Fomenko cm.-(Studies in the development of modern mathematics; v. 1) p. Translated from the Russian. Contents: pt. 1. Historical survey-pt. 2. Present state of the theory. ISBN 2-88124-702-4 (set): $350.00 (est.).-ISBN 2-88124-700-8 (pt. 1): $200.00 (est.).-ISBN 2-88124-701-6 (pt. 2): $200.00 (est.) 1. Plateau's problem. 2. Surfaces, Minimal. I. Title. II. Series. QA644.F66 1989 89-7443 516.3'62-dc20 CIP

No part of this book may be reproduced or utilized in any form or by any means, electronic and mechanical, including photocopying and recording, or by any information storage or retrieval sy:ltem, without permission in writing from the publisher. Printed in Great Britain by Billing & Sons Ltd., Worcester.

CONTENTS Preface

I

xi

HISTORICAL SURVEY AND INTRODUCTION TO THE THEORY OF MINIMAL SURFACES §1. The Origins of Multidimensional Variational Calculus.

§2. The Nineteenth Century, the Era of the Discovery of Basic Minimal Surface Properties. 1. Plateau's physical experiments and methods of forming soap films. 2. Physical principles underlying the formation of soap films. 3. The extremal properties of soap films and minimality of their area. The properties of the surface of separation between two media. 4. The surface of separation between two media in equilibrium is a surface of constant mean curvature. S. Soap films of constant positive curvature and constant zero curvature.

§3. Topological and Physical Properties of Minimal Surfaces. 1. Stable and unstable surfaces. 2. Plateau's experiments with stable columns of liquids. 3. The physical realization of a helicoid. 4. The physical realization of a catenoid and its restructuring as the boundary contour changes. The change of the topological type of minimal surface in accordance with its stability or instability.

§4. The Plateau Principles. Minimal Surfaces in Animate Nature. 1. Two-dimensional minimal surfaces in three-dimensional space and the first Plateau principle.

7

7 11

13

16 19

20 20 22 27 30 34

49

49

CONTENTS

vi

2. The area functional, Dirichlet functional, harmonic mappings and conformal coordinates. 3. Singular points of minimal surfaces and three Plateau principles. 4. The realization of minimal surfaces in animate nature.

II

III

51 55 60

SURVEY OF SOME IMPORTANT PUBLICATIONS IN MINIMAL SURFACE THEORY FROM THE NINETEENTH TO THE EARLY TWENTIETH CENTURY

65

§1. Monge (1746-1818).

65

§2. Poisson (1781-1840).

74

§3. Plateau (1801-1883).

83

§4. Some works of the Early Twentieth Century Rad6, Douglas).

122

§s. Minimal Surfaces in the Large.

123

SOME FACTS FROM ELEMENTARY TOPOLOGY §1. Singular and Cellular Homology Groups. 1. Singular chains and homology groups. 2. Cellular complexes, barycentric subdivisions. 3. Cellular homology and computation of the singular homology of the sphere. 4. Theorem on the coincidence of the singular and cellular homology of a finite complex. 5. The geometric determination of cellular homology groups. 6. The simplest examples of cellular homology group computations.

§2. Cohomology Groups and Obstructions to the Extensions of Mappings. 1. Singular cochains and the coboundary operator.

173 173 173 176 180 184 186 189

192 192

CONTENTS 2. The problem of the extension of a continuous mapping from a subspace to the whole space. 3. Obstructions to the extension of mappings. 4. The cases of the existence of the retraction of a space onto a subspace which is homeomorphic to the sphere. References and Bibliography

Index

vii

194 194 199 204 219

List of Contents of Part II

IV

MODERN STATE OF MINIMAL SURFACE THEORY §1. Minimal Surfaces and Homology. 1. Two-dimensional minimal surfaces in Euclidean space and in Riemannian manifolds. 2. The multidimensional Riemannian volume functional and second fundamental form on a submanifold. 3. Multidimensional locally minimal surfaces. 4. The global minimality of complex submanifolds. 5. The complex Plateau problem. 6. On various approaches to the concepts of surface and boundary of a surface. 7. The homology boundary of a surface and the role of the coefficient group. 8. Surprising examples of physical stable minimal surfaces, that nevertheless retract onto their boundaries. 9. When does a soap film spanning a frame not contain closed soap-bubbles?

§2. Integral Currents. 1. De Rham currents. Basic notions. 2. Rectifiable currents and flat chains. 3. Normal and integral currents. 4. Various formulations of the minimal current existence theorem. 5. Varifolds and minimal surfaces 6. The interior regularity for minimal surfaces and the structure of their singular points.

9 16 18 20 21 24 28 41 48

51 54 55

57 59 65

viii

CONTENTS 7. Regularity almost everywhere for the supports of elliptic-integrand-minimizing k-currents and k-varifolds. 8. The interior regularity for volume-minimizing hypersurfaces and the existence of minimal cones of codimension one. 9. Dimension estimates for the set of singular points of a minimal surface. 10. Other problems of minimal surface regularity.

§3. Minimal Currents in Riemannian Manifolds. 1. Minimal cones associated with singular points of minimal surfaces. 2. Multidimensional minimal cones. 3. Minimal surfaces invariant with respect to the action of Lie groups. 4. The Fermat principle, minimal cones, and light rays. 5. S.N. Bernstein's problem.

§4. Minimization of Volumes of Manifolds with Fixed Boundary and of Closed Manifolds. Existence of a Minimum in Each Spectral Bordism Class.

67

69 71 74

75

75 78 88 93 104

108

1. Bordant manifolds and the multidimensional Plateau problem. 2. The properties of bordant manifold classes. 3. The statement of the existence theorem for globally minimal surfaces in the spectral bordant manifold class.

123

§5. Generalized Homology and Cohomology Theories and Their Relation to the Multidimensional Plateau Problem.

126

1. The definition of generalized homology and cohomology. 2. The coboundary and boundary of a pair of spaces (X,A). '3. Surface variational classes. 4. The general existence theorem for globally minimal surfaces in an arbitrary class determined by a generalized spectral homology or cohomology theory. 5. A short sketch of the proof of Theorem 1.

108 115

126 129 130

132 135

CONTENTS

§6. Existence of a Minimum in Each Homotopy Class of Multivarifolds. 1. The functional multivarifold language. 2. Multivarifolds and variational problems in the classes of surfaces of fixed topological type. 3. Minimization problems for generalized integrands in the parametrization and parametrized multivarifold classes. 4. Criteria for the global minimality of surfaces and currents.

ix

142 142 150 154 158

§7. Cases where a Solution of the Dirichlet Problem for the Equation of Minimal Surfaces of High Codimensions does not Exist.

165

§8. Example of a Smooth, Closed, Unknotted Curve in R 3 , Bounding Only Minimal surfaces of Large Genus.

175

§9. Certain New Methods of Effective Construction of Globally Minimal Surfaces in Riemannian Manifolds.

182

1. The universal lower estimate of the volumes of topologically non-trivial minimal surfaces. 2. The coefficient of deformation of a vector field. 3. Surfaces of non-trivial topological type and of least volume. 4. On the minimal volume of surfaces passing through the centre of a symmetric convex domain in Euclidean space.

§10. Totally Geodesic Surfaces Realizing Non-Trivial Cycles, Cocycles, and Elements of Homotopy Groups in Symmetric Spaces. 1. Totally geodesic submanifolds in Lie groups. 2. Necessary information about symmetric spaces. 3. When does a totally geodesic submanifold realize a nontrivial cycle? 4. The classification theorem for totally geodesic submanifolds realizing non-trivial cycles in symmetric spaces. 5. The classification of cocycles realizable by totally geodesic spheres in compact Lie groups. 6. The classification of elements of homotopy groups realizable by totally geodesic spheres in symmetric spaces of type I.

182 185 186 197

201 201 203 203 207 208

211

x

CONTENTS

§11. Bott Periodicity and Its Relation with the Multidimensional Dirichlet Functional. 1. The explicit description of unitary Bott periodicity. 2. Unitary Bott periodicity follows from the properties of the two-dimensional extremals of the Dirichlet functional. 3. Orthogonal periodicity follows from the properties of the eight-dimensional extremals of the Dirichlet functional.

214 214 217 219

§12. Survey of Some Recent Results in Harmonic Mapping Theory.

221

References and Bibliography Index

234 249

Preface The Pleaeau Problem is a wide branch of modern mathematics embracing many different problems relating to the study of so-called minimal surfaces, that is, surfaces of minimal area. In the simplest version, we deal with the following problem: it is required to find the surface of minimal area spanning a given fixed wire frame in three-dimensional space. A physical model of such a surface is a soap film formed on a wire frame after it has been removed from a soap solution. From the mathematical point of view, such soap films are described by solutions of a partial differential equation of the second order; hence their behaviour is rather complicated and not yet fully investigated. Soap films or, more generally, surfaces of separation between physical media in equilibrium occur in many applied problems of chemistry, physics and animate nature. One well-known example is given by radiolarians, marine organisms whose skeletons are a good visual representation of special features of soap films spanning quite complicated boundary frames. Applications give rise not only to two-dimensional, but also multi-dimensional minimal surfaces spanning fixed closed "contours" in some multi-dimensional Riemannian space (manifold). It is convenient to consider such surfaces as extremals of the multidimensional volume functional, which makes it possible to apply powerful methods of modern analysis and topology for their study. It should be noted that to pose the problem of determining a surface ofleast area (or volume) with mathematical rigour requires the correct definition of such fundamental concepts as surface, its boundary, minimality of a surface, etc. It turns out that there have been several natural definitions of all these concepts, which enables us to study minimal surfaces by various methods complementary to one another. Within the bounds of a comparatively small book, it is practically impossible to embrace all the facets of the modern Plateau problem, to which an enormous amount of literature is devoted. The author, therefore, has set himself the goal of writing a book in accordance with the following principle: maximum of clarity and minimum offormalization. Certainly, this requirement can be fulfilled only to a certain extent and so, in some cases (mainly with regard to the last chapter of the book), we have had to dwell on particular non-trivial mathematical constructions whose formalization is necessary for a specific study of minimal surfaces. In the first chapter, a brief historical survey of the origins of the modern Plateau problem is given. We begin it with some papers from the early part of the eighteenth century and then give more attention to the nineteenth, when the basic properties of minimal surfaces were discovered. We consider still more thoroughly the famous physical experiments of Plateau (1801-1883), in which various observations of the behaviour of the surface of separation of two media were systematized. One of the results of this series of experiments was a precise definition of the so called Plateau principles which guide both the local and global topological behaviour of soap films. Together with an account of the mathematical and physical aspects of the Plateau xi

xii

PREFACE

problem, we speak about those mathematicians whose activity was most closely related to the topics in question. We also characterize the historical reasons for the emergence of certain mathematical, mechanical, or physical aspects of the Pleaeau problem. The second chapter contains fragments of some remarkable papers of the nineteenth and early twentieth centuries, devoted to the investigation of minimal surfaces. In our opinion, an acquaintance with them is very educative, since the ideas underlying the researches of Monge, Poisson, Plateau, Douglas, and Rad6 were subsequently developed into independent scientific disciplines. The third chapter is of an auxiliary character, containing information regarding homology and cohomology theories, necessary for the further investigation of multidimensional variational problems of the present day. Finally, the fourth chapter contains a description of the modern state of the Plateau problem. The book is designed for mathematicians, specialists in calculus of variations, topology, functional analysis, theory of differential equations, and Lie groups and algebras.

A. T. Fomenko

I HISTORICAL SURVEY AND INTRODUCTION TO THE THEORY OF MINIMAL SURFACES § 1. The Origins of Multidimensional Variational

Calculus. In this section we shall introduce the reader to the atmosphere in which one of the most important branches of modern mathematics, namely, multidimensional calculus of variations, was being created. The theory of minimal surfaces, which originated first in the remarkable researches of mathematicians and mechanical engineers of the eighteenth and nineteenth centuries, is developing today within variational calculus. The original development of this theory is inseparable from the personalities of the specialists who were involved in this research, and also from the specific historical environment the demands of which led to the fast growth of the theory with its concrete applications to problems of mechanics and physics. It is customary to single out the so-called one-dimensional and multidimensional variational problems in modern variational calculus. One-dimensional problems deal with the investigation of functionals defined, for example, on the space of piecewise smooth curves 'Y (t) in a Riemannian manifold. The classical examples of these functionals are the length functional 111' Idt and action functional II1'j2dt. The extremals of such functionals are, therefore, certain curves in manifolds. For example, the extremals of the length functional are geodesic lines parametrized by an arbitrary continuous parameter while the extremals of the action functional are geodesic lines parametrized by the natural parameter, that is, arc length measured from a certain fixed point of the curve. However, in many problems of physics and mechanics, important functionals arise which are defined on multidimensional objects and surfaces, for example, on the space of two-dimensional surfaces with fixed boundary. An important example is the area functional associating each of these surfaces with its area. Another example closely linked to the previous one is the so-called Dirichlet functional whose definition we give below. The relation between the area functional and Dirichlet functional is very similar to the well-known relation between the length and action functionals. In this terminology, the area and Dirichlet functionals can be called two-dimensional functionals. The main object under investigation in the present book is multidimensional functionals of dimension three and higher. It so happens that, in modern terminology, a certain shift of meaning has taken place. Since the case of two-dimensional functionals has been studied quite well by now, it is already considered as "classical" and currently, by the term "multidimensional functionals" one often means functionals defined on surfaces of dimension three and higher. However, at the

2

THE PLATEAU PROBLEM: PART ONE

beginning of the twentieth century, multidimensional functionals also referred to two-dimensional ones. It should be borne in mind each time the term "multidimensional functional" is encountered in the literature. A systematic study of one-dimensional functionals is usually associated with the name of Leonhard Euler (1707-1783) of Basle (Switzerland). His father Paul Euler intended the son for an ecclesiastic career, but the young Leonhard Euler took a great interest in mathematics under the influence of J. Bernoulli whose lectures he attended and who supervised Euler's work. It is a remarkable fact that Paul Euler also studied mathematics under Jakob Bernoulli's supervision l55 , 173. In 1725, J. Bernoulli's sons Daniel and Nicolas came to St. Petersburg and soon the young L. Euler received, on their recommendation, an invitation to work in the department of physiology at the St. Petersburg Academy of Sciences 155. He arrived in St. Petersburg in 1727 and worked in Russia for 14 years (until 1741) earning himself acclaim as the greatest mathematician of his time. The interests of Euler were extremely wide, and his scientific heritage enormous. He wrote more than 850 papers and a very great number of letters, many of which are actually separate mathematical investigations. In St. Petersburg Euler carried out a multitude of state orders, made up geographical maps, published papers in analysis, number theory, differential equations, astronomy, etc. In 1738, Euler went blind in one eye as a result of overtaxing himself(in 1766, he lost his sight altogether), but this ailment did not slow down his creative activity. Around Euler, a school of talented scientists was formed including Kotelnikov, Rumovsky, Fuss, Golovin, Sofronov et al.l55. However, he did not feel comfortable in St. Petersburg and accepted an invitation to move to Berlin in 1741 and work in the Academy of Sciences there. Even so, Euler worked in St. Petersburg again from 1766 to 1788173. He was married twice and had thirteen children. In the second St. Petersburg period of his scientific activity, Euler presented another 416 books and papers before the Academy. They were dictated to his disciples. Many problems solved or posed by Euler arose during his research in applied science and lunar orbit calculus, in particular, shipbuilding and navigation theories. About 40% of his papers are devoted to applied mathematics, physics, mechanics, hydromechanics, elasticity theory, ballistics, machine theory, optics, etc. Many discoveries made by Euler were re-discovered after his death, this being particularly noticeable in the theory of differential equations 155. In the history of differential geometry, it is usually assumed (see, for example,174) that in 1760 Euler discovered a new branch of geometry combining both purely geometric and differential variational methods. In this respect, particularly interesting is his well-known "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes" (1744), in which he gave methods of solution of isoperimetric problems and investigated the geometric properties of some remarkable curves, in particular, the catenary curve. This was the first ever account of variational calculus. It contained Euler's equations and many

THEORY OF MINIMAL SURFACES

3

applications. It was Euler, by the way, who discovered that a catenoid (surface formed by rotating a catenary curve, that is, a curve formed by a heavily sagging fixed-end chain) is a two-dimensional minimal surface. Soon after the paper "Recherches sur la courbure des surfaces" (Histoire de l' Academie des Sciences, Berlin, pp. 119-141) of 1760 Lagrange (1736-1813), then still a young professor in Turin, published in 1762 his well-known paper on variational calculus 105 • Besides the many remarkable results of 105 he reduced, in its supplement, the minimal surface (Le., extremals of the area functional) equation to a form, in which the functions p and q are determined from the following condition: the two differential I-forms pdx+qdy and JdY - 2q dx 2 should be total 1+ P +q differentials. These functions are precisely what defines a minimal surface. Nevertheless, one cannot help noticing the somewhat sceptical attitude of Lagrange towards differential geometry problems. Thus, for example, the following remark made in his letter to d' Alembert in 1772 has become widely known: "Does it seem to you that the Sublime Geometry tends to be a little decadent?". It is possible that this view accounts for Lagrange treating these geometric problems only as illustrations of the applications of the remarkable analytic methods developed by him. Thus, for instance, the minimal surface equation was obtained by Lagrange by applying the general methods of variational calculus. Meanwhile, he did not give a geometric interpretation of the results so obtained. Moreover, Lagrange did not pose the question as to how to find non-trivial examples or minimal surfaces in three-dimensional space. The next important step in the direction of the development of minimal surface theory (Le., surfaces of locally minimal area) was made by G. Monge (1746-1818). Many achievements in the above-mentioned branch of variational calculus are associated with this name. Before turning to an account of them, we dwell briefly on the biography of Monge and the period during which his unique talent took shape. Gaspard Monge was born in 1746 into the family of a pedlar who achieved rather high rank as the head of a trading corporation. This allowed Monge's father to give his three sons a good primary education, which resulted in their all subsequently becoming mathematics professors. Monge received his secondary education in the college of the French town of Beaune (dept. Cote-d'Or). After this, Monge entered a college of the Oratorians in Lyons, and left at the age of 18. Having declined an offer to take holy orders, he soon obtained a place in the military school at Mezieres, which prepared engineers of high qualification. From 1769, Monge had secured himself an independent position as professor of the Mezieres school, which gave him a chance to plunge into active scientific work. The scope of his interests was enormously wide, mathematics, physics, chemistry. His first discovery that became widely known had been the creation of descriptive geometry. The idea of projecting an object under investigation on to various planes and the methods of its reconstruction from a priori known

4


projections was not new, of course; it was applied by many authors beginning with the fifteenth and sixteenth centuries (see, for example, the research by A. Diirer 161 • However, in the papers of Monge, these methods were concretized in application to engineering problems, allowing D. Struik to remark that "his remarkable geometric intuition went hand in hand with practical engineering applications, to which his whole manner of thinking was always inclined"174. In particular, the development of descriptive geometry was stimulated by the demands of the theory and practice of fortification. The creation of descriptive geometry was acknowledged as a sufficient reason for electing Monge a member of the Academie. In 1780, Monge was elected adjoint geometre in place of Vandermonde 2o • Monge's universalism started manifesting itself at that time particularly clearly. For example, between February and May 1780, Monge presented the following reports: on a model of a carriage; on two mathematical memoirs of Legendre; on the pressure of the wind; on a machine to clean the ports; on the possibility of flying like the birds; on a machine for milling grain; on a system for lifting water. In December of the same year, he prepared a memoir on the movement of rivers and on probability calculus. Between January and May 1781, he reported about various types of pumps, an exceptionally severe frost of 1776, a machine for salvaging ships, windmills, methods of preventing destructions by mountain streams, axles for carts, on the supply of water in Beziers.J7, p. 38. In 1781, a new edition of the Encyclopidie methodique was initiated, and Monge took part in editing it as a physicist. In 1793, Dictionnaire de physique was started, and Monge not only worked as its editor, but also was the author of a number of articles. The biography of Monge's scientific activities is, probably, one of the most striking examples of the fact that urgent demands of production and technology bring forth remarkable scientists universalism similar to that of the scientists of the Renaissance when many great minds were not only mechanical engineers, mathematicians, or physics, but also poets, artists or sculptors. From 1783, Monge became interested in chemistry and metallurgy. He discovered (even before the experiments of Lavoisier) that water is a chemical compound of hydrogen and oxygen. Monge's priority in this respect was acknowledged by Lavoisier in his memoir (together with Laplace). Monge carried out research in aeronautics. He suggested hydrogen and carbon monoxide as a balloon filler. In 1785, Monge took up statics. When examining naval candidates, he became convinced that they were insufficiently acquainted with mechanics, and wrote (in 1786) a treatise" Traite elimentaire de statique a f'usage des colleges de fa Marine" (Paris, 1788). In particular, the treatise contained the theory of machines and conditions for equilibrium of levers of arbitrary shape and movable with respect to some of their points. He thoroughly investigated rectilinear levers (with and without weight), simple pulleys, blocks and tackles, windlasses, toothed wheels, lifting jacks both from the theoretical and practical points of view. This engineering approach is common to Monge


5

and many other specialists of that time. The mathematicians Monge and Vandermonde and the engineer Perier inspected blast furnaces and other metallurgical equipment. In 1786, Monge read a memoir (written in coauthorship with Vandermonde and Perier) in the Academie in which he collected the results of their experimental research into the changes of properties of iron, steel and cast iron in metallurgical processes. In the eighties, Monge was one of the greatest scientists of pre-war France. In 1790, at the age of forty-five, Monge was at the peak of his scientific fame: the foundations of most of his scientific theories had already been laid ... Monge was a universal scientist, and truly belonged to the eighteenth century. At that time, there was no strict distinction between mathematics, mechanics, and physics; they were usually taught by the same person. Chemistry was still in its initial stages and occupied an intermediate position between alchemy and physics. The limits of physics were very inexact: certain branches of mineralogy, botany and zoology were often ascribed to it. The problems of technology ... had to be solved by almost every academician. So, if Monge was outstanding among his contemporary scientists, then it was due only to his great talent, extremely active work, and unfailing enthusiasm"2o, p. 23). On 14 July 1789, the Bastille, the state prison and fortress, was stormed by the citizens of Paris. The Revolution had begun. Monge was taking an active part in the events. He entered the Sociiee patriotique in 1789, but did not forsake either his studies at the Academie, his professorship or his research work. Monge organized twelve schools for hydrographers in the most important of the French ports. In 1791, he entered the Societe patriotique de la section du Luxembourg and then the famous Jacobin society. In 1793, the Committee of Public Safety charged Monge, Vandermonde, and Berthollet with writing a steel-making manual for gunsmiths. Monge took up the production of guns and cannons, and guided the reorganization of smithies and iron foundries. He also organized the production of interchangeable parts. In particular, Monge worked at a new technology aimed at replacing earthenware by sand moulds, which considerably simplified and accelerated the technological process. In 1794, he published a bulletin "The Art of Manufacturing Cannon". Another of his defence activities was aerostats. He took part in the Comission d'aerostation created by the order of the Convention in 1793. The revolutionary era could not help affecting Monge's life; he had to overcome numerous difficulties relating to the fate of the J acobin movement of France 14 . In November 1794, the Jacobin club was closed down and, in May 1795, a warrant for his arrest was signed. He was in hiding in Paris for about two months, and it was only on July 22 that he succeeded in repudiating the charges 2o . Monge actively resumed his scientific and social activities. On January 21 1795, the Normal school was opened, in the organization of which Monge took part. However, this school functioned for only a short time, and closed in May 1795. But the Ecole Centrale des Travaux Publics did survive and, in September 1795, it was renamed the Ecole

6


Polytechnique. Monge paid much more attention to the latter: in his opinion, it should train engineers with serious scientific and practical schooling, and not teachers of mathematics. This school turned into the biggest scientific centre, and was able to maintain this reputation through the whole of the nineteenth century. Practically all great mathematicians living in France in the nineteenth century either graduated from the Ecole Polytechnique or belonged to the corporation of its teachers. The French Revolution had a strong impact on all branches of science. The rapid growth of industry stimulated the development of mathematics, mechanics, physics and chemistry. Monge was in the vanguard of the new scientific developments. In Napoleon's times, Monge became Minister for the Navy and participated in the expedition to Egypt. His main activity, however, was the directorship, of the Ecole Poly technique from its foundation to the fall of the Empire. For more information regarding the history of this school, see l4 . Monge's basic work in differential geometry, which generalized both his own results and the investigations of other geometers at the end of the eighteenth century was Application de l'analyse Ii la geometrie 123 , fragments of which relate to variational calculus and minimal surface theory; these are given by us in Chapter 2. The first version of this course was the transcript of Monge's lectures at the Ecole Polytechnique, which was published in a series. At the beginning of the book Monge develops the general theory of curves and two-dimensional surfaces. In particular, he analyses the properties of the two principal curvatures at an arbitrary point of a surface. In §XX, he considers separately the case where both radii of curvature (at each point of the surface) are equal and opposite in sign. Hence the mean curvature of the surface, i.e., the sum of the reciprocals of the radii of curvature, is identically equal to zero at each point of the surface. In other words, this surface is minimal and locally minimizes the area functional of the surface. In particular, Monge points out the following important property of this surface: if part of it is bounded by a continuous closed contour, then, of all surfaces passing through this contour, its area inside the contour is the least. The question of integrating the minimal surface equation was considered by Monge in124. Further success in this direction was achieved by Legendre in llO • Meanwhile Monge seemed not to be interested in the problem of describing concrete examples of minimal surfaces in three-dimensional space: in particular, he does not give any new examples in comparison with those already known (catenoid, helicoid). At that time, some topics of minimal surface theory were developed by Monge's pupil, J.-B. Meusnier. Jean-Baptiste-Marie-Charles Meusnier was born in 1754. Having left the military school at Mezieres, he was admitted to the Engineering corps of the French army and, before the Revolution, was Commander in chief of a division. He was elected a member of the Academie before the Revolution, after which he worked with his teacher Monge at the Commission on Weights and Measures. Then he was a general and fought on the front, becoming famous as a brave


7

warrior in the ranks of the Revolutionary army. During the heroic defence of Mainz against the Prussian forces, Meusnier was severely wounded and died soon after (in 1793). He was only 22 when he presented his Memoire sur la courbe des surfaces before the Academie in 1776. Meusnier paid much attention to minimal surfaces. We have already noted above that Lagrange obtained the equation of minimal surfaces analytically. Meusnier suggested a geometric derivation of this equation, and obtained some new corollaries to it. He also studied various properties of the catenoid and helicoid. One of the most outstanding pupils of Monge, who was also connected with the Ecole Polytechnique, was Simeon Poisson (1781-1840). Like his teacher, he was interested in applications of mathematics to mechanics and physics. In particular, his research in the theory of liquids and capillary effects was an impetus for the development of the mathematical theory of the separation surface of two media l44, 145. (See also the survey in 56 by H. Minkowski). Poisson also made an important contribution to the theory of minimal surfaces, about which we shall speak in §2 of the present chapter. Poisson is a typical representative of the Ecole Poly technique (he was a student, coach, and professor). He profitably reconsidered the following chapters of mathematical physics of his time capillarity (see the next section), lamina bending, electrostatics, Magnetostatics, heat conduction. The exceptional versatility of Poisson's activity is manifested by the fact that his name is associated with many a concept in the modern mathematics fund (e.g. Poisson bracket in Hamiltonian mechanics, Poisson equation, etc.). It is with Poisson (and some of Laplace's pupils) that the trend towards applications in the history of mathematical development in nineteenth century France is associated. Poisson is the author of more than 300 papers. His scientific outlook was guided, to a large extent, by the ideas of "atomistics in the spirit of Laplace" (see 104). Moreover, he often preferred to think that the derivatives and integrals used in physics were merely an abbreviated notation for the ratios of finite increments and sums. As professor of the Ecole Polytechnique he wrote the well-known work" Traite de Mechanique" which had considerable influence on the development of the mathematical aspects of mechanics and physics. For further details see in [14], [17], [20], [56], [90], [104], [105], [110], [123], [124], [135], [139], (144], [173], [174], [176].

§2. The Nineteenth Century, the Era of the Discovery of Basic Minimal Surface Properties. 1. Plateau's physical experiments and methods of forming soap films. Before studying minimal surfaces mathematically, we give a brief account of

8


those of their basic properties that can be demonstrated in the language of visual geometry. When the Belgian professor of physics and anatomy Joseph Plateau (1801-1883) started his experiments aimed at the study of the configuration of soap films, he could scarcely have believed that they would serve as an impetus for the emergence of a significant branch of science, growing fast up to the present day, and known as the "problem of Plateau". The problem of finding a surface ofleast area with given boundary seems to have been called the "problem of Plateau" by Lebesgue in his well-known paper109. Today, many different problems are combined under this title, which are often characterized by nonequivalent approaches even to the very definition of the problem of finding minimal area surfaces. Some of the physical experiments once performed by Plateau are extremely simple and well-known to the reader, since there is hardly anyone who did not entertain himself at one time in his life by blowing bubbles through a straw or constructing various soap films spanning a wire frame. It is well known that if a closed wire frame (homeomorphic to the circle) is dipped into soap water and removed with care, then a beautiful iridescent soap film bounded by the contour will form on it. The size of the film can be rather larger, but the greater the film, the easier and faster it will burst under the action of the force of gravity. If, on the contrary, the size of the frame is comparatively small, then the gravitational force can be neglected while studying some important properties of the soap film. We shall have recourse to this fact many times below. Minimal surfaces are mathematical objects which model physical soap films quite well. Conversely, many profound properties of minimal surfaces reveal themselves extremely visually in simple physical experiments with soap films. There exist many branches of mathematics, arising from concrete physical and applied problems; however, not one of them is related to so many of the various (in their apparatus) mathematical theories as the problem of Plateau is within the theory of minimal surfaces, such modern theories as differential equations, Lie groups and algebras, homology and cohomology groups, bordisms, etc., are interconnected. In the present section, we illustrate, with simple examples, some basic notions, technique, and results worked out in the different periods of development of the Plateau problem. Some methodical discoveries in the account of this material are due to T. Poston (see 146). Since exact mathematical constructions in this field sometimes require quite a ramified and refined technique, for the present we shall confine ourselves to visual constructions only and omit voluminous definitions and computations. In particular, we shall describe various generalizations to the multi-dimensional case (i.e. the multidimensional Plateau problem) in Chapter 4. Considerable attention to the mathematical study of the properties of soap films was paid as early as the eighteenth century by Euler and Lagrange (the problem of finding a surface of least area with given boundary). Exact solutions for certain, rather special boundary contours were found by Riemann, Schwarz, and Weierstrass even in the nineteenth century. The theory of minimal surfaces


9

emerged in the consideration of two kinds of soap films: viz., soap-bubbles and soap films spanning a wire frame. The first standard method of producing soapbubbles consists in blowing them through a straw dipped beforehand into soap water (Fig. 1). To stabilize the bubbles (which usually burst very soon), some glycerine is added to the water. These bubble-like soap films are characterized by their having been formed and kept in equilibrium by the internal pressure of the gas (air) trapped inside the film. The spherical shape of the bubble is easily

, (D "/

.~

.•

CD Figure 1 explained by the fact that this shape ensures the least area of the surface for a fixed volume bounded by this closed film. Another method of blowing soap bubbles is represented in Fig. 2. This time, a closed frame (e.g. a circle) should be taken, dipped into soap water, removed outside, and quickly displaced in space in the direction orthogonal to the plane of the frame (for simplicity, we assume that the frame is a plane curve). The film then bends under the pressure of the incoming flow of air, and several soap-bubbles are taken off one after another. It is clear that the same effect can be achieved by directing a jet of air on a plane soap film formed on a fixed contour. Yet another kind of soap film is obtained when we remove the wire frame from soap water and fix it in space without any sharp displacements. Then a stable soap film is formed on the frame, which normally has no soap-bubbles of the described type in its structure (they are formed if there is a difference in the pressures of the gas inside and outside the film). This film has the original wire frame as its boundary (Fig. 3). Note at once that this second type of soap film

10


[J) (J) (fJ (J)

o o

· W o "

~

o~

Figure 2

Figure 3

(bounded by a frame) is, for the present, the main object that can be studied by mathematical methods. This is due to the fact that in a great many applied problems, the minimal surface is attached to some fixed boundary, as it occurs, for instance, in membrane theory.


11

2. Physical principles underlying the formation of soap films. The physical principle underlying the formation of soap films and regulating their behaviour and their local and global properties is extremely simple: a physical system preserves a certain configuration only in the case when it cannot easily alter the configuration by occupying a position with less energy. In our case, the surface energy of the soap film, often described in terms of the surface tension of the liquid, is due to the existence of the attractive forces between individual molecules and the imbalance of these forces on the boundary of the surface. The existence of these unbalanced forces entails an interesting effect: the liquid film turns into an elastic surface seeking to minimize its area and, therefore, to minimize the tension energy per unit area. Here we again neglect the force due to gravity (in the case of bubbles and films with boundary) and air pressure (in the case of films without boundary). Consider in greater detail how the properties of the surface film change when soap is added to water. In Fig. 4, the surface separating two media, water and air, is represented schematically. The molecules of water are represented as black circles. Some of them are in the air, since the liquid evaporates, but we neglect this effect. Double-headed arrows denote the mutual attraction forces acting Gas

• .~.«• )lIe •'.• •• .~t~••• • • ~T'. • •

e. ·

Water molecules

Figure 4 between the polar molecules of water which are characterized, as is generally known, by the asymmetric position of the electric charge. It is clearly seen that it is these forces which are the cause of the existence of surface tension on the surface separating the media. The properties of the surface film of a liquid are formed exactly in this way. In contrast with the molecules of water, the molecules of soap are formed by long thin non-polar hydrocarbon chains with a polar oxygen-rich group at one end of the chain. When these molecules are added to water, they stream to its surface, and fill the surface of the liquid in a uniform

12


layer (Fig. 5). Meanwhile, each molecule of soap is oriented on the surface with its non-polar end outside. Driving the molecules of water inside the liquid, the molecules of soap thereby diminish the surface tension on the surface separating the media. This circumstance happens to impart additional elasticity to the surface, which manifests itself much more at the moment when we dip, and then Soap molecules

/

. . . _-:--.e4 . li!liilll i 1 i l ii .-------., ~

. , .1-' :: · e

•

•

Figure 5 remove the thin wire frame. This process is demonstrated in Fig. 6. In fact, when the wire rising from the liquid reaches its surface (Fig 6a), the surface bulges and envelops the contour (in the figure, its section is shown). This happens because the number of soap molecules decreases on the surface for the time being (Fig. 6a), that is, their number per unit area of the film decreases and, therefore, the surface tension at this place on the film increases. It is created by the molecules of water breaking through from below to the positions vacated, which makes the surface film in the immediate neighbourhood of the wire more flexible and elastic and, finally, leads to the formation of a soap film after the frame has been eventually removed from the water. The process of forming a film on the frame is shown in Fig. 6b. As the frame rises upwards, the flexible surface film is not torn by the wire, but envelops it, and is dragged behind out of the liquid. Ultimately, a soap film is obtained which spans the contour. The force of gravity restricts the dimensions of the minimal surface and, when some parts of the film are at a considerable distance from the boundary frame, the film bursts because the surface tension forces become insufficient to keep the film in equilibrium. A more detailed discussion of these and other effects can be found, for instance, in the paper of Almgren and Taylor12.

13


.J ......

•

•

~,.

(b)

••• (a)

+1++-\''\00 •

• T·•.1 . •• .: ......../. • ~.e'"

,-

,-,~- .. -. -.~.

- ..-

-j. .

• ...:::.-_-/.-./_JI . -::;;. .".,.,,-./ ..

••

Figure 6

3. The extremal properties of soap films and minimality of their area. The properties of the surface of separation between two media. It is precisely because of the process described above that a mathematical model of a soap film can be the so-called minimal surface, that is, a surface having the least possible area (locally) among all the surfaces with the same boundary. It is clear that the shape of the surface and its properties are determined, to a large extent, by the configuration of the boundary frame, that is, the boundary of the surface. Apparently, the concept of least area itself was introduced into geometry by Archimedes who not only noticed that the straight line realizes the shortest distance between the two points, but also well understood that the plane can be characterized in terms of minimal surface area. These ideas naturally arose from everyday life when, for example, for making a drum, a skin had to be drawn tightly on a fixed wooden skeleton to diminish its area bounded by the edge (boundary) of the drum. This made it possible to relate the idea of surface tension with the properties of surfaces having locally a minimal area. In general, the history of the development of the correct ideas about what surface tension is, is extremely interesting. Here we only touch upon some stages of this progress (see 146).

14


The next essential success in the development of these ideas (after a long stagnation) seems to have been achieved by Boyle who became interested in 1676 in the shape of drops of a liquid. Having paid attention to the fact that raindrops are approximately spherical, Boyle decided to find a relation between the size of a drop and its shape. To this end, he had to perform a rare experiment for the time. To study the behaviour of the drops for a sufficiently long time and to prevent their destruction, Boyle poured two liquids into a glass vessel: a solution of K 2C0 3 (a comparatively heavy and dense liquid, namely a concentrated solution of potassium carbonate) and alcohol (a light liquid). When the liquids came to rest, a clearly discernible surface of separation between the two media appeared (Fig. 7). Then a drop of a third liquid, of intermediate density and immiscible with the other two liquids already present in the vessel, was accurately placed on the surface of separation between the two media. The drop was in equilibrium on being immersed in the alcohol and touching the surface of K 2C0 3 with its lower point. This liquid (oil) has been chosen so as not to wet the surface of separation of the media, that is, so that the drop should not spread on the surface of separation, and have the shape shown in Fig. 7b. Drops of different size were then placed on the surface of separation beginning with small ones and gradually increasing their dimensions. As it happened, the shape of a

,

(a)

Figure 7

drop changed with its growth (Fig. 8). Visually, flattening of drops was noticeable when they were the size of a pea. Certainly, as we know today,


15

Figure 8

flattening of drops occurs in any case, even for small-size drops; the force of gravity acts on them. However, to observe this phenomenon, sufficiently accurate precision instruments should be available and Boyle, certainly, did not have them in 1676. Besides, it should be remembered that the ideas about the necessity for the shape of a drop to correspond to the least energy were only at the beginning of their development in the seventeenth century. Most processes of a similar kind were explained then in terms of phlogiston, which without fail further impeded the understanding of the true mechanism determining the shape of drops. And finally, in reality, the shape of a drop minimizes the combination of surface energy, while the force of gravity which, for a small sized drop, considerably complicates the mathematical side of the matter. In 1751 Segner came to a clear realization of this physical fact and understood that, for an accurate study of surface tension, the influence of the gravitational force should be eliminated as far as possible. He did this partly by studying freely falling drops, and also by placing them in a liquid of the same density. The drop and the embracing liquid had been chosen so as to be immiscible. In particular, he managed to show that a sphere had the least area among all closed surfaces bounding a fixed volume. Apparently, Segner was the first to understand the true role of surface tension in all the processes. However, the further development of these ideas (including that of surface energy) was greatly hampered by the domination of phlogiston theory still strongly upheld at the time.

16


4. The surface of separation between two media in equilibrium is a surface of constant mean curvature. An important step in understanding the interior geometry of surfaces of separation between media was made in 1828 by Poisson when he showed that the surface of separation between two media in equilibrium (provided the force of gravity is neglected) is a surface of constant mean curvature. We now dwell on this important result at greater length, and recount it in modern terminology with the use of simple notions of modern differential geometry. At this stage, a preliminary acquaintance with some basic properties of the second fundamental form of a two-dimensional surface embedded in three-dimensional Euclidean space may prove useful. See the corresponding theory, e.g., in 5o, 66. Let M2 be a two-dimensional smooth surface in R3, where the Euclidean space R3 is endowed with Cartesian coordinates x,y,z. Let the surface be determined by the radius vector r = r(u,v) = (x(u,v),y(u,v),z(u,v)), where the parameters u and v vary in a certain domain of the Euclidean plane and determine regular coordinates in a neighbourhood of a point P on the surface. Let n(u, v) be the unit normal vector to the surface at the point P(u, v). Consider the symmetric square matrix Q made up of numbers qij = u·,n>, where rUi'Uj are the second partial derivatives of the radius vector with'respect to the variables u and v (denoted here by u l and u 2' respectively). The Euclidean scalar product in R3 is here denoted by . If a and b are two tangent vectors to the surface at the point P, then the expression Lijqijaibj = Q(a,b), where a = (al'a 2 ),b = (bl'b 2), is a bilinear form defined on the tangent plane to the surface, called the second fundamental form of the surface. It is clear that the second fundamental form depends on the method of embedding the surface and, under a deformation of it will, generally speaking, also be changed. In the two-dimensional case, in which we are here interested, the second fundamental form is usually written as L du 2 + 2M dudv + Ndv 2, where L = , N = < rvv,n >, L, M, and N being smooth functions of the coordinates on the surface. Together with the second fundamental form we also consider the first. It determines the Riemannian metric induced on the surface by the underlying Euclidean metric. This form is usually denoted by Edu 2 + 2F dudv + Gdv2 where E = < r u' ru>' F = < ru' rv>' G = < r v' rv>' Denote the matrices of these forms by Q

= (~

~)and A = (~ ~), respectively.

Then (see 50 and

66)

the following two scalar functions, K =detA -I Q and H = Spur A - I Q, are important geometric characteristics of the surface, called the Gaussian and mean curvatures, respectively. We are now especially interested in the mean curvature, since this is what is most closely related to the geometry of the boundary of separation between media. It follows from the definition of mean curvature that 2MF . h i . _ F2+ EN . AS It appens, we can great' y sImp 1·r. hy t h'IS expreSSIOn H = GL -EG


17

by a choice of convenient local coordinates, which makes it possible to relate the mean curvature with the surface tension on the boundary of separation between media. Recall that curvilinear regular coordinates u, v on a surface M are said to be conformal (or isothermal) if the Riemannian metric ds2 = Edu2 + 2Fdudv + Gdv2 induced becomes diagonal, i.e., E = G, F = 0, and ds 2 = E (du2 + dv2). It happens also that if a metric on a surface is real-analytic, then, in a neighbourhood of each point on the surface, there always exist local conformal coordinates 50, 66. In other words, for any point PeM, there exists a neighbourhood U(P) such that coordinates p and q (which are real-analytic functions of the original coordinates), in which the metric ds is of the form >. (p,q)·(dp2 + dq2), can be introduced in it. We can assume that the surfaces of separation between media are real-analytic at all interior non-singular points. Therefore, without loss of generality, we can assume the existence of conformal coordinates in a neighbourhood of each regular point. Thus, let u and v be conformal coordinates in a certain neighbourhood of a point P on a surface M in R 3. We can assume that the matrix of the first form is the identity at the point P. This can always be achieved at one point by means of d:.e corresponding orthogonal transformation of conformal coordinates. In particular, we have at the point P:EG - F2 = 1. Since the coordinates are conformal, we see that E = G, F = in a neighbourhood of the point P.

°

Therefore, H

= ~

(L + N). The value of the mean curvature at the point P is

found in a still simpler way: H(P) = L + N, since E(P) = 1. Recall the explicit form of the coefficients Land N. We obtain: H = E-l( . Note that the coordinates x, yare not necessarily conformal on the surface, but they are orthogonal at the point P and, therefore, E(P) = G(P) = 1, F(P) = 0. Since, with respect to the coordinates x, y, z, the normal vector n(P) is written in the form iJ2f iJ2f n(P) = (0, 0, 1), H(P) = .:If = -2 +""!12. It is important for us that iJx uy the graph of the function f defines (locally) the surface M in a neighbourhood of the point P. To relate the mean curvature with surface tension, recall another interpretation of the Laplace operator. Let f(xl' ... 'xn) be a smooth function on Rn endowed with Cartesian coordinates (xl' ... 'x n). We define the s -local average of the function fon Rn as follows:

18


Figure 9

F, f{x) = _1_ JS,f, where S, is the Euclidean sphere in Rn of radius 5 and centre at

"Ie

the point x = (xl' ... 'xn) and "Ie is the volume of the sphere S,. In other words, we average the original function f over the boundary of the spherical neighbourhood of radius 5 with centre at the point x. If the function is defined on the plane, then S, is a circle and "Ie its circumference, i.e., "Ie = 211"5. We now define the "deviation of the function from its 5 -local average" by putting ~/ (x) = f (x) - F / (x). In the limit, as 5 ~ 0, we can construct a function g (x) = ,li!!}.o~, f (x). It is easy to verify (we leave it to the reader) that the function g(x) coincides, up to constant multiplier, with the function ~f(x). The function ,~o F, f (x) = F f (x) can be called the local average of the function f. Then ~f (x) = f (x) - Ff (x), i.e., the value of the Laplace operator on the function f equals the deviation of the original function f from its local average Ff. Such an interpretation of the Laplace operator is useful from many points of view. For example, iffis a harmonic function, that is, ~f = 0, then it coincides with its local average. In particular, a harmonic function never attains a strict local maximum at interior points of the domain where this function is defined (maximum is attained on the boundary). Coming back to the analysis of mean curvature, we see that the curvature H(P) = fxx + fyy = ~f can be interpreted as the deviation of the radius vector of the surface from its local average in the direction of the normal. Recall that the radius vector is completely determined by the function f (x, y). Assume that the surface is the boundary of separation between two media and that the attraction forces act between near points on it, as in the case of water molecules in the above example. For clarity, we can assume that we deal with a soap film. Then a chain of equalities can be written: the pressure exterted on the surface at the

19


point P equals the projection onto the normal to the surface of the resultant of all local attraction forces between the points near to P, which, in turn, equals (up to a certain constant multiplier not depending on the point) the deviation of the radius vector from its local average (deviation in the direction of the normal). Thus, the pressure at the point P equals a certain constant multiplied by the mean curvature of the surface at the given point. We can write the equality H = A· (PI - P2)' where H is the mean curvature, PI and P2 pressures in the media separated by the surface between them, and

~

the surface tension. The

constant emerging in this analysis, i.e., the coefficient of the mean curvature is called the surface tension. Hence, the result enunciated above follows; namely, that if a two-dimensional surface is a soap-bubble, or a system of such bubbles, or the boundary of separation between two liquids of the same density (so that the force of gravity can be neglected), or a soap film spanning a wire frame, then, provided the surface is in equilibrium, the pressure on each side of the surface is a constant function, i.e., does not depend on the point. Therefore, the mean curvature of the surface is constant. Q.E.D. (Laplace's and Poisson's theorem). This fact accounts for the spherical shape of soap-bubbles, which they acquire in free fall. In this case, the force of gravity can be neglected. Meanwhile, the pressure of the gas inside the sphere exceeds the outside pressure, and equilibrium of the film is reached as a result of the action of the surface tension forces stabilizing the spherical film. The case of a soap film spanning a wire frame is of special interest. Here, the difference of pressures on both sides of the film vanishes. Therefore, the pressure on one side of the film is the same as that on the opposite side in a neighbourhood of each point on the surface. Thus, the resultant of the forces equals zero (force of gravity is neglected again) and, therefore, the mean curvature of the film equals zero.

5. Soap films of constant positive curvature and constant zero curvature. Having discovered that soap films or "liquid surfaces" are surfaces of constant mean curvature, Poisson naturally posed the problem regarding a complete description of these surfaces. Since the problem of the possible shapes of drops of liquid, or small bubbles has not yet been solved completely, the researchers' attention was first concentrated on the case of positive mean curvature. It is clear that the standard Euclidean sphere is such a surface and, therefore, freely falling soap-bubbles, in fact, acquire the shape of a sphere. However, it remained unsolved whether there were other closed (i.e., without boundary and singularities) surfaces of constant positive mean curvature. Certainly, numerous physical experiments with soap-bubbles quickly convince us that the sphere is a unique possible surface of constant positive curvature in the class of smooth closed surfaces, but the mathematical proof of this fact requires some effort. Poisson proved that in the class of spheroids which are sufficiently close to the

20


sphere and are surfaces of revolution (i.e., slightly oblate at the poles, or a little prolate along some axis), the sphere is a unique surface with constant mean curvature. This was practically the only information until 1853 when Jellet proved that among closed, two-dimensional and star-shaped surfaces (i.e., each point of which is visible from a certain interior point, the "centre" of the surface), the sphere is unique having constant positive mean curvature. Finally, it has been proved in this century that a compact, two-dimensional, and closed surface of constant positive curvature without self-intersections is the standard sphere. Note that here we speak of surfaces embedded in, or immersed in R3. In particular, the second form and the mean curvature are defined only for surfaces immersed in a certain ambient space (Riemannian manifold). Thus, we have singled out two basic cases: if a soap film bounds a closed volume with the inside pressure exceeding that of the outside, then the surface has constant positive mean curvature; if, however, a soap film does not contain any closed volumes and spans a wire frame, then there is no difference of pressure, and the mean curvature is zero. The first fundamental experiments performed by Plateau were devoted to the study of the boundary of separation of liquids. The experiments were made with immiscible liquids of the same density; one liquid was immersed in another, which led to the formation of drops bounded by surfaces of constant positive curvature. Experiments with surfaces of zero mean curvature were performed later. Henceforward, we shall concentrate our attention on the case of soap films with boundary. These films have, then, the following three equivalent descriptions: (a) surfaces of zero mean curvature, (b) surfaces of minimal area, that is, minimal surfaces, (c) surfaces, at each interior regular point of which the principal curvatures are equal in absolute value and opposite in sign. The last characteristization follows from the equality H = AI + A2, where AI and A2 are principal curvatures. As it turns out, if a surface is minimal, then there always exists conformal coordinates in a neighbourhood of any of its points, from which it follows that if a minimal surface M2 in Rn is given in non-parametric form, i.e., as Xj = gj (u,v), where 3 ::s i::s n, u = xl' V = x 2' then the functions gj are always real-analytic functions of u and v. Other details see in: [12], [18], [34], [35], [56], [104], [105], [109], [110], [123], [124], [135], [139] - [141], [143], [144], [145], [146], [147], [148], [202], [209], [210].

§3. Topological and Physical Properties of Minimal Surfaces. 1. Stable and unstable surfaces. Even the simplest experiments show that different surfaces of separation between two media react to a small perturbation differently. Some of them oppose destruction, and prove stable, others are destroyed at once, and are unstable. Here, an important effect is revealed. It can be visually illustrated even


21

for scalar functions on a smooth manifold. Recall that if f (x) is a smooth function on a finite-dimensional manifold M, then a major role in the study of the behaviour of the function is played by its critical points, that is, those at which grad f = O. These points may be of different kinds. First, points of local maximum and minimum of a function are critical points. However, besides these there are the so-called saddle points. A typical point of this kind is schematically represented in Fig. 10. They are characterized by having two kinds of direction: along direction a (Fig. 10), the function is strictly increasing, and along the orthogonal direction b, it is strictly decreasing. The number of independent directions, along which the function decreases strictly, is usually called the index of the critical point. Nondegenerate local maximum points have maximal index.

Figure 10 A perfectly similar situation occurs when a function (or functional) is defined on an infinite-dimensional space. Such a space is (with the correct choice of formulation which we do not make precise here) the set of all two-dimensional surfaces, e.g. those spanning wire frames in three-dimensional space. Then the area functional associates each such surface with its area. A change, i.e. deformation of the surface generates a corresponding change in the area functional. If the two-dimensional film is in equilibrium, then this means that, considered as a point of an infinite-dimensional space, it is critical for the area functional. But, just as in the finite-dimensional case, this equilibrium position may be of different kinds: a local minimum, local' maximum, or saddle. Furthermore, saddles can also be of different kinds. In the last case, there exist sufficiently small perturbations of the film that diminish its area. For the physical realizations of stable films, we find that, in the case of a local minimum of the area functional, the film is stable; i.e. it opposes small perturbations and strives to return to the original position, that is, equilibrium. In the case of a saddle, the film is unstable and there exist perturbations, arbitrarily small in amplitude, that

22


decrease the area of the film which starts deforming spontaneously, acquiring a new configuration corresponding to a smaller value of energy. Meanwhile, the new configuration can be considerably different from the original, even from the topological point of view (see the examples below). The "infinitedimensionality" of the space of all surfaces also means that there exist infinitely many different independent directions of perturbations of the film.

2. Plateau's experiments with stable columns of liquids. Plateau performed a series of experiments with liquids. We have already mentioned some of them. Here, we briefly describe the following series of his experiments. Between two metal discs of the same radius, whose centres were placed in one straight line orthogonal to the plane of the discs, Plateau obtained a column of liquid in the shape of a right circular cylinder consisting entirely of liquid (Fig. 11). Since the boundary of the cylinder is a surface of constant positive mean curvature, theoretically such a column of liquid is a critical point in the space of all "surfaces". This case should not be mixed up with that of closed (!), compact and smooth surfaces of constant positive mean curvature. (As we know already, such a surface is necessarily a sphere.) A cylinder is not a compact surface if it is extended infinitely in both directions. If, however, it is bounded by two edges, i.e., circumferences (discs), then it is a closed, smooth manifold with boundary. The properties of a cylindrical column of liquid essentially depend on its height. In real physical experiments, a column of Disc

Height of the column is small Figure 11


23

liquid, "not very high", can be realized between two parallel discs. If the two discs are far away from each other, then the column of the liquid is destroyed. It is extremely interesting to observe the process of destruction of this column when the two discs, remaining parallel, gradually move away from each other and stretch the column vertically. As it happens, the column of liquid is stable only in the case when its height does not exceed approximately three times the diameter of the boundary disc. If, however, the height of the column approaches the value three times as great as the diameter (approximately 3.1 of the diameter) and exceeds this value, then the column begins to decompose. If the height of the column is increased extremely slowly and carefully, then the behaviour of the column in the vicinity of the critical value for its height can be observed visually. Its deformation occurs sufficiently slowly in this case, which makes it possible to represent the process of restructuring the column by means of Fig. 12. Thus, low cylinders are stable, and those too high unstable. The critical height is approximately three times the diameter of the disc. Certainly, these experiments are performed with discs of sufficiently small diameters, so that surface tension forces may successfully counterpoise the force of gravity acting on the column of liquid.

Figure 12

The process of decomposition of an unstable column, i.e., its qualitative restructuring, can be also demonstrated by another real experiment performed as early as in the seventeenth century. Consider a wire frame in the form of a regular plane circle and, dipping it into soap water, span it with a disc, that is, a

24


soap film. Then move the frame fast in the direction orthogonal to its plane that is, orthogonal to the soap film (Fig. l3). Under the action of the inflowing air jet, the soap film bends and first takes the approximate shape of a right circular cylinder (in any case, this occurs in the immediate vicinity of the frame). However, this cylindrical shape is soon destroyed, there appear waists, the film undergoes a qualitative restructuring, and soap-bubbles start breaking off it. They have a practically spherical shape. Obviously, this process can be investigated by fixing the frame in space and directing on it a jet of air, orthogonal to the plane of the frame. A cylindrical column of liquid was the first critical surface whose stability was proved by Plateau. Then Beer put forward the conjecture that all surfaces of constant mean curvature were local minima of the area functional. Plateau proved that this conjecture was incorrect. He considered a cylinder of height greater than critical. From the mathematical point of view, such a cylinder is still a critical point (surface) for the area functional in the space of all surfaces. First Plateau proved analytically that such a cylinder has, locally, the least possible area in the special class of all perturbations of this cylinder, preserving the area of each horizontal section. In other words, this class of perturbations is characterized by the absence of vertical displacements of the liquid. Horizontal displacements of the liquid preserving the area of any plane section which is orthogonal to the original axis of the cylinder are permitted. In other words, if we perturb the cylinder so as to slightly bend it, but, in doing so, the plane sections preserve their area, then the perturbed cylinder has a greater area than the original. Recall that if the height of

oQ)

o o

Figure 13


25

the cylinder is less than critical, then any (!) of its perturbations increases its area, that is, in this case the original surface is a point of local minimum for the area functional in the space of all surfaces. If, however, the height is greater than critical, then the cylinder is a saddle point in the space of all surfaces, there existing still infinitely many independent perturbation directions ex of this critical surface in the space of all surfaces increasing its area. The fact that the number of directions along which the critical surface is a minimum of the area functional is infinite, follows from the fact that there are infinitely many different deformations ex preserving the areas of horizontal plane sections. On the other hand, if the height of the cylinder is greater than critical, then there exists at least one surface perturbation direction {3 along which the area starts decreasing. As we already know, to create such a perturbation, it suffices to move the liquid vertically with the effect that a part of the cylinder will distend, and the other, conversely, will narrow (Fig. 12). A saddle is schematically represented in Fig. 14. Moreover, Plateau demonstrated by experiment, that a critical surface behaves as a saddle in the case when the height of the cylinder exceeds three times the diameter of the boundary disc. He obtained a column of liquid between two discs. The height of the column was just a little more than critical. The column started deforming slowly in the vertical direction while remaining a symmetrical surface of revolution. At that moment, Plateau rapped the column of liquid with a thin horizontal glass stick, that is, he gave the column a perturbation preserving the area of plane horizontal column sections. The column tilted slightly, but returned to the original axially-symmetric

Figure 14

26


position practically instantaneously, while continuing to deform slowly in the vertical direction. The experiment shows that the surface is stable with respect to perturbations of the indicated type Oi (Fig. 15). However, a deformation in the vertical direction is realized by perturbations of type [3. It is interesting that the decomposition of a sufficiently long column of liquid occurs in a wave-like manner. To demonstrate this process visually, take a thin thread covered by a

Figure 15

liquid cylindrical layer so as to make the thread the axis of this cylinder. Then this unstable cylinder gets restructured, as shown in Fig. 16. The cylinder becomes wavy, then each inflation turns into a ball fitted on the original threadaxis. It is clear that this phenomenon is the total result or the action of surface tension forces and the instability of long cylinders of liquid. Certainly, the process represented in Fig. 16 is somewhat of a simplification of the real phenomenon which is much more complicated, though, finally, a necklace of spherical drops strung on the original thread is nevertheless obtained. The process was studied especially meticulously, since the problem of covering a thin conductor (e.g., copper wire) by a uniform cylindrical insulation layer cropped up (and was solved successfully) in electrical engineering. Meanwhile, it was important to learn how to stabilize the long cylindrical layer wrapping the wire placed in the axis of the cylinder.


27

Figure 16

3. The physical realization of a helicoid. In 1842, Catalan proved that unique complete ruled surfaces of zero mean curvature are the plane and helicoid. A helicoid is obtained as a composition of two motions of a straight line, viz., translational with uniform velocity and rotary with uniform angular velocity in the plane orthogonal to the translation vector (Fig. 17). In other words, a straight line intersecting another straight line at right angles is taken, and displaced uniformly with respect to the latter while uniformly rotating it. Obviously, this surface is not compact. If we confine ourselves to the rotation of a line segment instead of the whole straight line, then we obtain a surface represented in Fig. 18. In coordinates (x, y, z) and (r, t), it is specified by the following radius vector: x = rcos wt, r = at, y = rsin wt, where a and ware the velocities of the translational and rotary motions, respectively. Plateau realized "one-half' of a helicoid, by taking a spiral of wire around an axial straight line (Fig. 19). However, an attempt to realize the complete helicoid by means of a film spanning a wire frame met with difficulties. Certainly, a part of a complete helicoid can be obtained by assembling two replicas of the film represented in Fig. 19. However, with such an approach, we deduce that, besides the boundary contour made up of two spirals with a common axis, to realize a soap film, the axis itself should be added, for example, given as a thin thread

28

Figure 17


Figure 18

along the axis of the construction (Fig. 20). This thread stabilizes the film, and opposes its restructuring into a minimal surface which is not a helicoid any more. In fact, if this axial thread is removed and only two boundary spirals are kept, then, in the case when the pitches of the spirals are not very large, i.e., when the spiral makes many turns in a comparatively small line segment along the axis, the helicoid is transformed into the surface shown in Fig. 21. It is clear that this surface is not ruled, that is, does not consist of straight line segments. Attempts to realize a part of a right helicoid as a soap film spanning a closed wire frame, given as two coaxial spirals with small pitch, fail. The contour was assumed not to contain any extra wire line segments placed strictly inside the film, and not

Figure 19

Figure 20


Figure 21

29

Figure 22

forming part of its boundary. The essential difference between a right helicoid and the surfaces encountered earlier is clearly revealed here. The point is that a helicoid is a non-compact surface existing (from the mathematical point of view) independently of any boundary contour. As it turned out, it is an unstable surface for small pitches. However, in practical experiments, we can only realize soap films spanning some or other compact wire frame. While imposing this extra requirement on the realization of a helicoid by a stable surface, we encounter the difficulties indicated above. If attempts to realize a helicoid with small pitch are renounced and we set ourselves the goal of realizing a helicoid with sufficiently large pitch of boundary spirals, then the construction of such a stable soap film is not hard. For this purpose, a contour made up of two coaxial spirals should be considered. It is clear that, for sufficiently large pitch, this contour bounds the soap film, which is a helicoid (Fig. 22). The proof that this film has a ruled structure can be given in the following visual way. It is clear that the helicoid in Fig. 22 is obtained from the minimal surface in Fig. 21 by stretching the boundary spirals in the direction of the axis. As the spiral pitch increases, the soap films deforms. Fix the axis of the spirals and the direction of viewing the film in space. We shall observe the deformation of the part h of the film placed in the immediate neighbourhood of the point where the spirals cross (Fig. 23). This part of the film is projected on the retina of the eye as a curvilinear triangle, evidently decreasing as the spirals stretch. At some moment, this triangle disappears having turned into a point, and we obtain a film drawn in Fig. 22. Since the projection of the film is now of the form shown in Fig. 23, when the film is rotated about its axis, the spirals start running, e.g., from left to right. The points where the spirals cross move in the same direction. Since the part of the film projected (on the retina of the eye) into each crossing point is a straight line segment joining a point on the nearest spiral to a point on the

30


farther, then the whole film is made up of these rectilinear segments orthogonal to the axis, which proves that the film is ruled and coincident with a helicoid.

Figure 23

4. The physical realization of a catenoid and its restructuring as the boundary contour changes. The following popular example of a minimal surface is obtained as follows. Consider a minimal surface of revolution formed by rotating the curve given by the equation y = a cosh .!, where a is an arbitrary non-zero constant, around the a axis Ox. It is generally known that this curve coincides with the shape of a heavily sagging chain fixed at two points (Fig. 24). The force due to gravity is directed here downwards along the axis Oy. A direct computation shows that the mean curvature of the surface of revolution obtained equals zero: 1 y" H =~ + ~ = Thus, we have obtained a 1 2 y-JI + (y')2 (1 + (y')2)3/2 minimal surface called a catenoid. It can be regarded as a soap film spanning two coaxial circles placed in parallel planes. If we rewrite the equation of the catenary line as I. = cosh .!, then we obtain, evidently, the result that all a

a

catenary lines and, therefore, all catenoids, are equivalent in the sense that they

31


x Vertex of the catenary line

Figure 24 can be transformed into one other by a motion of three-dimensional space and changing the scale. In this sense, catenary lines resemble parabolas which are also equivalent with respect to the motions of the plane and changes of scale (in contrast with hyperbolas and ellipses). A catenoid is obtained by rotating a catenary line only in the case where this line is at a certain distance from the axis of rotation. If the catenary line is shifted from this position, then the surface of revolution obtained is, generally speaking, no longer minimal. Thus, for example, if a catenary line with small sag almost touches the axis Ox, then on rotating it, the point of the line that is lowest and nearest to the straight line describes a circumference of small radius. Therefore, one of the curvature radii is very small, the other large and the sum of the principal curvatures at the points placed on the gorge of the surface of revolution is different from zero. Consider the dependence of the structure of catenoids on the size of the boundary contour. For simplicity, consider two coaxial circles of the same radius r. Let h be the distance between the parallel planes containing them (Fig. 25). We will assume that the radius is fixed. Introduce another parameter a, the radius of the gorge of the catenoid, i.e., the distance from the axis of rotation to the nearest points of the catenoid, which form a circle of radius a. It is clear that the parameters a and h are related. It can be assumed that h is a smooth function of a. We describe the characteristics of the function h(a) qualitatively. The dependence of h on a is determined by the circumstance indicated above: since

32


the mean curvature of the catenoid is equal to zero, the radius of curvature, the radius of the osculating circle at the vertex of the catenary line (and the point nearest to the axis of rotation), equals the radius of the circumference described by the vertex of the line when it rotates about the axis, that is, equals a.

Figure 25

Therefore, the smaller a, the greater the curvature of the catenary line at its vertex. The parameter a varies on the interval from zero to infinity, If a > r, then, obviously, no minimal surface spanning the two circumferences exists (Fig. 26). The section of the catenoid by a vertical plane passing through the axis of symmetry is represented by a thick line, and the osculating circle at the vertex of the catenary curve by a thin line. When a = r, it is obvious that h = 0, that is, the two circles osculate. When a < r, the distance h starts growing, the circles move apart and a minimal surface forms on them. As a increases further, the distance h continues to increase for some time and then, as seen from Fig. 26, after having reached a certain maximal value h max , starts decreasing. As a tends to zero, h also tends to zero and the catenary line approaches the axis Ox nearer and nearer. The graph so obtained of the function h(a) is given in Fig. 26. When a is close to zero, the radius of curvature at the vertex of the catenary line is also small, which makes this line approach the horizontal axis. The maximal value hmax is approximately equal to

i3 r.

When the distance h between the


33

I

I I

oL«

h o~

1.. :

hCci) ________.-____

Il.. __--

~

Figure 26

circumferences exceeds h max, the catenoid breaks, and the minimal surface turns into two plane discs spanning the boundary circles. In this case, no other soap films with this boundary exist. This transfer from one solution of the equation of minimal surfaces to another occurs in a jump which essentially changes the topology of the film. It is useful to consider the evolution of the catenoid as h increases from zero to h max • In Fig. 27, several consecutive positions of the soap film are shown. A curious and important fact (which we shall use below) is the availability for any h, 0 < h < h max, of two catenoidal soap films: interior and exterior. As h increases, these two films start sagging in the direction of the axis of symmetry and draw nearer. Finally, when h exceeds the critical value, both films merge, burst, and get restructured into a pair of discs spanning the circles.

34


DC{~

/

, ....

Disc

-

II

/

/'

......

.......

.......

....-

/1

I ~

,,

.....

Disc

Figure 27 Thus, for any value ofh in the interval 0 < h < h max, there exist three soap films with given boundary: two catenoids and a pair of discs. Plateau made the conjecture that the interior catenoid was unstable in contrast with the exterior one and both discs, but could not prove it. Attempts to realize the interior catenoid as a physical soap film fail. The conjecture regarding the instability of the interior film turns out to be correct. We now briefly describe a certain qualitative behaviour which does not claim complete mathematical rigour, but makes it possible to have a clear idea of the distribution of the stable and unstable positions of a soap film when its boundary is deformed.

5. The change of the topological type of a minimal surface in accordance with its stability or instability. We first formulate a certain general problem whose investigation turned out to be helpful from many points of view. Consider the space Coo (M, N) of smooth mappings f:M-N, where N is a fixed Riemannian manifold, and the M are all possible Riemannian manifolds with non-empty boundaries. Consider some functional F (f) on the space Coo (M, N), given in the form: F (f) = fM L (f) dow where L is a certain Lagrangian depending on the mapping f and its derivatives of various orders, and dO M is the Riemannian volume form on the


35

manifold M. As the main examples of functionals F, we shall consider the multidimensional volume functional volk f (M) and the Dirichlet functional. Consider the Euler-Lagrange equation I (f) = 0 for the functional F. The extremals, that is, the critical (stationary) points fo of the functional Fare solutions to the Euler-Lagrange equation. The extremals of the volume functional are locally minimal surfaces, and those of the Dirichlet functional are harmonic mappings f:M-N. To specify some or other extremal fo' boundary conditions should be specified, i.e., a mapping g. oM--+N given. For example, to specify a two-dimensional soap film in R3, some closed contour 1', i.e., a mapping of a circle (or system of circles), should be fixed in R3. Thus, the set of all boundary conditions is naturally identified with the space Coo (oM,N) of mappings g: oM-N. Recall that the specification ofa boundary condition g: oM ~ N defines, generally speaking, several extremals fo of the functional F. In other words, there are, generally speaking, many solutions fo of the Euler-Lagrange equation for a given fixed boundary condition g: oM---+N, i.e. such that fo IaM = g. We associate each boundary condition g: oM--+ N with the set Kg of the extremals fo: M--+N corresponding to it. For each extremal fo E Kg, we calculate the value F(Q, that is, the value of the functional F on the given extremal. Associating each boundary condition g: 0 M~ N with the set of numbers F(fo)' where fo E Kg, we obtain, generally speaking, a many-valued function g-+iF(Q,foEKgldefined on the function space Coo (oM,N). This function has branch points and other singularities. Thus, we have associated each funcitonal F with its graph F(Q, fo E Kg defined on the space Coo (oM,N). In other words, we have associated each Euler-Lagrange equation I (f) = 0 with the graph of the many-valued function g --+ F(Q. It is clear that different functionals (defined on the same space of boundary conditions Coo (oM, N) are represented by different many-valued functions, each of which is characterized by its own set of singularities, and to different Euler equations (i.e., different functionals F), there correspond, generally speaking, different types of singularities. Thus, we obtain the opportunity of associating each functional defined on the space Coo (M, N) with a certain set of singularities (of the many-valued function g -+ F(fo». A detailed study of this correspondence would make it possible to break the set of all functionals F into types, and make up their classification in terms of singularity theory. It is natural to assume that two functionals belong to the same type if they have the same singularity sets. This problem is also of special interest because those types of singularity characterize the Euler-Lagrange equations of the corresponding functionals. It would be interesting to clarify, for example, which types of singularity correspond to the equation of minimal surfaces and the equation of harmonic surfaces (mappings). In the present item, we shall recount an example which vividly shows the types of singularity that can arise in the graphs of many-valued functions of the form g-+F(fo). The description of the singularities "in general position" of functions of this kind is borrowed from the well known works of V.I. Arnold.

36


In a concrete investigation of the many-valued functions g~F(Q, the problem can be simplified at the initial stages by considering not the whole space Coo (oM,N), but only finite-dimensional submanifolds in Coo (oM,N), to which the function g--+ F(Q is restricted. In our example which we recount below, we shall study a two-dimensional submanifolds in Coo (0 M, N) for the case of the twodimensional area functional. In the present item, an interesting effect relating to the geometry of twodimensional minimal surfaces is described. This work arose as the result of the analysis of an extremely curious observation of T. Poston (see I46). This effect was investigated in detail by A.T. Fomenko and A.A. Tuzhilin. Meanwhile, we seem to have discovered some new properties of deformations and restructurings of two-dimensional minimal surfaces. Quite recently, an interesting paper M.J. Beeson and A.J. Tromba has been published, viz., "The cusp catastrophe of Thorn in the bifurcation of minimal surfaces" (see Manusc. Math., V.46, 1984, pp. 273-308), in which the bifurcations of the Enneper minimal surface have been studied. It has been known for a long time that, generally speaking, several different minimal surfaces can span the same contour. Some examples are already familiar to us, e.g., two catenoids and a pair of discs. The uniqueness of a soap film can be guaranteed, for example, for plane, closed, non-self-intersecting contours, i.e., realized as a system of curves embedded into one plane. It is clear that any film different from a part of the plane bounded by these curves has a greater area than the plane film. Moreover, two variational problems can be considered: finding films of least area for a given contour and finding minimal films of least topological genus. It cannot be excluded that a film with least possible area has, nevertheless, a topological genus that may by no means be minimal, that is, the film can be homeomorphic to a sphere with a number of handles. Recall that a surface of zero topological genus has no handles. On the contrary, a minimal film ofleast genus (for a given contour) may not realize a minimum of area in the class of all films with given boundary. In other words, the absolute minimum of the area functional can be attained on films whose topological genus is not minimal (Fig. 28). If the represented pairs of circles are near enough, then the area of the first film is less than the area of the second. At the same time, the first film is homeomorphic to a torus with a hole (Fig. 28), that is, has, in this sense, topological genus 1, and the second film is homeomorphic to a disc and has topological genus O. Therefore, we will speak, for the moment, of minimal films, meaning their local minimality and not fully taking up the question of their global minimality and the relation of absolute minima of the area functional with the topological genus of the film. Consider the frame in Fig. 28. It proves to be sufficiently convenient for the observation of non-trivial restructurings of minimal surfaces and a change of the topological type in accordance with the stability or instability of soap films. It is clear that this contour is homeomorphic to the circle and, moreover, can be


37

~

Contour on the torus

Figure 28

deformed, by a smooth deformation, into a circle standardly embedded into the plane without self-intersections. We assume that it is realized in R3 as a wire contour, the wire being sufficiently thin. Denote the distance between the upper parallel rings by u (Fig. 28), and that between the lower parallel rings by v. We call the contour with the values of u and v fixed, the state (u, v) of the contour. The set of all possible states (u, v) of the contour can be identified with the square [O,u max] x (O,v max ]' For each fixed state of the contour, there exist, generally speaking, several minimal surfaces (soap films) with given boundary. Our purpose is to investigate their behaviour and restructurings under a continuous deformation of the contour. Since a formal analytic investigation of the behaviour of such surfaces is, as a whole, extremely complicated, we were forced to resort to a concrete physical experiment, realizing the minimal surfaces which we are interested in, and also their deformations, as soap films spanning a wire frame. In our experiments, the following soap films were obtained: la, Ib, Ic, and Id (Fig. 29), films of type 2 (Fig. 29), of type 3a and type 3b (Fig. 29) and, finally, films of type 4 (Fig. 29). Let us explain the numbering of the films. It is clear that all films of type I are diffeomorphic to each other (i.e., films la, Ib, Ic, and Id) and to the disc. Films of type 1a and type Ib are obtained in the experiment when the values of u and v are not very small, though, from the

38


formal mathematical point of view, they can be considered also for arbitrarily small positive values ofu (or v). Films of type 2 are no longer manifolds, though, from the homotopy point of view, they are also trivial (as films of type I), that is, each of them is contractible in itself to a point. In our experiment, these films were obtained for small values ofu and v. Films of type 3 (i.e., 3a and 3b) are homotopy different from those of type I and type 2, since films of type 3 are homotopy equivalent to a circle. It is clear that films of type 3 can be obtained from films of type 2 by puncturing either the right-hand or left-hand disc in them (Fig. 29). And, finally, films of type 4 are not homotopy equivalent to films of previous types I, 2, and 3, since films of type 4 are homotopy equivalent to a wedge of two circles. Thus, films of types 1,2 and 3, 4 are homotopy different. Films of type 4 are obtained if both discs are punctured in a film of type 2, or one disc in a film of type 3. Note also that films of type 3 can be obtained from a film of type I by directing a jet of air on one of the two "great discs" that a film of type I possesses. Eventually, one of the discs will bend and, having touched the opposite and parallel disc, will be glued to it, thus obtaining either a film of type 3a or 3b. We shall restrict ourselves, for now, to the investigation of films of type I, that is, diffeomorphic to the disc and of topological genus O. T. Poston in 146 pointed out the following restructuring of films arising from a continuous deformation of the frame. Take a film of type la and deform the boundary frame, thus deforming the film spanning it in the way shown in Fig. 30. Draw the two lower circles apart by degrees. We can assume that this process approximately coincides with stretching a catenoid when its two edges move apart. Here, we use the fact that, for a sufficiently small original distance between the circumferences, the soap film is a band spanning each pair of circles and is sufficiently well approximated by a catenoid. In stretching the catenoid, at a critical moment, the catenoid collapses, gets restructured, and turns into a pair of discs spanning two circles moving apart from each other. However, in our case, these two discs are still connected with a narrow band spanning the two upper circles which are affected by the described deformation to a lesser degree. Ultimately, we obtain a film shown in Fig. 30. Now, moving the two lower circles in the opposite direction and returning them to the original position, we do not change the topological type of the film any more and obtain a film of type I b. Thus, we have constructed a continuous deformation of the contour under which a jump-like change of the film occurred, which has altered its type: from position la, we transferred to position lb. Using the contour symmetry, we can make the reverse transfer from a film of type Ib to a film of type la in a similar manner. It suffices to perform the operation described above, but only with the two upper circles, by gradually removing them from each other, and then returning to the original position. In the deformation process, the film will get restructured in a jump, and a film of type Ib will be converted into a film of type la. This "pendulum" can "swing" many times if the soap film is made from a sufficiently elastic liquid.

39


4-

Stable

Stable

Figure 29

Stable

Unstable

40


Figure 30

But, together with the described jump-like restructuring of the film, there exists another remarkable deformation of the boundary frame leading to a smooth change (i.e., without jumps) of the film with the same final result (!). In fact, consider a film of type la and, in contrast with the previous case, start moving the upper circles apart in the way shown in Fig. 31. In doing so, the topological type of the film remains unaltered and undergoes a homeomorphism. Then, we start moving the lower circles apart as in Fig. 31. The resulting symmetrical film is saddle-type: its centre is a saddle point. In one direction, the film rises upwards, and in the other, descends downwards. This deformation is smooth, jump-free. Since a film of type ld is symmetrical, by applying the same operation, we can pass from it to a film of type lb in a jump-free manner. Similarly, repeating all the steps in reverse order, we can smoothly transform a film of type lb into a film of type la in a jump-free manner and without topological restructurings. Thus, we obtain two essentially different paths joining positions la and lb in the space of all surfaces with homeomorphic boundary contours. The first path was realized by a smooth contour deformation in the course of which the film underwent a jump-like restructuring. The second path is realized by another smooth deformation during which the film is changed without jumps, but transferred, just as in the first case, from position la to lb.


41

Figure 31

To a first approximation, we can assume that the contour deformations so described can be specified by a change of two real parameters: u, the distance between the two upper circles and v, the distance between the lower ones. In other words, u and v are the widths of the upper and lower rings-bands (Fig. 29). Each position of the boundary contour can be assumed to be determined by specifying a pair of real numbers which, for better understanding, can be regarded as the coordinates ofa point in the two-dimensional plane. T. Poston1 46 suggested a certain schematic diagram representing these deformations (the socalled Whitney cusp, see Fig. 32). The author has decided to approach this problem from a somewhat different angle, namely, by investigating the properties of the real "graph", the graph of the soap film area. As it happens, the situation obtained is described by the socalled swallow tail (a well-known singularity). Let us formulate the problem in a more precise manner. Consider a square in the (u,v) plane representing all possible states of the boundary contour. For each point (u, v), we have the corresponding contour which, generally speaking, several minimal surfaces of type 1 can span. Calculating their areas, we obtain a set of numbers and, therefore, the graph of a many-valued function, the area of the minimal surfaces. Note that films can be of two types: stable and unstable. Thus, among the types of films listed above, a film of type lc is unstable, while the remaining are stable. We will consider unstable films more carefully with the

42


Figure 32 aid of one of the so-called Plateau principles, which states that a part of a minimal film bounded by an arbitrary closed contour traced on the film is also minimal with respect to this contour. For details regarding the Plateau principles, see the next items. In practical experiments, we obtained unstable soap films in the following way. The fact that a film is unstable implies the existence of arbitrarily small perturbations of it that destroy the film or make it appreciably change its position with respect to the boundary frame. Meanwhile, it is clear that if sufficiently many flexible threads are placed on the film in a convenient manner, their ends being attached to the frame, then we can make the film stable if the net of threads is sufficiently dense (see schematic diagram in Fig. 33a). In other words, we can attempt to keep an unstable soap film in a stable position by increasing the boundary frame and adding a net of threads to the original frame. This simple remark has proved useful in the study of concrete minimal surfaces. We begin with an example. Take an unstable film of type Ic, for instance (Fig. 33b), spanning a contour to which a thread is attached beforehand in the way indicated

THEORY OF MINI MAL SURFACES

43

...

Co

~ E E ,:!. 0_

.....-!!'~-~T

~ ~ - - - - - -- - ~

----

~o


45

Graph of area of minimal surfaces is represented by a many-valued function

Figure 33i-j

46


in Fig. 33b. Let this thread lie on the film of type Ic freely, that is, no forces act on it. We shall slightly deform the contour. While moving the right-hand circles apart, we see that a part of the film bends to the left, and the thread stretches tight. Similarly, on moving the left-hand circles apart, we see that the film deforms and the thread stretches tight while bending to the right. Meanwhile, the thread is transformed into a singular edge of the film, keeping it in the equilibrium position and not letting it deform further. In both cases, the pull on the thread leads to the appearance of a singular edge on the film. This procedure can be reversed easily. For example, beginning with the position when the thread is stretched tight and bent to the right, we start deforming the contour by moving the right-hand circles apart. Ultimately, a jump-like change of the film position occurs and the thread bends to the left, which corresponds to the transfer to a new state of the system. It is clear that there exists an intermediate position (which is already known to us, being a film of type Ic), in which the thread is not stretched tight and lies on the film freely. Such films are critical (extremal) for the area functional. If, in a critical position of a film (for small u and v), the thread is detached from one end (we might hold the second end of the film in the hand, and not fasten the thread to the contour with it), while giving it a small initial momentum, then the film will jump to state lao This implies the instability of the film of type Ic, which, under a small jumplike perturbation, alters its position, and changes into a film of type Ia. To show that, for each state of the contour (for small u and v), there exists an unstable film of type Ic, we shall proceed in the following way. Take a film of type 1a, attach a thread to a point of the contour in the manner shown in Fig. 33c, and start pulling the thread (Fig. 33d). We soon obtain a critical position of the film of type Ic. If the process continues, then a jump of the film will occur, and the thread will bend to the left. If, on the contrary, while it is in the critical position, the thread is released and given a small momentum, then the film will return to the original position Ia. In the critical position, the thread lies on the film freely. Thus, a film of type 1c is unstable, and we have investigated the process of its conversion into a film of type Ia or lb. We can say that a film of type Ic is a film of type Ia whose "separating line" is a saddle connecting the rings placed nearer to the centre and spanned by the film. The film of type Ia can be treated as a film spanning both left-hand and right-hand rings, but the right-hand ones are spanned weakly and, therefore, the saddle is placed far from the centre of the contour. Consider a state (u,v,Ia) of the system, where u and v are small. We shall increase v in a jump-free manner. The film of type Ia bends more towards the centre (Fig. 33e). Finally, there is a state such that on giving v another small increase the film starts moving independently under the action of surface tension forces and comes to state 1b in a jump, with the state of the contour almost unchanged. Now, if we decrease v to its original value, then the film undergoes no jumps. It will preserve its type 1b obtained as a result of the jump. Thus, if we start the process from the state (uo' v I' 1a)~ (uo' v I' 1b» and subsequently


47

decreasing v to the original value vo' we obtain another state (uo,vo>la) then, after the jump (on increasing Vo up to vI in the state (uo,vo,lb) of the system with the same state of the contour. It is clear that, on increasing v, the area of the film of type la increased. During the jump, when (as is seen from our experiment) the film was itself moving under the action of surface tension forces, the area decreased in a jump. Here, we use the fact that any free movement of a soap film leads to a decrease in its area. However, when we decreased v, the area of the film remained practically unaltered, since now the film is of type 1b. Therefore, the graph of the variation of the film area in the described process is of the following form (Fig. 33f). The value if corresponds to the state of the contour in which the area of the film of type la equals that of the film of type lb, which evidently happens when U o = if . Therefore, the graph in Fig. 33 corresponds to the initial condition Vo ], where E and G are the coefficients of the first fundamental form (Riemannian metric) induced on the graph of the function q> by the underlying Euclidean metric in R3. If r = r (u, v) is the radius vector of the surface, then E = < ru' ru >, G = < rv' r v >. Consider further the class of all functions q> continuous on the closure of the region G, piecewise smooth in G, and assuming

50


the same values on the boundary of the region as the function g. Then the problem of finding a function q> on which the Dirichlet integral D [q>] assumes the least possible value, has a unique solution. The same function is also a unique solution to the boundary problem for the Laplace equation.1q> = 0 with the boundary values g = g I"I on the curve "I given beforehand. In the first half of the century, a certain shift of emphasis in the approach to the solution of Plateau's problem occurred. Plateau himself formulated several principles which we shall list below. At this point, we only give the first of them. FIRST PLATEAU PRINCIPLE. Let a surface of zero mean curvature be given. The surface can be described by means of an equation for the radius vector, or by some geometric rule, as a helicoid, for instance. Consider an arbitrary piecewise smooth, closed, non-self-intersecting contour on it, that is, draw on the surface a closed curve without self-intersections. We will assume that the contour bounds a part of the surface, this part being stable. Make an iron wire frame which is an exact copy of the curve drawn, slightly oxidize it with dilute nitric acid, and fully immerse it in soap water with glycerine. If we remove the frame from the liquid, then, among the soap films that can appear on this contour, there is certainly a film coinciding with the part of the surface, bounded by the curve originally drawn. We see that Plateau proceeded from a minimal surface already given, and considered subsequently different contours admitting of the realization of some or other stable parts of the surface as soap films. With the development of variational calculus, the point of view according to which the original object is a contour and the problem consists in the study of various minimal surfaces that can span it, become dominant. The most important question is the existence of a minimal film spanning a given frame. Since the study of arbitrary contours is non-trivial enough, a rather special class of them was first singled out. Consider surfaces specified in R 3 by the smooth graph of a single-valued function z = f (x, y), i.e., having a smooth, orthogonal, one-to-one projection onto the twodimensional plane or a domain in it. Let a surface M be projected into a domain G with convex boundary "I, that is, the boundary curve (assumed to be piecewisesmooth) is convex. If the surface is given by the graph, then the area functional is of the form: A [f] = HG .../1 + fi + ~ dxdy. The Euler equation for the area functional, i.e., the equation describing the critical stationary surfaces is (1 + f~) f)ry - 2 fx fy fxy + (1 + f;) fxx = O. It is non-linear and fairly complicated, which corresponds to the intricate behaviour of its solutions, i.e., minimal surfaces, in relation to the nature of the boundary contour. As it happens, for any closed, piecewise-smooth contour "I' whose projection into the convex, closed curve "I on the plane is one-to-one, there exists a minimal surface with given boundary, projected, therefore, onto the domain bounded on the plane by the curve "I.


51

The existence and uniqueness of this solution in the indicated class of surfaces admitting of a one-to-one projection is a corollary to the boundary projection being convex. If the requirement that the domain G be convex is dropped, then the statement of the theorem becomes incorrect. In other words, there exist smooth contours "(' whose projections onto a non-convex curve "( in the plane are one-to-one, and such that there exists no minimal surface with boundary "(' whose projection onto the domain G bounded by the curve "( is one-to-one. The simplest example is in Fig. 34. A minimal film that can span this contour "(' exists but in the first place, is not projected inside the curve "( and in the second, cannot be specified as the smooth graph of a one-valued function defined on the plane. It is clear that these effects are consequences of the curve "( being nonconvex.

Figure 34

2. The area functional, Dirichlet functional, harmonic mappings and conformal coordinates. Further substantial progress was achieved by Douglas who considered smooth mappings of the standard two-dimensional disc D C R2 into R3. We refer to the disc by its Euclidean coordinates u,v. Then its mapping is given by the radius vector r = r(u, v). If E, F, and G are the coefficients of the first fundamental form of the surface, which is the image of the disc in R 3, them for the Dirichlet and area functionals, we obtain the following explicit expressions:

52

D[r] =


t HD2

HD2

(E + G) dudv; A[r] =

vlEG - F2 dudv. Extremal, critical

radius vectors for these functionals, i.e., solutions of the Euler equations are arranged as follows. The extremal vectors r (u, v) for the area functional are those and only those radius vectors for which the mean curvature of the corresponding surface equals zero, that is, minimal surfaces. Extremal radius vectors for the Dirichlet functional are those and only those vectors which are harmonic with

iJ2r

iFr

respect to u, v, that is, .:lr = -2 + -2 == O. If conformal coordinates are given

au

av

on a minimal surface, then its radius vector becomes harmonic. Thus, each minimal film is harmonic with respect to conformal coordinates. The converse does not hold; that is to say, a surface swept out by the harmonic radius vector need not be minimal, the simplest example being the graph of the real a (x, y) (or imaginary b (x, y)) part of a non-linear complex-analytic function f (x + iy) = a (x, y) + ib (x, y) defined on the plane, e.g., f (z) = zn, n ~ 2. In Fig. 35, a schematic diagram of the interaction of sets of minimal and harmonic radius vectors is represented. If some radius vector is minimal (i.e., the surface corresponding to it is minimal), then all the other radius vectors obtained from the original by a regular change of variables are also minimal. In these parametrization changes, the surface remains unaltered, and the mean curvature is invariant under a change of local coordinates on the surface. In each such class of equivalent (with respect to coordinate changes) minimal radius vectors, there must be one harmonic one. In Fig. 35, it is shown that each equivalence class intersects the set of harmonic radius vectors. Harmonic vectors

Figure 35


53

The area and Dirichlet functionals are related by the inequality D[r] ~ A[r], the equality holding if and only if E = G, F = 0, that is, when the coordinates u, v are conformal. This follows from the obvious inequality -JEG - F2

::5

E

~

G. A certain analytic advantage of the Dirichlet functional

over the area functional consists in its integrand not containing radicals. A similar situation occurs in the theory of one-dimensional path length and path action functionals, where the extremals of the action functional I < y, y> dt are geodesics, and those of the length functional < y, y> dt are all possible regular parametrizations of the geodesics. In the one-dimensional case, the length functional has "more" extremals than the action functional. In the twodimensional case, the situation is more complicated. The most important result proved by Douglas is the existence theorem for a minimal surface in a ftxed topological type surface class. Before stating the theorem, recall that each smooth, compact, closed, connected, and two-dimensional manifold is diffeomorphic either to a sphere with a certain number of handles or a sphere with a number of cross-caps glued to it (Fig. 36). A handle is an ordinary cylinder glued boundarywise to two holes in a two-dimensional sphere. A cross-cap is an ordinary Mobius strip glued boundarywise (i.e., along the circumference) to a hole in a sphere. Sometimes, for clarity, a cross-cap is realized by bending the edge of a Mobius strip so as to transform it into the standard plane circle with the consequence that the conftguration of the Mobius strip becomes more complicated, with self-intersections appearing. Now, we can give a precis of Douglas's results (the original text of part of his paper is given in the next chapter). Assume that, for a two-dimensional manifold M of given topological type (i.e., with a given number of handles or cross-caps) with boundary, there exists a smooth mapping f:M-+R3 with ftnite area, i.e., A [f] < 00, (or with ftnite value of the Dirichlet functional), the boundary of the surface being mapped onto a given set of closed Jordan curves in R3. Assume that the minimum (more precisely, inftmum) of the areas of such immersions of a surface in R 3 with ftxed boundary is strictly less than the minimum of the areas of all immersions in R 3 (with the same boundary) of all surfaces obtained from the original by discarding one handle or one cross-cap. Then there exists a smooth mapping fo: M~R3 such that it is minimal, i.e., realizes the minimum of area in the class of all immersions of the surface in R3 with given boundary. This surface has zero mean curvature at all of its non-singular points. Let us comment on the condition for the existence of at least one immersion of M in R 3 with ftnite area. A similar condition was imposed above in the formulation of the Dirichlet principle. The point is that there exist curves which can be spanned by no surface offtnite area (Fig. 37). Two orthogonal projections of a space curve onto two planes are depicted. The equation of the curve in standard spherical coordinates s, 8,
this function of the two quantities a, {3, the integral of the second equation will be at first (d)

y

+ (3x x

(e)

= q>(a, (3),

= q>".

For the same reason, if we represent by \jI another function of a and (3, then the integral of the second equation of the other characteristic will first be given by the two equations (f)

y + ax = 1/;{a,(3),

(g)

x = 1/;',

But the point of the surface must be determined by the intersection of the two characteristics and, hence, four equations (d), (e), (f), (g) must be valid for its projection onto the plane of x and y, and they cannot be valid unless the functions q> and \jI satisfy the following two: (h)

q> - {3q>" = \jI - a\jl',

(i)

q>"

= \jI',

which result from the elimination ofx and y, and whose integrals must serve to determine the forms of the functions q> and \jI. To integrate these latter two equations, it is first required to separate the functions, and for that, we shall differentiate them, regarding successively a, then {3, as unique variables; and, first eliminating all that depends on one of these functions, and then all that depends on the other, we shall find the two second partial differential equations. (a - (3) q>" 1 + q>' (a - (3)

\jI" 1

+

= 0,

\jI" =

0,

which can be transformed into the following q>'

d--

a-{3 = 0, d{3

\jI"

d--

a-{3 = 0, da

SURVEY OF MINIMAL SURFACE THEORY

71

and whose integrals are ' = (a - (3)" a, \II"

= (a - (3)'it" {3,

in which and 'it are two new arbitrary functions of a unique quantity, a for one, and {3 for the other, and which we accentuate because of the subsequent integrations. These two equations are themselves exact differentials taken, one while making only a vary, the other while making only {3 vary; and their integrals are = (a - (3)'a + a + F{3, = (a - (3)'it' {3 + 'it{3 + fa,

in which f and F are two new arbitrary functions of a unique quantity. Of the four arbitrary functions which are involved in these equations, only two are necessary, because it is voluntarily that we have differentiated two equations (h), (i), and that we have come to differentials of the second order. It is therefore required to determine the forms of two of these functions so that equations (h), (i) should be satisfied. Thus, we find from these integrals by differentiation that " = ' + F',

1/;' = 'it' + /" and, substituting for , ", \II, \II' their values in (h) and (i), we find that they are satisfied provided that F{3 = 'it{3, fa = - a,

which determines the forms of the two supernumerary functions F and f. Hence, substituting the expressions for these functions in the equations for , \II, ", \II', we have: = (a - (3) ' a - a + 'it{3, \II = (a - (3) 'it'a - a

"

+ 'it{3,

= - ' a + 'it' {3,

\II' = - ' a

+ 'it' {3;

72


finally, substituting these values in equations (d), (e) or in (f), (g), and finding the values of x, y, we have for the coordinates of the point of intersection of the projections of the two characteristics (1)

x = - (0) for r = f{0) and r = F(O) respectively. Outside the zone, that is, for values of r less than f{0) or greater than F(O) whatever 0 may be, the ordinate z will be subject to no limitation and can become infinite. But if the minimum area is to be all that portion of the surface the projection of which is bounded by the curve ABC, the values of r will extend from r = 0 to r = F(O) for every value of 0, and throughout this extent the ordinate z must be finite. We shall therefore suppress in this case that portion of the integral of H = 0 which would become infinite when r = 0; and the integral thus modified will be reduced to the degree of generality which the problem has; so that the single condition that z should be equal to «1>(0) when r is equal to F(O) will suffice for completing the solution of the problem. Thus the solution of the question of the minimum area and of similar questions, separates into two problems which are quite distinct so far as relates to the determination of the arbitrary functions. I only here indicate this distinction which I will take up on another occasion. If the required surface is closed on all sides, so that for example we have to find the surface of greatest area which incloses a given volume, the conditions for this relative maximum will not furnish any equation suitable for determining the two arbitrary functions which the complete integral of the equation H = 0 when applied to this problem will involve. It is by means of other considerations that this integral must be reduced so as to contain only three arbitrary constants, namely the three co-ordinates of the centre of the sphere which solves the problem; the radius of the sphere will be determined by means of the given volume. I propose to consider this particular question in another memoir. [It does not appear that Poisson ever returned to the two problems which he proposed in the above section to consider at a future period.] 118. In the remaining three sections of the memoir Poisson discusses an example. In an addition to the work entitled Methodus inveniendi lineas .... Euler determines the figure of the elastic lamina, properly so called, by means of a principle communicated to him by Daniel Bernoulli, namely, that the integral ds 2 taken throughout the length of the curve should be less than for any p


81

other curve of the same length; ds being the differential element of the sought curve and its radius of curvature. In order to give an example of the employment of the preceding formulae, we will extend this principle by induction to the figure of equilibrium of an elastic lamina which is curved in every direction and the points of which are not acted on by any given force. Thus denoting by e and ~ the two principal radii of curvature at any point of this surface, or more generally the radii of curvature of two normal sections at right angles, and by dO" the differential element of the surface, we shall suppose that among all surfaces of the same area the elastic surface is that which gives a minimum value to the

t)

integral

IS (~ +

integral

IS (~+

dO". [This is what Poisson says, but he really takes the

2

tY

dx dy; the two however coincide to the order of

approximation which he finally preserves.] 1 1 By the theory of the curvature of surfaces we know that the sum Q + ~ has the

same value for every pair of normal sections at right angles passing through the same point. With the notation already adopted, we have

+ Z,2) Z"__- 2z'z,z,'__+ (1 + z'2) Z" _1 + _1 = (1 __

e

~

~~

~

~~~

~~~-L~

(1+z'2+z,2)~

or, which is the same thing,

1

1

,

- + - = u +v e ~ "

where u =

z' z, v = -r.==;;===;< .Jl + z,2 + z2' .Jl + z,2 + z2' , ,

We have also do = .Jl + z,2 + z,2 dxdy.

Let c denote an undetermined constant, and put

v=

(u' + v.f + 2c.Jl + z,2 + z,2;

then the question amounts to making the integral IS V dx dy an absolute minimum. (see Art. 104.) The quality N of the nineteenth section (Art. 102) will be zero, and P, Q, R, S, T, will have for values P = 2 (u' + v,) (dU' + dV,) + 2cu, d'7.' dz'

82 Q


= 2 (u' +

dV) v,) ( -dU' + ' + 2cv, dz, dz,

dV) R = 2 (u' + v) ( -dU' + ' ,

S

= 2 (u' +

dz"

dU' v) ( - ,

dz,'

dz'"

dv ) + -'dz,"

dU'- + dV) T = 2 (u' + v,) ( ' . dz"

dz"

It will be sufficient to substitute these values and their first and second differential coefficients with respect to x and y in the equation H = 0 of the twenty-first section (Art. 104), in order to obtain the indefinite equation to the elastic surface; this equation will be a partial differential equation of the fourth order. We must also substitute these same quantities in the equations of the twenty-sixth section, in order to obtain the equations relative to the perimeter of the elastic surface in all the cases which can occur. We will confine ourselves to writing these equations for the case where the elastic surface differs but little from a plane figure parallel to the plane of x and y; and we shall neglect consequently the terms in V of the fourth degree with respect to partial differential coefficients of z. Thus the values of P, Q, ... and therefore the equations in question will be exact as far as quantities of the third order. Thus we have, simply

v=

(z" + z,,)2 + 2c +

C

(z'2 + z,2),

from which we obtain

N

= 0, P = 2cz',

Q

= 2cz" R = T = 2 (z" + ZIt)' S = 0;

thus the equation H = 0 will become z"" + 2z;; + z"" - (z" + z,,) =

o.

If we denote by r a new variable, we may replace this equation by the following system of two equations of the second order: zIt

+

ZIt

=

r, f' + r" = cr .................... .

(a).

In consequence of these values ofR, S, T, the quantity z of the twenty-fourth section (Art. lO7) will be equal to 2r. In order to fix our ideas, I will suppose that the limits of the elastic surface in equilibrium are curves fixed and given, but that the tangent plane to this surface is not restricted by any condition throughout the


83

perimeters of these curves; hence it will follow from the second case of the twenty-sixth section (Art. Ill), that we must combine with the two equations of each limiting curve the equation Z = 0 or r = 0, in order to form the two systems of simultaneous equations, which with the given area of the elastic lamina will serve to determine the constant c and the arbitrary functions contained in the integrals of equations (a). The area of the lamina cannot differ much from that of its projection on the plane of x and y; denote the area of the projection by A, and that of the lamina by A (1 + g) so that g is a very small positive fraction; we shall have A(1 + g) =

H-v'l + Z,2 + Z,2 dx dy,

or to that order of approximation which we have adopted Ag =

t

H(Z'2 +

Z,2)

dx dy .................... .

(b).

§3. Plateau (1801-1883). The present section is particularly important for this book. We give in it some fragments of the famous paper by Plateau 140, which we have already talked a great deal about in Chapter I of our book. It was in this work that Plateau laid the foundations of the study of the main qualitative effects which regulate the structure of two-dimensional minimal surfaces. It was in 140 that he described his most interesting experiments with soap films and gave a qualitative explanation for the phenomena of stability and instability of films. We quote the following fragments from 140: pp. 91-107 (§§55-65) and pp. 312-363 (§§179-203)1.

J.Plateau

Statique experimentale et thiorique des liquides soumis aux seules forces moliculaires. §§55. We now consider the figures which are formed when the rings are farther apart than required to obtain a cylindrical body. If, after having formed between the two rings a vertical cylinder whose height is much less than that corresponding to the stability limit, we lift the upper ring a little, then we see the cylinder shrink slightly in the meridional direction so as to form a waist; if the ring is lifted higher, then the waist becomes still more noticeable, while the figure is always perfectly symmetrical on both sides of the gorge circle which is situated, eventually, in the middle of the interval between the rings. If, in the cylinder from which we proceeded, the ratio between the height and diameter is of a suitable value, then we can, by continuing the process, make the waist very pronounced; and then the meridional 1

Translated from the French (tr.).

84


line changes the sense of its curvature and shifts towards the rings, in such a way that two inflection points are situated at equal distances from the two sides of the gorge circle; furthermore, the bases of the figure preserve their convex shape, and even their curvature augments more or less. In this experiment, there is always, as can be imagined, a limit for moving the rings apart beyond which equilibrium is no longer possible; if it is surpassed, the waist spontaneously narrows and breaks; the figure separates into two portions; but, if we move the rings apart less than the limit in question, then the equilibrium is stable. The cylinder which seems to be able to represent the above results in the most pronounced manner is one with the ratio of the height to the diameter of approximately 5 to 7: for example, by employing rings of 70 mm in diameter, it is required to form a cylinder of about 50 mm height; the upper ring can then be lifted until it is at a distance of approximately 110 mm from the other, and we thus obtain a figure in which the gorge circle has diameter of only about 30 mm. The experiment performed in this way calls for great accuracy: the densities of the two liquids should be equal and the homogeneity of the oil perfect, while in approaching the limit for drawing the rings apart ones actions should be as careful as possible. But, proceeding so as to keep the axis of revolution horizontal, we can succeed without difficulty: the vertical 70 mm rings should be placed in advance at a distance of 110 mm from each other, each fixed with its lower part to a vertical iron wire, while also fixing the lower extremities of the wires to an iron plate which supports the whole arrangement; finally, the wires themselves should be covered with cotton so that the oil cannot cling (§9)1. First, a cylinder is formed between the two rings, and then the volume of the mass is gradually diminished with the aid of a small syringe. If, when the gorge circle is no more than 30 mm in diameter, we take care to remove the oil with only very small portions at a time, we succeed in reducing this diameter to 27 mm and obtain the result represented in Fig. 46.

~J \. :"/"-,." .'

.

..

.-

=' ... "".... -

-' ,' ._ . ...

I

·1"·

.....

,

.

. ...

'"'

~.

~.

,.

Figure 46 Thus, it is evident that all these waisted figures with convex bases, which can, as with those studied in the previous sections, differ as little from the cylinder as we please, are also portions of the unduloid, but taken, on the contrary, in the indeterminate unduloid: while the equator of the inflation is in the middle of the former, the middle of the others is occupied by the gorge circle of the waist; the largest of the former consist of the whole inflation between two half-waists (Figs. 21, 22) (in the original text-tr.) and that represented in Fig. 46 consists of a whole waist between two portions of the inflations. §56. Now, take our horizontal rings again, and suppose that we can place the upper 1 This operation is performed by the method indicated in §46, i.e., by attaching the mass first to one of the fixed rings and then drawing it towards the other by means of a movable ring of the same diameter, and, finally, absorbing the superfluous oil.


85

one closer or farther away from the other at will; in addition, form a cylinder between them, and then, without changing the distance between the rings, gradually remove the whole mass of the oil. If the ratio of the distance between the rings to their diameters is much less than that in the last experiment of the previous section, then the curvature of the bases, instead of augmenting with increase of the waist, on the contrary, decreases, 2

and if this ratio does not exceed approximately 3' then it becomes necessary to render the bases absolutely plane. If the ratio is still small, it is possible to go even further: continuing to absorb the liquid, we see the bases become concave; thus, for example, they form between our rings of 70 mm in diameter as cylinder of height 35 mm (Fig. 47); gradually absorbing the oil, we shall see the bases deform as the waist shrinks and, finally, lose all their curvature. We shall thus obtain the result represented in Fig. 48. If we continue to employ the small syringe, then the bases will acquire a concave curvature; but for the moment, let us dwell on the case when they are plane.

"""""".'1" ~ : ..

.:: ':....'" ; ...• ··.F

.....:. - ."

.. ~

Figure 47

J2 - 'lr2)dw, Y = Rji(cf>2 + 'lr2)dw, z = Rj2cf>'lrdw, where cf> and 'Ir denote single-valued analytical functions of w = u T iv in Iwl < 1. PROBLEM P 4• Solve problem P 3 under the supplementary condition that cf>, 'Ir do not have any common zero in Iwl < 1.


125

PROBLEM P s. Suppose that the orthogonal projection of the given JORDAN curve r* upon the xy-plane is a simply covered JORDAN curve r. Denote by R the JORDAN region bounded by r. Solve problem Plunder the supplementary condition that the solution admits of a representation S: z = z(x, y), (x, y) in R, where z(x, y) is single-valued and continuous in R and analytic in the interior of

R. Besides the problem of PLATEAU, we shall have to consider the problem of the least area, which requires the determination of a continuous surface S bounded by a given JORDAN curve r*, such that the LEBESGUE area U (S) of S is a minimum when compared with the areas of all continuous surfaces bounded by r*, it being understood that only surfaces of the type of the circular disc are considered*• Then comes the simultaneous problem, which requires the determination of a common solution of the problem of PLATEAU and of the problem of the least areat . According to the different statements of the problem of PLATEAU, we have, strictly speaking, a number of simultaneous problems. As a matter of fact, only the statements P 2 and P 5 of the problem of PLATEAU have actually been used in this connection. Anyone of the preceding problems gives rise to the question as to whether or not the solution is unique. Furthermore, there arises the question as to whether or not some of these problems are equivalent. The existence theorems concerned with the problems listed above will be discussed in subsequent Chapters. The other questions raised in this section will be considered in the present chapter. Sections III.6 to III.16 contain a number of special facts which are then coordinated in the sections III.17 to II1.20. II1.6. The following definitions prove useful for the sequel. Let there be given, in a JORDAN region R of the uv-plane, a continuous function g(u, v). The g(u, v) will be called a generalized harmonic function in R if for every interior point (u o' vo) ofR the following condition is satisfied. There exists a neighbourhood Vo of (uo' vo) and a topological transformation u = U o (0:, (3), v = Vo (0:, (3) ofVo into some region '10 of an 0: (3-plane, such that the function g [uo(o:, (3), Vo (0:, (3)] of 0:, {3 is harmonic in '10. Variables 0:, {3 with this property will be called local typical variables for the point (u o' v0). Suppose that g(u, v) is a generalized harmonic function in R and that g(u, v) vanishes at an interior point (u o' vo) of R. Let (0: 0, (3o) be the image of (uo' vo) under the transformation u = U o (0: 0, (30)' v = vo(o:, (3), where 0:, {3 are local typical variables for (uo' vo). Suppose that all the partial derivatives of g with respect to 0:, {3 vanish at (0: 0, (30) up to and including a certain order (n - 1) ~ 0 while at least one of the partial derivatives of order n is different from zero. Then * Surfaces of different topological types will only be considered at the end of Chapter VI. t This problem has been called the problem of PLATEAU by LEBESGUE: Integrale, longeur, aire. Ann. Mat. Pura Appl. Vol. 7 (1902) pp. 231-359.

126


(uo' vo) will be called a zero of order n of g(u, v). This notion is independent of the particular choice of the lcoal typical variables ex, {3. 111.7. If gl (u, v), g2(U, v) are generalized harmonic functions in R, then gl + g2 generally is not a generalized harmonic function. On the other hand, several important properties of harmonic functions remain valid for generalized harmonic functions. The property expressed by the principle of maximum and minimum clearly remains valid. The following lemma is an immediate consequence of well-known properties of harmonic functions which remain valid for generalized harmonic functions. LEMMA *. Let g(u, v) be a generalized harmonic function in a (simply connected) JORDAN region R. Suppose that g(u, v) has a zero (uo' vo) of order n ~ 1 in the interior ofR. Then g(u, v) vanishes in at least 2n distinct points on the boundary ofR. 111.8 Consider now a surface S:r = r(u, v), (u, v) in R that is a minimal surface (in the sense defined in III.3) bounded by a JORDAN curve r*. If ex, {3 are local typical parameters (see 111.3) for an interior point (uo' vo) of R, then the components x, y, z of r as functions of ex, {3 are harmonic functions. Hence, if a, b, c, d are any four constants, then ax + by + cz + d is also a harmonic function of ex, (3. Thus if a, b, c, d are any four constants, then the function ax(u, v) + by(u, v) + cz(u, v) + d is a generalized harmonic function in R. The following theorems are immediate consequences of this remark, on account of the facts referred to in 111.7. 1. If a convex region K in the xyz-space contains the boundary r* of a minimal surface (see 111.3) then the whole surface is contained in K**. 2. The tangent plane, at a regular point (see 111.4) of the minimal surface, interacts the boundary curve in at least four distinct pointst. 3. Every plane passing through a branch-point of order n (see 111.4) of the minimal surface intersects the boundary curve r* in at least 2(n + 1) distinct pointstt. III. 9. The last theorem permits us to exclude the possibility of branch-points in certain cases. Suppose there exists, in the xyz-space, a straight line 1 such that no plane through 1 intersects the boundary curve r* in more than two distinct points. Then the minimal surface cannot have branch-points, as follows immediately from theorem 3 in 111.8. The assumption is satisfied, for instance, if * T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 1932) p. 10, where the lemma is stated for n = 2. • ** The reviewer learned about this theorem from L. FEJER.

The theorem is also true for surfaces with negative curvature. This fact played an important role in the work of S. BERNSTEIN on partial differential equations of the elliptic type. See for references L. LICHTENSTEIN: Neuere Entwicklung usw. Enzyklopadie der math. Vol. 2(3) pp. 1277-1334. tt See T. RAD6: The problem of the least area and the problem of PLATEAU. Math. Z. Vol. 32 (1930) p. 794, where the theorem is stated for n = 1.


127

r* has a simply covered star-shaped JORDAN curve as its parallel or central projection upon some plane. If the projection has the stronger property of being convex, then a much stronger conclusion can be drawn. Suppose that the parallel projection of r* upon some plane is a simply covered convex curve. Choose a plane perpendicular to the direction of projection for xy-plane. Then the orthogonal projection of r* upon the xy-plane is again a simply covered convex curve which we shall call r. Let S: x = x(u, v) y = y(u, v), z = z(u, v), (u, v) in R be the minimal surface under consideration. From theorem 3 in 111.8 it follows that S has no branch points. From theorem 2 in 111.8 it follows that S has no tangent plane perpendicular to the xy-plane. From this it follows that S has a simply covered xy-projection in the small. Hence the equations x = x(u, v), y = y(u, v) define a transformation with the following properties. 1. The transformation is one-to-one and continuous in the vicinity of every interior point (u o' vo) of R. 2. The boundary of R is carried in a one-to-one and continuous way into the JORDAN curve r. On account of the so-called monodromy theorem in topology *, it follows then that the transformation x = x(u, v), y = y(u, v), (u, v) in R carries R topologically into the JORDAN region bounded by r. Hence u, v can be expressed as single-valued continuous functions of x, y in R, and there follows then for the minimal surface S a representation S : z = z(x, y), (x, y) in or on r, where z(x, y) is single-valued and continuous in and on r. Since S has no branchpoints, S is a minimal surface in the sense of differential geometry. Hence (see 11.16) z(x, y) is analytic in the interior of r and satisfies there the partial differential equation (1 + q2)r - 2pqs + (1 + p2)t =

o.

(3.1)

Similar conclusions may be obtained if the boundary curve r* is supposed to have a simply covered convex curve as its central projection upon some plane. 111.10. Summing up, we have the following resultst . Let S be a minimal surface (in the sence of 111.3) bounded by a JORDAN curve r*. If r* has a simply covered star-shaped JORDAN curve r as its parallel or central projection upon some plane, then S has no branch-points, that is to say S is a minimal surface in the sense of differential geometry. If the projection r is convex, then S does not intersect itself even in the large. If the orthogonal projection ofr* upon the xy-plane is a simply covered convex curve r, then S can be represented in the form S : z = z(x, y), (x, y) in or on r, where z(x, y) is single-valued and continuous in and on r, and satisfies in the interior of r the partial differential equation (3.1) . • See, for instance, KEREKJART6: Vorlesungen iiber Topologie I, p. 175. t See T. RAD6: The problem of the least area and the problem of PLATEAU. Math. Z. Vol. 32 (1930) pp. 763-796. - T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

128


111.11. We are going to consider now certain uniqueness theorems. A first important fact in this connection is the uniqueness theorem for the partial differntial equation (3.1). Let there be given, on a JORDAN curve r in the xyplane, a continuous boundary function 4> (P) of the point P varying on r. If then Zl(X, y), Z2(X, y) are solutions of (3.1) which both reduce to Ql (P) on r, then Zl(X, y) z2(x, y) in the whole interior of r*. Denote then by r* the JORDAN curve, in xyz-space determined by the equation Z = 4>(P). The above uniqueness theorem asserts that r* cannot bound more than one minimal surface that has a simply covered xy-projection. This statement is rather unsatisfactory; indeed, we shall see (111.17) that the boundary curve of a minimal surface may very well have a simply covered xy-projection, while the minimal surface itself does not have this property. On the other hand, if the xy-projection of the boundary curve r* is convex, then every minimal surface bounded by r* also has a simply-covered xy-projection, on account of 111.10. A similar argument holds in case r* is known to have a simply covered convex curve as its central projection. This results in the following uniqueness theoremt .

=

If a JORDAN curve r * has a simply covered convex curve as its parallel or central projection upon some plane, then r * cannot bound more than one minimal surface (this term being used in the general sense defined in IIl.3). 111.12. The assumptions of the preceding uniqueness theorem are obviously satisfied if r* is a skew quadrilateral. For this case, the uniqueness theorem has been stated without proof by H.A. SCHWARZ. For the purpose of an application to be made later on we mention the following consequence of the uniqueness theorem. Suppose a JORDAN curve r* is invariant under reflection in a certain plane p. Every minimal surface, bounded by r*, is carried by the reflection into a minimal surface bounded by r*. Hence, if it is known that r* bounds just one minimal surface S, then it follows that S is also invariant under the reflection. Repeated application of this remark to the case when r* consists of the edges AB, BC, CD, DA ofa regular tetrahedron with vertices A, B, C, D, leads to the result that the minimal surface bounded by r* passes through the centre of the tetrahedron (this fact has been verified by SCHWARZ by using the explicit formulae for the surface). 111.13. Let us consider now the relation between the problem of the least area and the simultaneous problem (see III. 5). If a solution S of the problem of the least area satisfies the assumptions made in the classical Calculus of Variations, then S is a minimal surface (see 11.5). Without those assumptions this conclusion * See, also for references, the beautiful treatment of this theorem and of related subjects by E. HOPF: Elementare Bemerkungen tiber die Liisungen partieller Differentialgleichungen zweiter Ordung )!om elliptischen Typhus. S.-B. preu {3. Akad. Wiss. 1927 pp. 147-152. See also A. HAAR: Uber reguliire Variations-probleme. Acta Litt. Sci. Szeged Vol. 3 (1927) pp. 224-234. . t T. RAD6: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

129


does not in general hold. The following example is a slight modification of one given by LEBESGUE in his Thesis*. Let the given JORDAN curve r* coincide with the circle r* : x2 + y2

= 1, z = o.

Then the area U (S) of any continuous surface S (of the type of the circular disc) bounded by r* is ~ 1r. Consider then the surface y = 2n(r - tfsin cf>, y= 0

z = 0 for t ~ r ~ 1, z = (1 - 4r2)n for O~r~

t,

where r, cf> are polar coordinates in the uv-p1ane, and n is a given positive integer. Using the relations u = r cos cf>, v = r sin cf>, we have the equations of S appearing in the form S: x = x(u, v), y = y(u, v), z = z(u, v), u 2 + v2 ::5 1, where x(u, v), y(u, v), z(u, v) are easily seen to have continuous partial derivatives up to and including the order n - 1. S consists of the simply covered disc x2 + y2 ::5 1, z = 0 and of the spine x = 0, y = 0,0 ::5 z ::5 1. The area U (S) is found to be equal to 7r by computing the integral H(EG - F2)t. Hence the area ofS is a minimum. Still, S is not a minimal surface, not even in the general sense defined in III.3. Indeed, a minimal surface is contained in every convex region that contains its boundary curve (see III.8), and S obviously does not satisfy this condition. Instead of using one spine as above, we can disfigure any given surface S by putting on it any finite number of spines, without changing its area. We shall see (Chapter VI) that the problem of the least area is solvable for every JORDAN curve. From these two facts it follows that the problem of the least area has infinitely many solutions for every JORDAN curve r *, and that a surface S which solves the problem is not in general a minimal surface. 111.14. Thus the solution of the problem of the least area does not imply the solution of the problem of PLATEAU. Neither does the solution of the problem of PLATEA U imply the solution of the problem of the least area; in other words the area of the minimal surface, bounded by a given JORDAN curve r*, is not necessarily a minimum. This fact had been recognized at the earliest stage of the theory. For the case of doubly connected minimal surfaces bounded by two given curves, the catenoids offer simple examples of the lack of the minimizing propertyt. For the cast of minimal surfaces of the type of the circular disc, bounded by a given curve, H.A. SCHWARZ obtained very general examples in the following way*. * Integrale, longuere, aire. Ann. Mat. pura appl. Vol. 7 (1902) pp 231-359. t See, also for references, the beautiful Chapter IV in the little book ofG.A. BLISS: Calculus of Variations (No.1 of the Cams Mathematical Monographs). Gesammelte Mathematische Abhandlungen Vol. 1 pp. 151-167 and 223-269.

*

130


Consider, in the u + iv = w plane, a JORDAN region R bounded by an analytic JORDAN curve. Denote by u(w) a function which is analytic and different from zero in R. Then the equations w

X

=

RJ (I

- w2) Jt(w) dw,

w

y

= RJi (I

+ wZ) Jt(w) dw,

(3.2)

w Z

= RJ2wJt(w) dw,

where w varies in R, define a regular minimal surface'. The area of this surface certainly is not a minimum if the inequality (2.4) in 11.7 is not satisfied. The lefthand side of that inequality reduces in the present case to

8A J J[Au + Av. I- +( u + v) 2

R

2

2

2

2 2

]

du dv

(3.3)

Hence the area of the minimal surface certainly is not a minimum if this integral can be made negative by substituting a function A (u, v) that has continuous partial derivatives of the first order in R and that vanishes on the boundary of R. This criterion, curiously enough, does not depend upon the function Jt(w) which determines the minimal surface; the criterion is concerned solely with the region that R. SCHWARZ based the discussion of this situation in the study of the characteristic values of a certain partial differential equation". One of the results he obtained states that if the region R contains the unit circle u 2 + v2 :5 1 in its interior, then the integral (3.3) can be made negative. On account of the geometrical meaningt of the variable w in the formulas (3.2), this assumption concerning R means that the spherical image of the minimal surface defined by (3.2) completely covers half the unit sphere. The result of SCHWARZ can also be obtained in the following elementary way*. Let r be a positive parameter and define a function A (u, v; r) by

A(U,

V;

u2 +v-r r) = u2 + V + r for 0 ~ u2 + v~r.

Put J(r) =

u2

H[A~ + A~ - (1 + :;2+ vI ] du dv

+ v. (w) = w in (2.28). **Gesammelte Mathematische Abhandlungen Vol. 1 pp. 241-269. t w is the stereographic projection of the spherical image of the surface. See DARBOUX: Theorie generale des surfaces Vol. I pp. 347-348. T. RADO: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Szeged Vol. 6 (1932) pp. 1-20.

*


131

Partial integration gives J(r) =

-JJA [~A + (J + :; +1If ] du dv u2 + 11 1, such that J(r) < 0 for 1 < r < o. Suppose then that a JORDAN region R contains the disc u 2 + v2 ::51 in its interior. Then r can be determined such a way that 1 < r < 0 and that the disc

u 2 + v2 ~ r2 is interior to R. Define a function A (u, v) by the formulae A(U v) =

,

I

A(U, v; r)

0

in 0 fOT

~

u2 + 11·~ r2, u2 + 11 > r2.

Then A (u, v) vanishes on the boundary of R and makes the integral (3.3) negative. The first partial derivatives of A (u, v) are discontinuous on u 2 + v2 = r2; this edge however can be rounded off by a familiar process. It is thus proved that ifR contains the disc u 2 + v2 ::5 1 in its interior, then the formulae (3.2) define, for every choice of the analytic function Il(w) a minimal surface the area of which is not a minimum. 111.15. In order to obtain a clear-cut example, the function Il(w) in (3.2) should be chosen in such a way that the resulting minimal surface is bounded by a JORDAN curve. It can easily be shown that for Il(w) == i this condition is satisfied, provided R is a disc u 2 + v2 ::5 r2 with r < ..[3. Then the following explicit example is obtained*. The equations * T. RAD6: Contributions to the theory of minimal surfaces. Acta

(1932) pp. 1-20.

Litt. Sci. Szeged Vol. 6

132


x = u +

UV2 -

y = -

V -

z = u2

-

U 2V

t

U\

+ t

V\

}

U2

+

V2 ~

r2, 1 = 0, 'I' = 0 for w = w0 = U o + ivo. Since problem P 4 requires that 4> and 'I' have no common zeros, we see that in problem P4 the solution is not permitted to have branch-points. It is easily seen that if a solution of problem P 2 does not have branch-points, then it admits of a representation as required in problem P 4 • In other words, if we add the condition EG - F2 > 0 for u 2 + v2 < 1 in the statement of problem P 2, then we obtain problem P 4, that is to say the classical statement of the problem of PLATEAU in parametric form *. The question arises whether or not problem P 4 is always possible. The impossibility, in general, of the problem would be demonstrated by exhibiting a single JORDAN curve r* for which it could be proved that every minimal surface (of the type of the circular disc) bounded by r* necessarily has branchpoints. Such a curve has not yet been exhibited; the conjecture that any knotted JORDAN curve would serve the purpose can readily be refuted by examples of knotted JORDAN curves which do bound minimal surfaces (of the type of the circular disc) free of branch-points. More explicitly: there exist knotted JORDAN curves for which the classical problem P 4 is possible, and no JORDAN curve is known at present for which problem P 4 is impossible t . Combining the existence theorem in Chapter V with the theorems in 111.10, we obtain existence theorems for the classical problem P 4 which seem to be the most general known at present. For instance: if a JORDAN curve r* has a simply covered star-shaped curve as its parallel or central projection upon some plane, then problem P 4 is solvable for r*. 111.20. While problem P 4 excludes branch-points altogether, and while problem P 2 does not imply any restriction as to branch-points, the geometrical interpretation of problem P 3 is less clear-cut. Differentiating again the equations (3.6), we obtain

~u - ~v

= 2(4)4>' - '1''1''),

Yuu - iyuv

= 2i(4)4>' + '1''1''),

} (3.7)

zuu - izuv = 2(4)'1'' + 4>''1'). From (3.6) and (3.7) it follows readily: if, for a solution of problem P 3' we have , * See H.A. SCHWARZ: Gesammelte Mathematische Abhandlungen Vol. 1. t Cf. VI.35.

136


EG - F2 = 0 at an interior point U o + ivo = w o' then the partial derivatives of the first and second order of x(u, v), y(u, v), z(u, v) all vanish at that point. In other words: the branch-points of a solution of problem P J are at least of order 2 (see 111.4 for the definition). On the other hand, examples show that problem P 2 may have solutions with branch-points of order one. Thus it follows that the problems P 2 and P J are not equivalent. Problem P 3 will be seen to be solvable for every JORDAN curve that is not knotted (Chapter V). Again, it is not known at the present time whether there do or do not exist curves for which problem P 3 is impossible. III.21. We have seen (1I1.l8) that if the given JORDAN curve r* is situated in the xy-plane, then the problem of PLATEA U in parametric form (in anyone of the statements P 2, P 3, P 4 of 111.5) reduces to the problem of mapping the JORDAN region bounded by r* in a one-to-one and continuous fashion, and in the interior, in conformal fashion onto the unit circle u 2 + v2 ::5 1. This situation played an important role in the theory. The most direct illustration is given by the case when r* is a polygon. One of the earliest ideas for dealing with the problem of the conformal mapping of the unit circle onto the region bounded by a plane polygon was based on the principle of symmetry and led to the socalled formulas of SCHWARZ and CHRISTOFFEL. The method used by SCHWARZ in his classical investigations on the problem of PLATEAU is clearly a generalization to 3-space of the plane method'. In a general way, the reader will find, in the Chapters IV, V, VI dealing with the existence theorems, many instances where the theory of the conformal mapping of plane regions clearly served as a model for the development of the theory of the problem of PLATEAU. An important remark, due to J. DOUGLAS, should be mentioned here. Since the problem ofOSGOOD-CARATHEODORY (conformal mapping ofa plane JORDAN region upon the circle) is included in problem P 2, it follows that if we have a method of solution of problem P 2 which does not make use of the solution of that problem, then we have a simultaneous solution of the problem of OSGOOD-CARATHEODORY and of the problem of PLATEAU. J. DOUGLAS ~mphasizes the fact that this is the case with his own method( III.22. Instead of considering only existence theorems, the relation between conformal mapping of plane regions, that is to say, analytic functions of a complex variable, on the one hand and minimal surfaces on the other, might be discussed on account of its own intrinsic interest. The theory of minimal surfaces appears then as a generalization of the theory of analytic functions of a complex variable. While Chapters IV, V, VI will review numerous facts and methods • Compare, for instance, the following two papers of H.A. SCHWARZ: Uber einige Abbildungsaufgaben. Gesammelte Mathematische Abhandlungen Vol. 2 pp. 65-83 and Bestimmung einer speziellen Minimalflache. Gesammelte Mathematische Abhandlungen Vol. I pp. 6-125. See also DARBOUX: Theorie generale des surfaces Vol. I pp. 490-601. t See J. DOUGLAS: Solution of the problem of PLATEAU. Trans. Amer. Math. Soc. Vol. 33 (1931) pp. 263-321.


137

which might be interpreted from this point of view, it might be useful to present to the reader some specific illustrations. III.23. If f (w) = x(u, v) + iy(u, v) is an analytic function of w for Iwl < 1, then x" = yv' x., = - Yu (CAUCHY-RIEMANN equations). from these equations it follows that (3.8) Conversely, it follows from (3.8) that either y is conjugate harmonic to x, or x is conjugate harmonic to y. Let us call two harmonic functions related by (3.8) a couple oj conjugate harmonic Junctions.

On the other hand, the theorem of WEIERSTRASS (see 11.17) leads one to consider triples of harmonic functions related by the equations

We shall say that x, y, z form a triple oj conjugate harmonic Junctions. We shall review now a few facts which develop further this analogy. 111.24. Suppose f (w) is analytic in Iwl < 1 and even on Iwl = 1, for the sake of simplicity. Put f'(w) = g(w). Then the area of the image of Iwl ::;; 1 by f(w) is given by 127r

U = f f Ig(re iB )I2r dr de, o 0

(3.10)

while the length L of the image of Iwl = 1 is 2".

L= flg(eiB)lde. o

(3.11)

If f (w) gave a simply covered image of Iwl ::;; I, then we could assert, on account of the isoperimetric inequality, that U ~t1TU,

or, on account of (3.10) and (3.11), that (3.12) CARLEMAN* proved that (3.12) holds regardless of whether f(w) gives a * Zur Theorie der Minimalfliichen. Math. Z. Vol. 9 (1921) pp. 154-160.

138


simply covered image. From this he inferred that between the area A of a minimal surface and the length L of its boundary curve the isoperimetric inequality also holds. To prove this*, suppose, to simplify the discussion, that the minimal surface is given by

S : x = x(u, v), y = y(u, v), z = z(u, v), u2 + v2

~

1,

(3.13)

where x, y, z form a triple of conjugate harmonic functions (see II1.23) which remain analytic even on u 2 + v2 = 1. Then x, y, z are the real parts of analytic functions:

and if we put f; = gl' f2 = g2' f; = g3 and use the equation gy + g~ + g~ = 0 (see II.18), then the isoperimetric inequality is expressed by

t~

j 2rlgk(rei8Wrdrde~t.,.. [2{0

k-100

(t ~

k-l

Igk(e i8

)F) !de] 2

(3.14)

To prove (3.14), we observe that on account of the inequality of MINKOWSKIt we have

t E (Ylgk(ei8)lde)2~ [Y·(t k=l 0

0

E Igk(ei9W)tde]2.

k=l

(3.15)

Hence (3.14) follows immediately from (3.15) and (3.12). Some further discussion shows that the sign of equality in (3.14) holds if and only if the minimal surface reduces to a simply covered circular disc. If we suppose that the minimal surface has a minimum area, then the theorem of CARLEMAN is almost trivial, as has been observed by BLASCHKE:\:. Indeed, consider a cone consisting of the straight segments which connect a flxed point of the boundary curve with a variable point on that curve. Since cones are developable, the isoperimetric inequality holds for this cone, and hence a fortiori for the minimal surface, since the area of this latter is by assumption not greater than the area of the cone, while the boundary curve is the same for either surface. This remark of BLASCHKE shows that the point of the theorem of CARLEMAN is that the theorem is true even if the area of the minimal surface is not a minimum (cf. III.14). II1.25. Further inequalities between the area A of a minimal surface and the • We follow the simple proof given by E.F. BECHENBACH: The area and boundary of minimal surfaces. Ann. of Math: Vol. 33 (1932) pp. 658-664.

t See for instance POLYA-SZEGO: Aufgaben und Lehrsiltze Vol. 2 p. 14.

* T. CARLEMAN: Zur Theorie der Minimalflilchen. Math. Z. Vol. 9 (1921) p. 160.


139

length L of its boundary curve have been obtained by BECKENBACH*. Suppose that the minimal surface is again given by the equations (3.13). Suppose also that at the origin we have E = 1 (that is to say, the linear ratio of magnification at the origin is unity). Then U

~ 11",

That is to say, A is at least equal to the area and L is at least equal to the perimeter of the unit circle. The sign of equality holds if and only if the minimal surface is a simply covered circular disc. These and similar theorems are proved by BECKENBACH by using FOURIER expansions. 111.26. A minimal surface S being again given by (3.13) where x, y, z are supposed to form a triple of conjugate harmonic functions, we shall call (x2 + y2 + Z2)! the norm of S and we shall write (x2 + y2 + Z2)! = lSI. Then lSI is the generalization of the absolute value of an analytic function f(w) of w. If f(w) is analytic, then log If(w)1 is a harmonic function ofu, v, and this accounts for many important facts in the theory of functions of a complex variable. While log lSI is not harmonic, it can be shown to be subharmonict . A function g (u, v) is subharmonic in a domain D iffor every point (uo' vo) ofD the inequality g(uo' vo) ~

21r

!". I g(uo + o

ecosS , Vo +

e sinS) dS

is satisfied for sufficiently small values of e*. If g has continuous' partial derivatives of the second order, then this condition is equivalent to ~g = g"u + ~ ~ 0 and log lSI is easily shown to satisfy this latter condition. It is sufficient to consider the situation in domains D where lSI > o. Put log lSI = g. Direct computation gives ~g = (r~ + rt,)r 2 - 2«rrj + (rry)

ISI 4

'

(3.16)

where r = r(u, v) is the vector equation of S. Since the components of r form a triple of conjugate harmonic functions, we have

* See E.F. BECKENBACH: The area and boundary of minimal surfaces. Ann. of Math.

Vol. 33 (1932) pp. 658-664.-The theorems in III.24 and III.25 also hold for surfaces of negative curvature, given in isothermal representation. See E.F. BECKENBACH and T. RAD6: Subhannonic functions and surfaces of negative curvature. To appear in Trans. Amer. Math. Soc. t III.26 to III.29 are taken from E.F. BECKENBACH and T. RAD6: Subharmonic functions and minimal surfaces. To appear in Trans. Amer. Math. Soc. See F. RIESZ: Sur les fonctions subhannoniques etc. Acta Math. Vol. 48 (1926) pp. 329-343.

*

140


At those points where r~ = r; = 0 we have ~g = o. At those points where = r; > 0, ru and rv are different from zero and are perpendicular to each other. Using a unit vector ~ perpendicular to ru and rv we can write r~

where a, b, c are scalars. It follows that

Substituting in (3.16) we get

Thus ~g ~ 0 always. 111.27. The fact that log lSI is subharmonic makes it possible to extend a great number of theorems on analytic functions to minimal surfaces, namely those theorems that depend essentially on the fact that the product of analytic functions is again an analytic function. While there is no direct analogy for minimal surfaces, the subharmonic character of log lSI permits one to extend the proofs. We just mention two examples. Let the minimal surface S be given by the equations (3.l3) where x, y, z from a triple of conjugate harmonic functions. Suppose that 1. x(O, 0) = 0, y(O, 0) = 0, z(O, 0) = 0, 2. (x 2 + y2 + Z2)!

= lSI

:5

1 in u 2 + v2 < 1.

Then

and the sign of equality holds if and only if the surface is a simply covered circular disc. This generalizes the lemma of SCHWARZ. 111.28. Suppose this time that the minimal surface is given by v u-

S : x = x(u, v), y = y(u, v), z = z(u, v), 0 < arctg -< a, where x, y, z form a triple of conjugate harmonic functions. Suppose that these functions remain continuous for v = 0, u > 0, and suppose that x(u, 0), y(u, 0),


z(u, 0) approach definite finite limits

o ::5

arctg ~ ::5 a u

E, E

141

xO, y;, z; as u - 7 + O. Then, in every angle

> 0, the functions x(u, v), y(u, v), z(u, v) approach the

limits x;, y;, z; as (u, v) -+ (0, 0). This generalizes the well-known theorem of LINDELOF. The proof follows by an extension of the so-called multiplication method!'. 111.29. The theorem ofll1.26 can be completed as follows. Threefunctions x(u, v), y (u, v), z(u, v) continuous in a domain D, form there a triple of conjugate harmonic functions if and only if log [(x + a) 2 + (y + b 2) + (z + c) 2] IS subharmonic for every choice of the constants a, b, c.

Jesse Douglas Minimal Surfaces of Higher Topological Structure Chapter 1 GEOMETRIC BACKGROUND AND STATEMENT OF RESULTS § 1. Introduction 1

About two years ago, the author published a solution of the Plateau problem for minimal surfaces of general topological form: k contours, each with an assigned sense of description, given in n-dimensional space; prescribed genus h or characteristic! r, and assigned character or orientability (two-sided or onesided).!· The publication was in the form of two papers, the first ofwhich 2 stated the results and outlined the methods used, while the second [2] gave details and proofs. Subsequently, an alternative method was outlined by R. Courant [11], who has so far published the details for the case of genus zero [12]. 3 Some years ago, the author disposed of the important special cases ofa doublyconnected minimal surface with two given boundaries [3], and of a Mobius strip I'See for instance POLYA·SZEGO: Aufgaben und Lehrsatze Vol. I p. 138 problem 277. I See the paragraph following (2.1) for definition. 10 This general form of the Plateau problem was first explicity formulated by the author in Bull. Amer. Math. Soc., v. 36 (1930), p. 50, where also is broadly indicated the method of solution here finally elaborated. 2 (I] of the bibliography at the end of this paper, references to which will be made by numbers in square brackets. 3 An independent presentation of the same method, as far as the one-contour case is concerned was given by L. Tonelli [13]. According to Tonelli, this method coincides substantially with that used by the author throughout his work on the Plateau problem. The chief point of difference consists in the use of the vector Dirichlet functional in its original form (5.12), without transformation into A(g, R).

142


[4]. The most fundamental case of a simply-connected minimal surface bounded by a given arbitrary Jordan curve had been previously solved by the author [5], and by an alternative method, with rather less generally, by T. Rad6 [14]. Although the title of the paper [3] specified "two contours," the formulas of that paper were developed for the general case. 4 The treatment of the general problem given in [1], [2], was based on these formulas and on the theory of abelian integrals and 8-functions on a Riemann surface of any finite genus. The present paper gives, first, a self-contained treatment of the problem, reviewing and considerably amplifying the essential features of the papers [1, 2, 3]. Further, it provides certain simple supplementary considerations,4a needed only in the case of characteristic greater than one, which serve to complete the presentation of the paper [2]. This is the case where "alter-symmetric" circuits 5 are present on the basic Riemann surface R, and the Green function of R acquires certain simple complementary terms. The effect of these is easily traced, 6 with the same final result of the existence of the minimal surface. Many of the notions and formulas concerning Riemann surfaces, which are developed incidentally in the course of the analytic treatment, are new. In our main theorem, 11, 7 the existence of the minimal surface Mr is established solely on the basis of the functional A(g, R) introduced by the author-closely related to the Dirichlet functional-without any reference to area or the use of the theory of conformal mapping. In theorem I, the least area property of Mr is then proved by employing the conformal mapping of polyhedral surfaces. In § 19 an alternative method of proving the combined theorems I and II is given, which uses this conformal mapping from the start. These theorems suppose the given contours capable of bounding some surface of finite area. Finally, § 17 contains a proof of a previously announced theorem, 8 expressing the solution of the Plateau problem for the case where the contours are perfectly general Jordan curves. Solely to fix the ideas, we may suppose that the given contours do not intersect one another, but our theory applies with practically no change to the case where mutual intersections are permitted. 9 The conformal mapping of multiply-connected plane regions, which, as throughout the author's work, is included in the Plateau problem as the case n = 2, is discussed in §18. The classic mapping theorem of Schottky is reestablished by the author's method, and combined with a proof of a topological correspondence between the boundaries. This new form of treatment of the Schottky theorem was stated by the author in [l]l0 and proved in [2]. §18 of the 4 (3), p. 324. 4a(Added in proof.) These have already been published in the preliminary note (31) to the present paper. 5 See §6.1. 6 See §12. 7 See §5. 8 [2], p. 108, Theorem IV. 9 Cf. §3.3, also [8]. 10 As Theorem II of that paper.

143


present paper is concerned with the details of proof of a certain essential inequality, (18.5). 2 The O-functions of the Riemann surface R enter through the expression of the Green function of R in terms of these functions. In a future paper, I intend to give a simpler and more generally applicable treatment, which will use the Green function in an intrinsic way without employing for it any explicit formula. lOa

3 For the definition of a minimal surface, we adopt the formulas given for n = 3 by Weierstrass: (i = 1, 2, ... , n)

(1.1) n

(1.2)

(w = u

+ iv).

These express only the first variational condition in the problem of the surface of least area with given boundaries, so that a minimal surface in this sense mayor may not have the least area for its boundaries. The minimal surfaces whose existence is established in the present paper will actually have minimum area for their boundaries and topological form. Thus, analytically expressed, our problem is to find a Riemann surface R of appropriate topological type, and n uniform harmonic functions .1t Fj(w) on it that obey (l.2) and represent a surface bounded by preassigned contours. IfE, F, G denote the fundamental quantities of the harmonic surface (1.1), then n

.1:

(1.3)

1

= I

F:2(w) = (E - G) -2iF. 1

Hence condition (1.2) is equivalent to (1.4)

E

= G,

F = 0,

expressing conformal representation of the harmonic surface on the Riemann surface R. It is to be observed that only .1t Fj(w) is required to be uniform; in general, Fj(w) itself will have p = r + k - 1 pure imaginary periods corresponding to various circuits on the Riemann surface. These are interpreted as translation lOa

(Added in proof.) The essential features of this alternative treatment have already been published in [32]. A detailed presentation will soon appear in the American Journal of Mathematics. See also [33].

144


periods of the adjoint minimal surface xi =:/{F,(w). One may consider the relation between the catenoid and its adjoint surface, the helicoid, where p = and there is a single translation period. 4

If the required minimal surface Mr be considered as having two faces, corresponding to opposite senses of the normal, then it becomes a closed surface, for the two surfaces are united at the boundaries. Regarded in this way, Mr can be represented on a closed Riemann surface R, whose genus, equal to that of the double-faced M r, is seen to the p = r + k - 1. When the number k of contours is arbitrary, then even in case r = 0, the genus p of R is capable of taking arbitrary values. For this reason, nearly all the essential features of the theory for surfaces of general topological form are already present in the case r = 0, i.e., where the topological type is that of a sphere with k perforations. The rather secondary nature of the modifications that result from a higher topological structure, as will appear in the sequel, constitutes perhaps one of the advantages of the present method.

5 Illustrations of minimal surfaces of higher topological type are given in the following diagrams. All are bounded by single contours. In fig. BOa, the minimal surface has the form of a torus from which a rather large simply-connected region has been removed. II Fig BOb shows a one-sided minimal surface of the form of a Mobius strip. Fig. BOc represents a standard form of surface of higher topological type; 12 here the genus h = 2. The contour is supposed to be almost in one plane. By increasing the number of double-bridges such as AIBp A2B2, we can produce a minimal surface of arbitrary genus. Fig. BOe is a standard representation l2 of the one-sided surface; here r = 3. The half-twisted strip S produces one-sidedness and contributes unity to the topological characteristic. More contours can be introduced arbitrarily by means of additional simple bridges, or by perforations in the surface. All these examples can be realized by the soap film experiments of Plateau. The same contours also bound simply-connected minimal surfaces. Thus in BOa, we have a minimal surface consisting approximately of two horizontal circular discs joined by a vertical catenoidal strip.13 There is a second simplyconnected minimal surface, where the circular discs are vertical and the catenoidal strip horizontal. Fig BOd indicates a simply-connected minimal surface bounded by the contour of fig. BOc. This consists, very approximately, of the circular disc + the heel-shaped regions PIBIQp P 2B2Q2 + the strips AI' 11

12

13

Cf. [2], p. 122. We first gave this example in our invited address to the American Mathematical Society, Oct. 1932, published as [6] (where, however, fig. 80a is not included). [20], p.153. Cf. fig. 91 b, with the narrow connecting part removed.

145


a

b

c

d

e

Figure 80

+ the shaded heel-shaped regions HI' H 2, which are to be imagined slightly above the plane of the paper. We can also visualize a minimal surface of genus h = 1 with the same contour by performing the modifications indicated in fig. BOd on only one of the double-bridges. Supposing HI> H 2, we have, very approximately, in the notation of(5.1): A2

=

(1.5)

a(r, 1)

(1.6)

a(r, 2) = a(r, 1) - 2H 2,

a(r, 0) - 2HI

which illustrate the sufficient conditions of Theorem I ( 5): (1.7)

a(r, 2) < a(r, 1) < a(r, 0),

that assure the existence of the minimal surfaces described. §2. Representation of Surfaces of General Topological form 1

This section is based on the fundamental and highly suggestive tract ofF. Klein, listed as [16] in the bibliography.14 14

Throughout this paper, the text-books [16-21] of the bibliography will be considered as current references.

146


Let R denote a Riemann surface having an inverse conformal transformation T into itself, where T associates the points ofR in pairs, so that T2 = 1 (involutory transformation). Such a Riemann surface, following Klein, is called symmetric. The locus of points fixed under T consists of one or several closed curves, called curves of transition. If R is considered as the Riemann surface of an algebraic curve P(x, y) = 0, with real coefficients, then T may be taken to be the interchange of conjugate complex points, and the curves of transition are the real branches of the algebraic curve. The transition curves may separate R or not; following Klein, R is accordingly said to be of the first kind or second kind. If we regard a pair of T -equivalent or conjugate points as a single geometric element, then R becomes a geometric manifold R' called a semi-Riemann surface, and R' is two-sided (orientable) or one-sided (non-orient able), according as R is of the first or second kind. The transition curves of R are the boundaries of R'. Often, when there seems to be little chance of ambiguity, we shall speak interchangeably of R and its semisurface R', but quite as often it will be important to distinguish carefully between the twO. 14a Any orientable or non-orientable surface S may be put into correspondence with a symmetric Riemann surface R of the first or second kind respectively, so that a point p of S together with its antipodal p correspond to a pair of conjugate points of R. (The antipodal point of p means the point I> geometrically coincident with p, but regarded as lying on the opposite side of S, i.e., with reversed sense of the normal.) The distinction between the first and second kind of Riemann surface corresponds to the fact that on a two-sided surface we cannot pass from any point to its antipodal without crossing the boundary, whereas it is characteristic of one-sided surface that such a passage is possible. In the form of the general symmetric Riemann surface R, we are provided with a complete system of standard domains of any finite genus or topological characteristic, with any finite number of boundaries, and either orient able or non-orientable. The completeness is from the standpoint of conformal mapping, that is, the 3p - 3 real conformal moduli (p = genus ofR) are capable of varying arbitrarily in the system. From our point of view, conformally equivalent Riemann surfaces are identical, and one may replace the other if convenient for any purpose. The conformal equivalence must respect the symmetry of the surface, that is, convert conjugate points into conjugate points. What is tantamount, the corresponding real algebraic curve P(x, y) = 0 is subjct to a real birational transformation, without change of anything essential to our discussion. 14.

Remark on notation: Thus, following our papers [1, 2), we use R in this paper to represent any complete, or closed, symmetric Riemann surface of finite genus, while R' denotes the corresponding semi-surface. This convention is followed as a rule with a few occasional exceptions, that need cause no real ambiguity. In subsequent papers [31-33), we have denoted the complete symmetric Riemann surface by an Old English or script 11. and the related semi-surface by R.


147

We choose R so that its semi-surface R', which results by identification of conjugate points, has exactly the topological form prescribed for the required minimal surface M, i.e., characteristic rand k boundaries, and also agrees with M in character of orientability. Then the genus of the complete surface R is (2.1)

p=r+k-l.

The definition of the topological characteristic r is: the maximum number of linearly independent circuits, where a zero circuit is considered as one which separates the surface. For two-sided surfaces, r = 2h is always even; for onesided surfaces, r may be any positive integer odd or even. IS The characteristic is also equal to the excess of the connectivity over the number of boundaries: c = r + k. Connectivity c means that there exist c - 1 successive cross-cuts which leave the surface connected, whereas every succession of c cross-cuts disconnects. A cross-cut is an arc each of whose end-points lies on a boundary or previously drawn cross-cut, while all the other points of the arc are interior points of the surface. We shall indicate the one- or two-sided character of our surfaces and quantities pertaining to them by the presence or absence of a bar. Thus we shall speak of surfaces Mr or Mr of the topological type r or F. Hereafter, we shall use r or F most frequently instead of the genus h, which applies only to orientable surfaces. Thus we may speak of the problem (r, r) or (r, F), putting in evidence the set (r) of given contours and the topological type of the required minimal surface. The following figure illustrates the case of a two-sided surface M, or symmetric Riemann surface R of the first kind, with k = 4, r = 4, therefore by (2.1), p = 7. Here the transition curves C = C I + C 2 + C 3 + C 4 separate R into two conjugate halves, and we may take R' to be the upper half. The inverse conformal transformation T may be pictured as the reflection in the plane containing C. 2 The one-sided case may be illustrated by the Riemann surface indicated in fig. 82. This is represented in the standard way as a double-sheeted surface over the z-plane. There are two branch-cuts, one C along the real axis, the other qq joining two conjugate complex points. We take the inverse conformal transformation T to be reflection in the real axis combined with passage to the other sheet. The points p. Ii in the figure represent T-equivalent points, if we take p in the upper, Ii in the lower sheet. It is evident that T leaves invariant every point of the branch-cut C, and no other points. C is therefore the unique

15

Examples: Mobius strip, r = I; with h handles, r = 2h + I; Klein surface (see footnote 25), r = 2; with h handles, r = 2h + 2.

148


Figure 81

g··7 Figure 82

l

'la_

~I

'I.

'1.

'Is

Figure 83 curve of transition; it is to be regarded as a closed curve, proceeding from a to b in one sheet and back to a in the other. Obviously, C does not separate R, for we can pass from p to p via the branchcut qq without traversing C, as indicated in the figure. If we identify all conjugate point-pairs pp, then R represents a Mobius strip bounded by C. Point-pairs such as pp correspond to a point on the Mobius strip together with its antipodal, and the one-sided nature of the strip is expressed exactly by the possibility of passing from p to p without crossing the boundary C. By adding more branch-cuts like qq joining conjugate complex points, and branch-cuts like ab along the real axis, we can obtain models for one-sided surfaces with any characteristic r and any finite number k of boundaries. The following figure represents such a surface with two boundaries and characteristic


149

three, where only the branch-cuts are indicated, along which the two sheets of the Riemann surface are joined. More generally, we may illustrate both the orientable and non-orientable cases with Riemann surfaces of any number of sheets, which admit an inverse conformal transformation T into themselves, consisting of the conjugate transformation: z into :l , combined with a substitution S of period two (S2 = 1) on the various sheets. 3 An alternative representation of one-sided surfaces Mr is based on the use of a two-sided covering surface, in the form of a semi-Riemann surface 1\' of the first kind16 with an inverse conformal transformation U into itelf, where U associates the points of R' in pairs and leaves no point of 1\' fIxed. Each pair of Uequivalent points of R' will correspond to the same point of Mr. This one-two correspondence between Mr and R' implies that the characteristic of1\' is 2r - 2 and that R' has 2k boundaries formed ofk pairs ofU-equivalent curves Cl' q; C 2, q; ... ;Ck, Ci,. The genus of the complete Riemann surface 1\ of which 1\' is one of the conjugate halves is (2.2)

p = 2r + 2k - 3.

In any parametric representation of the k contours r 1' ... ' r k on the transition curves C, each point of r j corresponds to a pair ofU-equivalent points on C j' C; respectively. For instance, a Mobius strip may be pictured in this way as a zone on the unit sphere x2 + y2 + Z2 = 1 bounded by the planes z = ± h, where diametral points of the sphere are U-equivalent.J7 Generally, we may picture 1\ as a Riemann surface of any number of sheets spread over the complex Z-sphere and consisting of two conjugate halves 1\',1\" separated by the 2k transition curves CI' q; ... ;Ck, Ci,. The inverse conformal transformation U of period two may be pictured as a diametral transformation z' = - liz combined with a substitution of period two on the sheets which compose 1\. The transformation U converts each half 1\', 1\" or 1\ into itself; besides, 1\ has an inverse conformal involution T which interchanges its two halves. T may be pictured as the reflection in the prime meridian plane, z' = z, combined with a substitution of period two on the various sheets of 1\. The transformations U and T together generate a four-group I, U, T, TU = UT of direct and inverse conformal transformations of 1\ into itself. 1\ may be regarded as a covering surface in two-one correspondence with the 16 17

That is, of the general type of the upper half of fig. 81. Cf. [4), p. 734.


150

previously used surface 1\ (art. 2, this section), where each point of 1\ corresponds to a pair ofT-equivalent points of1\, and V-equivalent points of 1\ correspond to conjugate points of 1\. By means of the preceding representation, the Plateau problem (r, r) is referred to (r, - r, 2r - 2), where - r denotes r with the reverse sense of description; that is, we have 2k contours consisting of both orientations of each of the k given ones. We seek a semi-Riemann surface 1\' with a Vautomorphism, 2k boundaries and characteristic 2r - 2, and upon R' a system of harmonic funcions.1U'i(w), which take equal values at U-equivalent points, obey n

. 1: FP(w) = 0, 1

= I

and

transform

the

boundaries

of 1\'

in

pairs

into

the opposite orientations of each contour r. Then xi =.'R.F.1.w) or M 2r _ 2' will coincide with the desired one-sided surface Mr of type F bounded by (r). The two-side~surface M 2r _ 2 is, in fact, a covering surface in two-one correspondence with Mr' The simplest illustration is the case of the Mobius strip (r 1'1), which we solved in the paper [4] by making it depend on the previously solved case [3] ofa doubly-connected surface with two given contours: (r I' - r I' 0). Here the two contours were the two opposite orientations of the single boundary assigned for the Mobius strip. The procedure there illustrated is completely typical of the general case. By these remarks, we are enabled, as far as the analytic treatment is concerned, to confine ourselves to the two-sided case, with merely due regard to preserving the V-automorphism of 1\ when the Riemann surface is varied.

4.

Reduction of R If the symmetric Riemann surface 1\ varies continuously, it may degenerate in various ways by the coalescence of branch = points, whereby certain branch-cuts disappear. IS In this way the semi-surface R' may separate into a number of disconnected parts, or its characteristic r may be decreased. In case of separation, the total characteristic of the components is evidently::;; r. We shall term either type of degeneration a reduction of 1\ or 1\'. Every reduction can be obtained by composition of primary reductions. A primary reduction of 1\ consists of the following. (1) In the two-sided case, the disappearance of a single pair of mutually symmetric branch-cuts. (a) If the disappearing branch-cuts are the only ones that unite two given sheets, the R' separates into two disconnected parts: R' = R; + R;, with no decrease of the total characteristic: r = r l + r 2. The k transition curves C, or boundaries ofR', are distributed between R; and R;: k = kl + k 2. We may have 18

Alternatively, two branch-cuts may units to form a single one, with the same effect.


151

k2 = 0, in which case R~ is a closed surface, and, as will be obvious, may simply be ignored for our purposes. If R~ is closed, we shall not call the reduction primary unless r2 = 2; ifr2 = 0, it may be considered that no reduction at all has taken place. (b) Ifbranch-cuts remain which join the same sheets as the disappearing ones, then the characteristic of R' is decreased by exactly two, or r - 2. (2) In the one-sided case a primary reduction will mean: (a) the disappearance of a single pair of mutually symmetric branch-cuts, provided that this produces separation of R':R' = R~ + R;. Here again r = r 1 + r 2. Then at least one of the components must be one-sided and both may be. (b) the disappearance of a single self-symmetric branch-cut, i.e., one which joins a pair of conjugate branch-points. This reduces the characteristic by one, to r - 1. If r - I is even, the reduced surface may be two-sided or one-sided; if r - I is odd, it must be one-sided. We shall denote primary reduction by an accent: r', r, and the general type of reduction by r", r', interpreting the latter so as to include the former. In a composite reduction of a one-sided surface, there may be any distribution of one or two-sided components. In terms of the corresponding algebraic equation P(x, y) = 0, the separation of R is equivalent to reducibility in the usual algebraic sense: (2.3) The lowering of the genus of R corresponds to the acquisition of new double points produced by the coalescence of branch-points.

5.

R' as a circular region

A useful alternative representation of a semi-Riemann surface R' is as the fundamental region of a group L of linear fractional transformations of z or i;. The k boundaries ofR' correspond to circles C p C 2, ... , C k. If the genus h = 0, R' is simply the region bounded by these k circles, and L is generated by the inversions in them: Sp S2' ... , Sk. IfR' is two-sided and h>O, we must adjoin to L a set of h linear fractional transformations of z. This gives in the fundamental region h pairs of circles Kp Kj; ... , K h, Kit, whose points are coordinated by the corresponding linear fractional transformations. If R' is one-sided, of characteristic r = 2h + I or 2h + 2, then we adjoin to Sp S2' ... , Sk a set of h linear fractional transformations of z and, respectively, one or two linear fractional transformations T of z such that T2 = 1. The latter contribute to the fuindamental region their fixed circles, whose points are coordinated by T. The complete symmetric Riemann surface R is obtained by adjoining to R' its inverse image in anyone of the bounding circles C p C 2, ... , C k, which form then the curves of transition.

152


For instance, a torus may be represented by a circular ring, on whose bounding circles radial points are coordinated. The group L is here generated by z' = qz, if the circles are about the origin with radii I, g. Any number of boundaries on the torus may be represented by circles interior to the ring; we adjoin to L the inversions in these circles. A Mobius strip is represented by a circular ring, where diametral points of one of the bounding circles are coordinated. Equivalent to this is a zone on a sphere, where one of the bounding circles is a great circle on which diametral points are coordinated. Adjoining to this the diametral zone, we have a representation of the two-sided covering surface 1\ previously spoken of, on which V-equivalent points are diametral points and correspond to a single point of the Mobius strip. This circular representation is quite simple and concrete, but we may observe the following advantages of the many-sheeted Riemann surface. The first consists in the fact that any separation of the Riemann surface is represented directly in a visible way, while in the circular region this separation corresponds to an approach to the same point of a number of circles. Thus in the latter case, only one of the components remains visible, the other disappears; in the former, both are on a par. This circumstance is important in defining certain basic functionals depending on R in the case when R reduces. Again, in the Riemann surface representation, we have the choice of an extremely wide variety of conformally equivalent forms of a given surface, for the fundamental group is then the total real birational group, with an infinite number of parameters. In the circular representation, we have only the sixparameter group of transformations z' = (az + b)/(cz + d). We can employ this greater freedom in the former case to avoid disadvantageous types of singularity, such as reduction to a point of a curve of transition, or real branch. For the group of real birational transformations is sufficiently large and effective to enable us to neutralize any such tendency by applying, roughly speaking, a kind of balancing magnification. In the circular representation any such attempt reduces some of the previously non-degenerating circles to points (sometimes at infinity). For instance, in the two-contour case, we may use for R' the circular ring bounded by C 1: Izl = I, C 2: Izl = q < 1. There are two forms of degeneration of R', namely q = 0, q = 1. In the former case, the circle C 2 disappears, in the latter the ring itself vanishes. In contrast, we have the representation of R' as a semi-Riemann surface of a real algebraic curve of genus one with two real branches. A system of such Riemann surfaces, complete from the standpoint of conformal and symmetric equivalence, is represented, for instance, by the family of cubic curves: (x 2 + y2 - l)(x - 2) =m, (2.4)

153


In this range of values of m, the curve has no double point; therefore, being cubic, it is of genus one. There are always two real branches, on which the two given contours may be represented parametrically. The modulus m corresponds to the ratio of the radii q in the circular representation. At one extreme, m = 0, the curve is reducible, separating into a circle and a straight line, each of genus zero with a single real branch. This corresponds to q = 0. At the other extreme, m = mo' the two real branches touch and the curve acquires a real double point. This corresponds to q = 1. We note that in both singular cases, the Riemann surface remains completely tangible in all respects. The figure shows the extreme curves m = m = mo drawn full, and an intermediate one dotted. We may observe that in the representation as a many-sheeted surface over the complex x-plane, one of the real branch-cuts or transition curves disappears, shrinking to the point x = 2, and the same happens to the corresponding component of the reducible surface. But if we pass to the representation over the y-plane-as we may do with preservation of conformality and symmetry-then there are two actual simplyconnected components: one with branch-points at y = ± 1, the other a simple yplane, and the second transition curve becomes the entire real axis of this plane. This illustrates how easily undesirable types of degeneration may be avoided by conformal or birational transformation of the many-sheeted Riemann surface, where this is not possible for the circular representation. The presence of infinite real branches can be avoided, if desired, by means of an inversion.

°,

Figure 84

19

Cf. [18], pp. 410-444; also [19]. The essential feature of this normal form is that all branch-points are of first order, like that of.Jz.


154

Finally, if two of the contours r i rj have a point in common, we can represent this in the circular form by two tangent circles CiCj. But, obviously, we cannot in this way represent more than two contours with the same point in common. On the other hand, in the representation by means of a Riemann surface or algebraic curve, we can arrange for real multiple points of any order of multiplicity. For all these reasons, we shall employ in the sequel the representation by means of a Riemann surface. We may even restrict ourselves to the Clebsch-Liiroth normal form. '9 This is apart from a few occasions where the use of circles seems more convenient, but in every case the situation can be recast in terms of Riemann surfaces. Besides the many-sheeted form of R, we may consider the two-dimensional manifold S2 which is the locus in real four-dimensional space x" x2' y" Y2 of the equation P(x, + iX2' y, + iY2) = o. incidentally S2 is itself exactly a minimal surface in the four-space. Its othogonal projection on the x = x, + iX2 plane or the y = y, + iY2 plane is a Riemann surface over that plane, on which S2 is represented conformally by the projection.

§3. 1.

Orientation of the Given Contours The 2k -

1

different cases

The contours (r) must be given not only by themselves, but also each with a definite sense or orientation; in fact, a given set of k contours (r) gives rise to 2k - , distinct forms of the Plateau problem, corresponding to that number of essentially different orientations of the set. Consider first the two-sided case of a minimal surface Mr. Suppose the boundaries (C) = (C" C 2, ... , C k) of R' are oriented so that R' is on the left. Then we may orient each of the contours r" r 2' ... , r k in a definite way, and in representing (r) parametrically allow only those topological representations of each r on the corresponding C which preserve the sense, as preassigned. This is equivalent to demanding that after choice of a definite sense of the normal of Mr (which is possible, because of its two-sidedness), the contours r" ... , r k in their preassigned orientations, shall have Mr on their left, from the viewpoint of one standing on the side of the surface indicated by the directed normal, and facing in the sense assigned to each contour. It is evident that if we reverse the direction of the normal to M r , and also that of all the contours r" ..., r k simultaneously, we still have a minimal surface of topological type r bounded by the given contours, and lying on their left. Therefore, it makes no essential change in the problem to reverse the sense of all the given contours simultaneously; this has only the effect of reversing the sense of the normal on the required minimal surface, and on all the surfaces whose areas are admitted to comparison. However, if we change the sense on some but not all of the contours r" ... , r k'


155

then, whichever direction we choose for the normal to M r , this surface no longer lies to the left of all the contours r I' ... , r k; i.e., it no longer furnishes a solution of the Plateau problem for the changed orientation of the given set of contours. It is evident that we may keep flxed the sense on one of the contours r I' but there are then exactly 2k - 1 different forms of the Plateau problem co:mected with the given set (r), associated with that many possible combinations of sense on the other k - 1 contours r 2 ... , r k' These 2k - 1 problems are really quite different; for instance, a given s~t of contours (r) may bound a proper minimal surface of a prescribed topological type in one system of orientations but not in another. The simplest illustration is k = 2, r = O. Here we have two given contours r I' r 2' and two essentially different Plateau problems, according to the sense which we assign to r 2 after that of r 1 has been flxed arbitrarily. Take, for instance, two co-axial circles in planes whose distance apart is sufficiently small as compared with the radii. Then if the circles are sensed as in flg. 85a, the minimal surface which is bounded by them, on their left exists, properly, being the catenoid with normal directed as in the flgure. If, on the contrary, the circles are sensed as in flg. 85b, then no proper doubly-connected minimal surface exists which they bound on their left; the only possible solution is the pair of circular discs with normal directed upward.

a Figure 85

If, for simplicity, we admit to comparison only surfaces of revolution, the difference between the two problems may be expressed as follows. In flg. 86a, the points P, Q are given on the same side of the X-axis; we require a curve joining these points, which, when revolved about the X-axis will generate a minimum area. The solution is the catenary PQ. In flg. 86b, we have the same problem with the points P, Q' on opposite sides of the X-axis, the point P being in the same position as before, where Q' is the reflection ofQ in the X-axis. Here the solution is the broken line P ABQ', as can easily be proved by elementary methods, with ise of the observation that the length of any curve joining P and Q' is greater than PA + Q'B. It is evident that if we revolve about xx' any curve

156

THE PLATEAU PROBLEM: PART ONE p

p

IJ

X'

B

A

x

x'

x

A

fJ..' b

a Figure 86

joining PQ or PQ' respectively, we obtain in either case a surface of revolution bounded by the same two circles, with respective centers A, B and radii PA, QB = Q'B. The situation may also be expressed, quite concretely, as follows. Let us take a flexible and extensible tube, as in fig. 87a, whose ends are fixed to, say, two metal rings r I' r 2 clamped in position in space. In this way a surface ~ is formed bounded by r I' r 2 and on their left, with the orientations of r I' r 2 and the normal to ~ as in the figure. Now release the connection of the tube to r 2' keeping it fixed to r I' and then stretch and bend the tube as in fig. 87b, bringing it down from above to be again attached to r 2' We obtain a surface ~' bounded on the left by the same contours r I' r 2 but in a different orientation: + r I' - r 2' Now we may put to ourselves two entirely distinct problems, namely: (a) of all surfaces derivable from ~ by continuous deformation, 20 the boundaries r I' r 2 remaining fixed, to find a minimal surface M of least area; (b) the same for all surfaces derivable similarly from ~'. In the process of deformation, we may also allow the boundaries of ~ to vary, provided that they return to their original positions induding sense. This enables us to turn a surface "inside out," e.g., to pass from ~ to ~" (fig. 87c). The normal of~" is directed towards the interior of the tube. If r I' r 2 are the two circles considered in fig. 85, then the solution of problem (a) is the catenoid, while that of problem (b) degenerates into the sum of the two circular discs. Quite as often, however, we obtain two distinct proper solutions to the problems (a), (b). Consider, for example, two circles in perpendicular planes, with coincident centers and unequal radii. Form the single contour consisting of the line-segment AB, the upper semi-circle of r 2' the line segment CD, the forward semi-circle of r l' This contour bounds a minimal surface of least area, Mi', lying entirely in the first quadrant, as we may call it. Let this surface be rotated through 180 0 about the diameter AD; then we get a minimal surface Mi', lying in the third quadrant, bounded by AB, CD and the complementary semi20

Allowing any type of self-intersection in the process.


b

a

c Figure 87

Figure 88

157

158


circles of r l' r 2' According to the Schwarz symmetry principle for minimal surfaces, 21 M~ is the analytic prolongation of of M~ across AB and CD. Therefore Mj + M~ = MI is a single analytic doubly-connected minimal surface, bounded by the circles r l' r 2' If these circles are oriented as indicated by the arrows marked MI in the figure, th~n the surface MI' with its normal properly directed, is on the left. Let MI then be reflected in the plane of the vertical circle. We obtain a minimal surface M 2, also bounded by r l' r 2' and while the sense r I is preserved, the sense of r 2 must be reversed as indicate ~ by the arrow mark-d M 2, in order to keep on thl left the surface M2 directed as obtained by the reflection from MI' If the vertical circle be moved to the right till it interlaces with the horizontal one, the preceding construction gives two (inversely congruent) doublyconnected minimal surfaces bounded by these interlacing contours. This illustrates the general theorem given by the author in [3], p. 351, according to which any two interlacing contours bound a doubly-connected minimal surface. In the present example, contrary to what one might be inclined to guess, neither of the minimal surfaces is self-intersecting. The reader will find it easy to illustrate the 22 = 4 cases corresponding to the different orientations of three given contours. 2.

One-sided surfaces Consider next the case of one-sided surfaces. The figure illustrates the two ways in which we may interpret the problem of a surface of least area with characteristic one bounded by two given contours r l' r 2 (Mobius surface with two boundaries). Here there is a proper solution only in the case of fig. 89a; in fig. 89b the solution degenerates to a Mobius strip r I and a circular disc r 2' However, if r 2' supposed to be a circle is placed with its center on the Mobius strip determined by r l' then, as I have proved, there exist proper one-sided solutions of characteristic one in both cases (a), (b). Here there can be no question of ascribing a sense to each contour so that the

a

Figure 89 21

[28], p. 175; [29], p. 397. Actually, a stronger form ~fthis sym~etry pri~c!ple is necessary for the present purpose; see J. Douglas, The analytIC prolongation of a mInimal surface over the rectilinear segment of its boundary, Duke Mathematical Journal, Dec., 1938.

159


surface M is on the left, when the normal to M is properly directed, since there is no way of distinguishing over the entire surface M one direction of the normal from the other; indeed, this fact constitutes exactly the definition of a one-sided or non-orientable surface. The two senses of the normal are interchangeable when followed continuously over a properly arranged circuit. We may say here that each contour occurs with both orientations. In the representation by means of the covering surface R' with k pairs ofU-equivalent boundaries C p Cj; ... ; C k, Cit we have to decide which orientation ofrj shall go with C j and which with This gives again 2k - 1 essentially distinct possibilities, since the first choice is indifferent.

cr.

3.

Contours with common points For definiteness, we shall suppose that the given contours (r) are k Jordan curves which do not interest one another. However, our theory and results are easily adapted to the ease where the contours have points in common, provided no separate contours can be formed by combining arcs of the given ones. That is, we may have a situation like fig. 90a but not like fig. 90b. All we need do in this case is to require the symmetric Riemann surface R to have the system of its transition curves (C) topologically equivalent to the system (r) of given contours, that is, for example, in fig. 90a, the corresponding algebraic curve P (x, y) = 0 must have exactly two real branches, one with a triple point and a double point corresponding to Q3 and Q2 respectively, the other with a double point corresponding to Q2' We have already shown in detail how this works out in the case of two contours with a single point in common, and where the minimal surface is required to be simply-connected. 22 The corresponding algebraic curve might be taken to be,

a

b Figure 90

22

[8]

160


e.g., a lemniscate, to whose real double point the common point Q of the two contours always corresponds. Actually, we used as semi-Riemann surface a parallel strip of the z-plane, whose infinite point corresponded to the intersection point Q.23 An alternative method of regarding fig. 90a is to consider that we have two contours r 1 + r 2 + r 3 + r 4 and r 5 + r 6' which, however, are not Jordan curves, but have multiple points. These can be taken account of by means of the sufficient condition (5.11) and (5.28), with the same final result as to the existence of the required minimal surface. In short, we require the semi-Riemann surface R' to be topologically equivalent to the required minimal surface Mr in all respects, including transition curves (C) as compared with given contours (r); we may say that R' is of type (r, r). If then, the k curves r l' r 2' ••• , r k which compose (r) have no multiple points either individually or mutually, then (C), as referred to the corresponding algebraic curve P (x, y) = 0, shall consist of k real branches without multiple points. On the other hand, in case the system of contours (r) presents multiple points, these must be represented as points of equal multiplicity on the system of real branches (C). §4. Surface of Assigned Topological Type Bounded by Given Contours 1.

Intuitive considerations Let the surface Sr bounded by the contours (r) vary continuously in an arbitrary way, while the boundaries remain fixed. Then, in the limit, this surface may resolve itself into separate parts, or its topological characteristic may be reduced, for example, by the closing up of an opening or by a handle of the surface shrinking to a curve. Also a closed part of Sr may simply break off, giving a closed component without relation to the contours (r). These processes may occur several times successively or in any combination, so that we may obtain most generally as limit of a surface Sr bounded by (r) a surface S" consisting of m parts having a total characteristic ~ r. If either m > 1 or the characteristic is really < r, we say that S" is the r~uced type r" and write it S; Similarly, a one-sided surface Sr boul!..ded by (r) may reduce, in varying continuously with (r) fixed, to a surface S" r consisting of one or several parts, whic!! may be one-sided or !wo-sided, having a total characteristic ~ r. We say that S" r is reduced, or type r", if it is really reduced, i.e., consists of more than one part or has a total characteristic less than r. 2.

More exact definitions Suppose given a set ofk contours (r), each with a definite sense of description. We define a surface Sr (or Sr) of characteristic r (or t) bounded 23

Cf. x 10.2.


161

by (r) as follows. Let R be any symmetric Riemann surface whose semi-surface R' has the characteristic r (or t) and k boundaries (C). First, as stated in §3.3, these must be topologically equivalent to (r); for definiteness, let no two contours r and no two transition curves C intersect. On R then define any onevalued continuous vector function r (u, v) which takes equal values at conjugate points of R, and which transforms the transition curves (C) in a monotonic continuous way into (r). This means that the correspondence established by r between (C) and (r) is either one-one continuous, or deviates from this, at most, by allowing any number 24 of partial arcs of one or more transition curves to correspond to single points on the related contours. The reference is to the depiction of the correspondence by a monotonic graph as in fig. 21 (in the original text-tr.). If the correspondence is one-one in any particular case, this will be especially proved. Then the locus of the point r (u, v) in the n-dimensional space will be called a surface of characteristic r (or t bounded by (r). This surface Sr (or Sr) is two-sided or one-sided according as R is of the first or second kind in the Klein sense. The stipulation as to the coincidence of values of r (u, v) at symmetric points makes r a one-valued function on the semi-sur...face R', and we may regard r as transforming R' topologically into Sr (or Sr)' This definition gives Sr (or Sr) in a particular representation. From this, its most general representation may be derived by a topological transformation of R into itself or a homeomorphic Riemann surface with preservation of conjugate points; in other words, the semi-surface R' undergoes a topological transformation. If R is reduced, of type r' or r" , then in case of separation of R, the continuity of ~ (u, v) applies, of course, only to each component part. We have then defined a surface of the reduced type r' or r" -i.e., primary reduced or general reduced-bounded by (r). Similarly we have in the one-sided case reduced surfaces of type F', F" bounded by (r). 3.

Reduction of surfaces The simplest types of reduction are illustrated in the next figure. Fig. 91a illustrates a doubly-connected surface bounded by r I' r 2 about to reduce to the

a Figure 91 24

Finite or denumerably infinite.

b

162


sum of two separate simply-connected ones. Fig. 91 b shows a surface of torus type about to reduce into a simply-connected one. In fig. 91c, we start with the Riemann surface for .JZ and delimit a branched portion by means of the contour r. Then we replace the branched part near the origin, bounded by 'Y, with a Mobius surface bounded by 'Y, as indicated. This gives a Mobius surface bounded by r. If now 'Y shrinks to the branch-point, this Mobius surface reduces back to the original simply-connnected branched surface.

Figure 92 A more general type of reduction is illustrated in the next figure, which represents schematically the reduction of a surface S6 bounded by five given contours into three separate parts with reduction of the total characteristic to two. One of the three handles, which gave the original characteristic six, has shrunk to a curve; the opening which produced another has shrunk to the point P; the third handle remains. The connections of the different parts have shrunk to curves, which may simply be removed, as well as the withered handle. We may also imagine additional components in the form of closed surfaces attaching at isolated points to those pictured, and these closed components may simply be broken off, without producing any new boundaries.

4.

Reduction of one-sided surfaces Starting with either a two-sided or a _ one-sided surface Sr of Sr' we may convert it into a one-sided surface Sr + 2 with an increase of two in the characteristic, by attaching a l.!.andle joining opposite sides of Sr' 25 or simply any handle in the:... case of Sr' already one-sided. Conversely, if this handle degenerates, Sr + 2 goes back to Sr or f with a reduction of two in the characteristic. Often, however, it is desirl!Ple to reEuce by exactly one unit the characteristic of a one-sided surface: _ Sr to Sr _ I or Sr _ I' and to do this by continuous deformation of Sr'

25

Fig. 93 illustrates a Klein surface with a single boundary. A Klein surface is one obtained from a sphere by attaching a handle, one end of which is joined to the outside, the other to the inside of the spherical surface. A closed Klein surface is necessarily self-intersecting, in three dimensions. See [21].


163

Figure 93

Figure 94

We first consider, conversely, how to build Sr _ 1 or Sr _ 1 up to Sr' This can be accomplished by the constf!:!ction of a cross-cap, 26 as follows. We pinch up a small portion of Sr _ 1 of Sr _ 1 into the form of a cone. Then we split this cone part way down along two elements, and join the opposite edges of the two slits cross-wise in the manner of a Riemann surface. The figure indicates the successive stages of the process. It is evident that by traversing the branch-cut, we can pass from any point on the surface to its antipodal. Thus the surface has become one-sided, if it were not already so, and the characteristic has been increased by unity, corresponding to the addition of exactly one new circuit, leading from any point to its antipodal via the branch-cut. Conversely, if the cross-cap disappears by continuous variation-the split reducing to a point-the characteristic is reduced by unity, and the surface may be converted from one-sided to two-sided. This will happen if the one-sided nature is due solely to the cross-cap under consideration, and not also to topological phenomena elsewhere on the surface, i.e., other cross-caps or reverse handles. Fig. 95 illustrates, by means of cross-caps, a Klein surface with two given boundaries. The next figure shows all the forms of primary reduction. In order, these are (a) simply-connected surface r 1 + Klein surface r 2' (b) Klein surface r 1 + simply-connected surface r 2' (c) Mobius surface r 1 + Mobius surface r 2' 26

Kerekjart6 (20), Kreuzhaube.

164


Figure 95

a

c

b

d

Figure 96

- ....'

.~.~ ." ~.>.'. /'.~:':~'~".'~>~" ~\ ~ ....

....•. -:""',,::- .-: and \V of the complex C into C' is given (where


174

if a collection of homomorphisms D k : C k Ck+ I is given so that, for any k, we have: Dk _ I dk + dk+ I Dk = Q>k - \Ilk. Two chain mappings Q> and \II connected by a chain homotopy are sometimes said to be chain homotopic. It follows from the definition of homology groups that chain homotopic mappings induce the same mappings of homology groups Hk (C) into Hk (C'), where Hk (C) = Ker dk/lmd k + I· To compute homology, the so-called exact sequence of the pair is useful. Let topological spaces X, Y be given, where Y is a closed subspace of X. It is clear that C k (Y) C C k (X), and we can consider the group of relative chains Ck (X)/C k (Y) = C k (X, Y). Since the boundary operator acts as follows:

it induces an operator C k (X,Y)--+Ck _ I (X,Y) which, for simplicity, we denote by the same symbol dk. The groups Ker dk = Zk (X, Y) ::J Bk (X, Y) = 1m dk (relative cycles and relative boundaries) can be defined, which makes it possible to consider the factor group Hk (X, Y) = Zk (X, Y)/Bk (X, Y)

called the group of relative k-dimensional homology of the space X modulo the subspace Y. It can be easily verified that, for relative homology groups, topological and homotopy invariance again hold. We now pass to the construction of the new operator d: Hk (X,Y)~Hk _ I (Y). Let zk €Ck (X,Y) be a relative cycle, and Z k € C k (X) an arbitrary representative of it in the coset. Since ok Zk = 0, dk Z k € C k _ I (Y). We denote this absolute cycle by dZk€ Zk - I (Y). The homology class of this cycle does not depend on the choice of representative from the homology class of the cycle zk. We can therefore define in this way a homomorphism (operator) d: Hk (X,Y)--+Hk _ I (Y) which we shall also denote by d, and call a boundary (homomorphium operator Fig. 98). Further, denote the embedding by i: Y --+ X. Then it induces a homomorphism i.: Hk (Y)~ Hk (X). Since any absolute cycle can be considered to be relative (modulo the subspace Y), this gives rise to the natural mapping:

THEOREM I. The following sequence of groups and homomorphisms is exact, i.e., the image of an antecedent homomorphism coincides with the kernel of the subsequent homomorphism at each of its terms: Hk

+ I

(X,Y) 0 Hk (Y) i* Hk (X) j* Hk (X,Y) 0 Hk -

I

(Y)

Proof. Let us verify the exactness, e.g., at the term Hk (X,Y). If 001 = 0,01 € Hk

FACTS FROM ELEMENTARY TOPOLOGY

175

x

Figure 98

(X, Y), then oZk is homologous to zero in Y (where Zk is the representative of ex, i.e., Zk € Zk (X, Y)). But then, by adding to zk a k-dimensional chain that realizes this homology in the subspace Y, we obtain a k-dimensional chain which is a cycle already from the point of view of the space X, i.e., we have represented the element ex as the image of an element {3, i.e. ex = i. {3, {3 € Hk (X) (Fig. 99). Thus, Ker 0 C Imi •. The reverse inclusion follows from the fact that any absolute cycle can be considered as a relative one in X, having zero boundary in Y. The exactness of the sequence for other terms is verified by a similar argument. For the sequel, it is useful to know the following properties of exact sequences (we leave the proofs to the reader): (1) The sequence O..... A..... O is exact ifand only

Figure 99


176

if A = O. (2) The sequence O--.,A~B--.,O is exact if and only if the groups A and B are isomorphic, and the homomorphism a is an isomorphism. (3) The sequence 0- A.J.B.lItC_O is exact if and only if the group A is a subgroup of the group B, the homomorphism i: A ~ B being an inclusion (monomorphism), C = BfA (factor group), and 11": B~BfA the natural projection onto the factor group. It turns out that relative homology can be reduced to absolute homology.

2. Cell complexes, barycentric subdivisions. IfX,Y are path-connected, then Ho (X,Y) = o. A topological space X is called a cell complex if it is representable as the union of disjoint subsets ok called cells, the closure of each cell ok being the image of a closed k-dimensional disc Dk under a certain continuous mapping (called characteristic) which is a homeomorphism onto the interior of the disc. Further, the boundary of each cell (i.e. the image of the boundary of the disc under the characteristic mapping) must be contained in the union of a finite number of cells of lesser dimensions, i.e., not exceeding k-l. Finally, it is required that a subset in X should be closed if and only if all full inverse images (under the characteristic mappings) of the intersections of this subset with all the cells are closed. A cell complex is said to be finite if it consists of a finite number of cells. The union of cells of dimensions not exceeding k is called the k-dimensional skeleton of the complex. We will say that a pair of spaces (X,Y) is a cell pair if X and Yare cell complexes, and the closed subspace Y is a cell subcomplex in X. We confine ourselves to considering finite complexes. Denote the space obtained from X by identifying the closed subspace Y with one point by X/Y. THEOREM

2.

Let (X, Y) be a cell pair. Then Hk (X,Y) = Hk (X/y) when k

:1=

O.

Proof. Denote the cone over Y by CY, i.e., space obtained from the cylinder Y x I by contracting the upper base to one point (Fig. 100). Construct a new space X U CY, i.e., identify the subspace Y in X with the base of Y in the cone CY (Fig. 101). Since X,Y are finite complexes, by contracting the cone CY on itself to a point, we obtain a homotopy equivalence: X U CY "'" XIY (Fig. 102). Thus, to prove the theorem, it suffices to establish that, for k > 0, the relations hold: Hk(X, Y) = Hk (X U CY). It follows from the exact sequence of the pair (X U CY, *), where * is a point, viz., the vertex of the cone, that Hk (H U CY) = Hk (X U CY,*) for k > 0; therefore, we have to prove that

for k > O. Before the proof, we shall need an argument relating to barycentric subdivisions.


177

-

·..·:~:2:··~r:?}~~z(~::-~· y Figure lOO

Figure 101

Figure lO2 Let Ak be the standard simplex. Its barycentric subdivision (3 Ak is defined by induction as follows. If k = 1, then (3 A I is obtained by adding a new vertex in the middle of the line segment AI, Ifk = 2, then (3 A2 is obtained by joining the centre of the triangle to its vertices and the mid-points of its sides (Fig. 103). And, finally, for an arbitrary k, the subdivision (3 Ak is obtained as follows: mark

178


pAi O------~O~-------O

Figure lO3 the centre of .:;lk, and break the simplex .:;lk into pyramids with the vertex at this centre and the bases-simplexes of the barycentric subdivision on the boundary of the simplex .:;lk. Now let f: .:;lk~X be an arbitrary singular simplex of the space X. We denote the chain equal to the sum of all singular simplexes obtained by restricting the original mapping f to the k-dimensional simplexes of the barycentric subdivision f3.:;lk of the original simplex .:;lk by f3f. Associating each singular simplex f (Le., elementary chain in C k (X)) with the singular chain f3f, we obtain the homomorphism 13k: C k (X)-+C k (X). We claim that the collection of mappings 13 = {13k} defines a chain mapping 13 of the chain complex C(X) into itself; moreover, it is chain homotopic to the identity. That the mapping 13 is chain follows from the fact that the restriction of the formal sum of singular simplexes to the barycentric subdivision equals the formal sum of these restrictions. We now construct the chain homotopy D connecting the mapping 13 with the identity explicitly. Define the division of the prism .:;lk xl into the sum of simplexes for each k > 0 as follows. This division is shown in Fig. 104 for k = 0, 1, 2. The inductive process of division of .:;lk x I can be described as follows. If the division is already defined for q < k, then it should be taken on a part of the boundary of .:;lk x I, viz. the union ("cup") (.:;lk x 0) U (a.:;lk x I) so that .:;lk x 0 may be the standard simplex, and a .:;lk x I divided in accordance with the inductive process when q < k. After that, we decide the whole prism into kdimensional simplexes whose bases are (k-l)-dimensional simplexes of the indicated division of the part of the boundary, and the vertex is the centre of the upper face. It is clear that the upper face will then undergo a barycentric subdivision. Now, let f: .:;l~X be a singular simplex. We denote by Dk f E C k + I(X) the (k + 1)-dimensional chain, that is the sum of all (k + 1)-dimensional singular simplexes obtained by restricting the mapping : .:;lk x I Y, where (x, t) = f{x), to the simplexes of the above-constructed division of the cylinder .:;lk x I. We obtain the homomorphisms D k: C k (X)~Ck + I (Y) which define the required chain homotopy connecting the identity mapping of the complex C(X) onto itself with the barycentric mapping 13. The proof that the mappings {Dk} give a chain


179

Figure 104 homotopy is left to the reader. We now pass to the proof of the theorem. Consider the pair-embedding mapping (X, Y)~ (X u CY, CY). It induces the homomorphism a: Hk (X, Y)-+Hk (X U CY, CY) = Hk (X U CY, *). We have used the fact that CY .., *, i.e., the cone is contractible to a point. We next prove that a is an epimorphism. Let z EO Zk (X U CY, CY) be some cycle. We have to find in the group Zk (X, Y) a cycle which will be carried into a cycle homologous to z in Zk (X U CY, CY) under the indicated mapping. Represent the cone CY as the union of two subsets: a cone A, i.e., part of CY consisting of points, for which t consisting of points for which t

~

~

!

!,

and a truncated cone, i.e., part of CY,

(Fig. 105) Reducing the singular simplexes

making up the cycle z by means of barycentric subdivisions, we each time obtain cycles homologous to it, but consisting of ever finer and finer simplexes. Since

180


Figure 105

the number of simplexes is finite, and X, Yare compact, there exists a sufficiently fine barycentric subdivision such that if any singular simplex of this subdivision such that if any singular simplex of this subdivision intersects A, then it wholly lies in the cone CY. Here we consider the image of the standard simplex under the mapping specifying a singular simplex. This cycle z' is homologous to z. Discard all those simplexes which intersect A from the cycle z' . Since this operation is performed within the cone CY, from the point of view of relative homology the new cycle z" remains in the same homology class as z'; hence, z also. Thus, z" E Hk (X U B, B,). At the same time, because of the homotopy invariance of homology groups, we have: Hk (X U B, B) = Hk (X, Y), since (X U B, B) "" (X, Y) (Fig. 106). Thus, we have explicitly exhibited a certain cycle z" E Hk (X, Y), which is carried into the cycle z EHk (X U CY, CY) by the homomorphism CY. The surjectivity of CY is thereby proved. The proof of monomorphicity is performed in an analogous way, and is left to the reader. See Fig. 107.

3. Cellular homology and computation of the singular homology of the sphere. We have described above one of the ways of introducting homology groups, using singular simplexes. However, for cell complexes, the so-called cellular homology can be defined, which coincides, as it happens, with singular homology, but possesses one important advantage: cellular homology is essentially easier to compute. Even in the simplest cause where the space consists of one point, the computation of singular homology requires a certain (but elementary) argument. If, however, the space is arranged in a more complicated way, then the computation of its singular homology rapidly becomes more complicated. In practice, cellular homology is used particularly often precisely for this reason.


181

Figure 106

Figure 107 We introduce it on the basis of the concept of singular homology already known to us, which will then enable us not only to prove the coincidence of singular and cellular homology groups, but also specify an explicit scheme for their computation. First, we compute the singular homology of the sphere. Since the zero-

182


dimensional sphere consists of a pair of points, we obtain: Hk (SO) k ;:0, and Ho (SO) = ZESZ.

=

°

when

LEMMA 1. The singular homology groups of the n-dimensional spheres sn, where n > 0, are of the form: Hk (sn) = when k ;:0, n, and Hk (sn) = Z when k = 0,

°

n.

The proof follows from the consideration of the exact homology sequence of the pair (Dn + I, sn) and from the fact that a disc is contractible to a point. Recall the definition of the wedge of two topological spaces. Let two points xo and Yo' respectively, be distinguished in two spaces X and Y. Construct a new space X V Y by identifying these points (Fig. 108). No other points are identified.

y

x XvY Figure 108

LEMMA 2. Let X = ViSf be the wedge of the n-dimensional spheres Sfwith the subject i, where 1 :s; i :s; N. If n > 0, k > 0, then the isomorphism Hk (X) = i Hk (SP) = z ... Z (N times) holds. For the proof, it su.ffices to consider the exact sequence of the pair (Vi Df; Vi oDf) with Vi (DfIODf) = Vi Sf(Fig. 109). Note an important fact for the sequel. We can take as a generator in the group Z = Hn (D,sn - I), the homology class of the simplest singular chain l·f, where f: Lln Dn is a homeomorphism of the simplex on to the disc. Therefore, the orientation of the sphere can be specified by fixing a generator in the group


183

Figure 109 Z = Hn (sn). A change in orientation is equivalent to the replacement of the element 1 by the element - 1. We now define cellular chain groups. Let X be a finite cell complex. Let us try to compute its singular homology in terms of cells and their characteristic mappings, i.e., in those terms in which the complex is given. The set of all kdimensional cells of a complex X will be denoted by X k. Let Xk be the kdimensional skeleton of the complex X. We shall assume that the orientations of all the cells are fixed. Number all the k-dimensional cells, and let Ak be the set of all subscripts. Then, due to Lemma 2, we obtain: H. (X k Xk - 1) = H. (V Sk) = { 0 when i +k I , I (tEA a Pk(X) when i = k. k

Here, P k (X) denotes a free Abelian group whose generators are in one-to-one correspondence with Ak. Since the elements of this group are naturally identified aaa~, where the a~ are the k-dimensional with linear combinations of the form cells of the complex X, the group Pk(X) is finitely generated. We call this group the group of k-dimensional chains of the space X. The groups P k (X) and Ck (X) are not isomorphic in the general case. Before going further, we consider the socalled exact homology sequence of the triple, which is a variant of the sequence of the pair. Let (X, Y, Z) be three spaces, where Y and Z are closed in X. Consider the two embeddings (Y, Z)--t(X, Z) and (X, Z)--t(X, Y); let

F

a:

Hk (X, Y)---tH k _ 1 (Y, Z)

be the boundary homomorphism generated by the homomorphism

a:

Hk (X, Y)--t Hk -

1

(Y)

(whose definition is given above) and also by the fact that each absolute cycle from Hk _ 1 (Y) can be considered as relative modulo Z, i.e., as an element of the group Hk _ 1 (Y,Z). Then we obtain the sequence


184

Hk (X, Y) a Hk _ 1 (y, Z) Hk _ 1 (X, Z) Hk _ 1 (X, Y) a ... The verification of its exactness is left to the reader. Returning to the groups P k (X) = Hk (Xk/Xk - I) and Pk _ 1 (X) = Hk - 1 (Xk - I/X k - 2), we can consider the exact sequence of the triple (Xk, Xk - I, Xk - 2). For the moment we only need from it the homomorphism Hk (Xk, Xk - I) ~Hk _ 1 (Xk - I, Xk - 2), which is written in our notation thus: P k (X)~Pk - I(XJDenoting it by iJ k, we obtain the chain complex k (X), iJkJ, viz., ...-Pk (X) .....k Pk _ 1 (X)~..... Just as for any chain complex, its homology groups are defined, i.e., the groups Ker iJ/ImiJ called the cellular homology groups of the complex. As it happens, there exists a canonical isomorphism between the homology of the chain complex described and the singular homology of X. This is the basic statement of the present item, enabling one to reduce the computation of singular homology to the computation of the homology of a considerably simpler chain complex. As it happens, this reduction is so effective that most concrete computations of homology are based just on this theorem. In particular, it immediately follows that cellular homology groups are homotopy invariant, and that singular homology groups of a finite complex are always finitely generated.

lp

4. Theorem on the coincidence of the singular and cellular homology of a finite cell complex. THEOREM 3. For a finite cell complex X, the singular homology groups Hk (X) and homology groups of the chain complex {Pk (X), iJ k} i.e., the groups KeriJk/ImiJ k + 1 (the so-called cellular homology groups) are isomorphic. First, we prove certain auxiliary statements. LEMMA 3. When k

> 1, the following isomorphism holds:

Proof. Consider the triple of complexes (Xk corresponding exact sequence:

+

I, Xk - 2, xk - 3) and the

Hk (Xk - 2, Xk - 3) Hk (Xk + I, xk - 3) Hk (Xk + I, Xk - 2) Hk _ 1 (Xk - 2, xk - 3). Its extreme terms are equal to zero, i.e.,

Repeating the argument for the triple (Xk + 1, Xk - 3, Xk - 4), we obtain


185

Proceeding further with respect to dimension, we obtain the following chain of isomorphisms: Hk (Xk + I, Xk - 2)

= Hk (Xk + I, Xk -

3)

= Hk (Xk + I, Xk - 4)

= ... = Hk (Xk + I, XO) = Hk (Xk + I)

when k > 1. If the skeleton XO consists of only one point, then the equality is valid also for k = 1. The point is that any finite cell complex (connected) is homotopy equivalent to a finite complex such that its zero-dimensional skeleton consists precisely of one point. For this, it suffices to consider all the zerodimensional cells of the original complex, and join each of them to one distinguished vertex (zero-dimensional cell *); moreover, all these paths must lie in the one-dimensional skeleton XI. Then we carry out the homotopy shown in Fig. 110, by contracting all the zero-dimensional cells to one.

Figure 110

LEMMA 4. The following isomorphism holds: Hk (X) = Hk (Xk + 1). < k + 1, the equality holds: Hi (Xk + I) =

Proof. We prove that, for any i Hi (Xk + 2).

In fact, consider the exact sequence of the pair (Xk + 2, Xk + 1):

Then the required equality follows from this; in particular, Hk (Xk + I) = Hk (Xk + 2). Going back to the proof of Lemma 3, we obtain:

which completes the proof. LEMMA 5. The following isomorphism holds: Ker 8k/lm8k + I = H (Xk + 1,

186


Xk - 2), where Ok are homomorphisms defining the complex of cellular chains. Proof. Consider the commutative diagram:

Here, the row is a segment of the exact sequence of the triple (Xk + 1, Xk, Xk - 2), while the column is a segment of the exact sequence of the triple (Xk, Xk - 1, Xk - 2) and the homomorphisms i*, i* are induced by the corresponding embeddings of the pairs i,i. The commutativity of the diagram means that Ok + 1 = i*o. Recall that Per (X) = Her (Xer, Xer - 1). Since the row and column are fragments of exact sequences, i* is an epimorphism, and i* a monomorphism. Hence, Hk (Xk + 1, xk - 2) = Hk (Xk, Xk - 2)/Ker i* = Hk (Xk, Xk - 2)/lmo. Since i* is a monomorphism, Hk (Xk, Xk - 2)/lmo = i* Hk (Xk, Xk - 2)/i* Imo = Imi*IImi* = Ker ok/1m i* = Ker Ok/1m Ok + 1 Here, we have used the identities 1m i* = Ker Ok (because of the exactness) and i* = Ok + 1 (commutatively of the diagram). Thus, Hk (Xk + 1, Xk - 2) = Ker 0klIm Ok + " and the lemma is proved.

°

°

°

Proof of Theorem 3. We obtain from Lemmas 3 and 5 Hk (X) = Ker ok/1m Ok + 1 = Hk (Xk + 1, Xk - 2), thus completing the proof.

5. The geometric determination of cellular homology groups. We need to clarify the geometric meaning of the operator Ok in the chain complex {Pk (X), Ok}' Consider two cells Ok - 1 and Ok in X. We will assume that their orientations are fixed and that the characteristic mapping x: D k X, x: D k - 1 X are compatible with these orientations. Consider the continuous mapping. ODk = Sk -

1

~ Xk - l/Xk - 2,

i.e. we map the boundary of the cell into the factor space of the (k - 1)-dimensional skeleton with respect to the (k - 2)-dimensional skeleton.


187

This factor space is homeomorphic to the wedge of (k - 1)-dimensional spheres. Since we have distinguished the cell Ok - I in the (k - 1)-dimensional skeleton Xk - I, it is mapped onto a certain sphere Sk - I from the wedge Xk - IlXk - 2 under the indicated factorization Xk - I a Xk - IlXk - 2. Thus, one sphere is distinguished in the wedge of spheres, which makes it possible to define the natural projection of the whole wedge onto this sphere. That is to say, the distinguished sphere is not displaced, and all the others are mapped into the base point (Fig. Ill). The mapping so constructed of the boundary of the ball Dk into the wedge of spheres with subsequent projection onto the distinguished sphere defines a continuous mapping Sk - I Sk - I, and, in particular, determines a homomorphism of the group Z = Hk _ I (Sk - I) into itself. Each homomorphism Z -Z is uniquely determined by an integer m, viz., the image of the identity element of the group Z. This number is called the degree of the mapping. In the case where the mapping constructed by us is smooth (it will be so in many examples considered below), the number m coincides with the usual degree of a smooth mapping, defined for mappings of orientable closed manifolds of the same dimension [50].

Figure III

We have associated each pair of cells Ok and Ok - I with a certain integer called the incidence coefficient of the cells and usually denoted thus: [Ok: Ok-I]. It can be seen from its definition that this number depends on the orientations of the cells chosen by us, and changes sign on changing one of the orientations. THEOREM 4. Let Ok be an arbitrary generator of the group Pk (X) = Hk (Xk, Xk - I). Then the action of the boundary operator 0 on this generator is given by the formula OOk = E[Ok:Ok - 110k - I, where the sum is taken over all the (k - 1)-dimensional cells Ok - I of the complex X. This statement provides us with a clear geometric interpretation of the boundary

188


operator a introduced above in the algebraic language for cellular chain groups. If the cell Ok - 1 does not intersect the closure of the cell Ok, then [Ok: Ok - I] = o.

Proof of Theorem 4. Consider two triples: (Dk, Sk - 1,0) and (Xk, Xk - I, Xk - 2), and the continuous mapping (Dk, Sk - 1,0) (Xk, Xk - I, Xk - 2), where Dk Xk is the characteristic mapping of the cell Ok, and Sk - 1 Xk - 1 is the restriction of this mapping to the boundary of the ball. The exact sequences of these triples can be naturally organized into the following commutative diagram:

o

z

z

I I" Hk (Dk)-+Hk (Dk,

1 i.

Sk -

. "Hk _ 1 (Sk - 1)---+0

I)~

1~.

Hk (Xk, Xk - I)~ Hk _ 1 (Xk - I, Xk - 2)

"

Pk (X)

"

Pk

- 1

(X)

Consider the element 1 EO Z = Hk (Dk, Sk - I). Under the homomorphism i., this generator is mapped into the cellular chain 1.o k EOP k (X), and after applying is transformed into 1.a Ok. Let us follow the displacement of this generator along the upper side of the square. Under the mapping j, the element 1 is transformed into 1 EO Z = Hk _ 1 (Sk - I) (since j is an isomorphism). Under the subsequent mapping ~., the generator of the group Z is transformed into a certain element of the group

a,

To each generator of the group P k _ 1 (X) = Hk _ 1 (VSk - I), there corresponds a certain cell Ok - I. It is clear that the coefficient of this cell in the image of the identity element under the mapping ~. is exactly equal to the degree of the composite mapping Sk - 1 Sk - I, i.e. the coefficient [Ok: Ok - I]. The theorem is thus proved. So, we have obtained a rather simple rule for computing the singular homology groups of a cell complex. For this, it suffices to consider the chain complex of cellular chains uniquely determined by the cell structure of X, and then write out the boundary operators explicitly, for which it suffices to compute the incidence coefficients of pairs of cells of consecutive dimensions. Then, the homology groups of the complex of groups so obtained must be computed. This construction is so vivid (and easily computed in many cases) that it sometimes forms the basis for the definition of cellular homology groups.


189

DEFINITION 2. Let X be a finite cell complex, Pk (X) cellular chain groups, and a k: P k (X) P k - 1 (X) homomorphisms defined by the formula a k Ok = E [ak:a k - I] ok - I. Then the homology groups of this complex, i.e. the groups Ker a k + 1 are called the cellular homology groups of the complex X. To compute these groups, the singular homology of X need not be known, since all the objects involved in Definition 2 admit of a purely geometric description (cells, characteristic mappings, incidence coefficients). Theorem 3 can be reformulated thus: the singular and cellular homology groups of a finite cell complex are isomorphic, with the corollary that if one space X is represented as a cell complex in two ways, then the cellular homology groups of X do not depend on the cellular decomposition, since they are isomorphic to the singular homology of the space. Thus, to compute the homology of the space X, we must choose its representation as a cell complex in as simple a way as possible, and then compute the cellular homology.

6. The simplest examples of cellular homology group computations. EXAMPLE I. The sphere sn admits of the simplest cellular decomposition: 0° U an, where 0° is a point, and an its complement in the sphere. It is clear that, for n~ 1, we have: Hi (sn) = Z when i = 0, n, and H j (sn) = when j #=0, n.

°

EXAMPLE 2. The real projective space Rpn. Recall that one of its realizations is the set of sequences of the form x = (xo> xl''''' xn), where Xi are real numbers and at least one coordinate is non-zero, the sequence being considered up to a nonzero multiplier. The simplest cellular decomposition ofRpn is arranged thus: for the cell Ok, all sequences x should be taken, for which xk #=0, xk + 1 = ... = xn = 0. Then, in each dimension k, we obtain exactly one cell Ok, i.e., Rpn = 0° U 0 1 U ... U an. Therefore, Pk (Rpn) = Z. It remains to compute the boundary operator ak:z Z. Fig. 112, the closure of the cell Ok, i.e., Rpk, is represented as a k-dimensional ball whose boundary, viz. the sphere Sk - 1, is factorized with respect to the action of the group Z2' i.e., the generator of this group is represented by the transformation x - x, the reflection of the sphere in the origin. In other words, Rpk is obtained from the ball Dk by identifying diametrically opposite points on its boundary. Meanwhile, the boundary of the ball Dk is mapped into the factor Rpk - l/Rpk - 2 = Sk - 1 as follows. Represent Sk - 1 as the union of three disjoint subsets S~ - I, Sk - 2, Sk- 1 _, where S~ - 1 and SI: - 1 are open hemispheres (upper and lower), and Sk - 2 the equator. The mapping aDk = Sk - 1 Sk - 1 = S~ - I/Sk - 2 is arranged thus: x x ifx E S~ - I, X * when x E Sk - 2, -x x when -x E Sk- I. Thus, we have obtained a mapping h: Sk - 1 Sk - 1 = S~ - 1/Sk - 2 which is a diffeomorphism on each of the subsets S~ and S~· It remains to fmd the degree of this mapping. It is clear that

190


the inverse image of each point x EO S + consists of two points: the point itself and the point diametrically opposite to it on the sphere Sk - 1. Therefore, the required degree either equals two or zero according to whether the orientation of the sphere Sk - 1 changes or not under the mappings x ~ - x. Therefore, it is required to find the degree of the auxiliary mapping a: Sk - 1-+ Sk - 1, where a (x) = -x. LEMMA 6. The degree of the mapping a (x) = - x of the sphere Sk equals (-I)k.

1

onto itself

Hence, it immediately follows that the degree of the mapping h equals 2 for even k, and zero for odd k. Thus, for odd k, the coefficient [Ok: Ok - 1] equals zero, and two for even k. This means that the boundary operators in the chain complex {Pk (X), «J k} have the form «J0 2-Y = 202-y - 1 = o. Thus, we have proved the following statement. Proposition I. The singular (and cellular) homology groups ofRP have the form:

Hn

o

for even n Z for odd n.

In conclusion, we prove Lemma 6. Consider the sphere Sk - 1, and fix an arbitrary tangent orthogonal frame e (x) = (ep ... ,ek _ 1) at a point x. Under the


191

mapping a: x ~ - x, this frame will be transformed into the frame e ( - x) = (- el"'" - ek _ I) (we assume that the sphere is standardly embedded into Euclidean space). We need to compare the orientations induced on the sphere by these two frames. Join the points x and - x with a meridian 'Y such that its velocity vector at the point x is the vector ek _ 1 (Fig. 113). Carry out a smooth deformation (transfer) of the frame e(x) from the point x to the point -x by moving along the path 'Y so that the vector ek _ 1 remains tangent to 'Y. Then we obtain two frames at the point - x: - el"'" - ek _ 2' - ek _ 1 and - el"'" - ek _ 2' + ek _ l' It is clear that their mutual orientation is determined by the sign of (- 1)1t - 2, thus the proof is completed.

Figure 113 EXAMPLE 3. The two-dimensional, compact, connected, closed, and orientable manifold M~ of genus g is homeomorphic to a two-dimensional sphere with g handles, and admits of the cellular decomposition a O U (~12g at> U a 2. It follows that the homology of M~ is of the form:

(2g terms), H2 = Z. The homology groups defined above do not embrace the whole set of "homological invariants" that sometimes enable one to distinguish between cell complexes. A homology theory can be constructed with the use of chains with coefficients in an arbitrary Abelian group A. This means that chains should be considered as linear combinations with coefficients in A. It is clear that all the

192


constructions are automatically transferred to this case, enabling one to define the groups Hk (X, A) called the homology groups with coefficients in the group A. The results relating to cell complexes are valid for the group A = Z (considered above) as well. In particular, these groups can be defined as the homology of the chain complex made up of the groups of the cellular chain groups {Eaj OJ' ajE A} = Pk (X,A) with coefficients in A. The homology groups Hk(X) studied by us above are written in the new notation as Hk (X,Z). For brevity, we will often omit the designation of the coefficient group except in those cases where the final result depends on the choice of the group A. EXERCISE. For Examples 1,2,3 (see above), calculate the homology groups with coefficients in the groups A = R, Q, Z2' Zp' where R is the field of real numbers, Q the field of rational numbers, and p a prime number.

§2. Cohomology Groups and Obstructions to the Extensions of Mappings. 1. Singular cochains and the coboundary operator. Let X be a cell complex, and C k (X) the group ofk-dimensional singular chains of the space X. Let A be an Abelian group. DEFINITION I. A homomorphism of the group C k (X) into the group A is called a singular cochain of the space X with coefficients in the group A. The natural operation of cochain addition transforms the set of cochains into an Abelian group denoted by Ck (X,A), and called the group of cochains of the space X. Consider the boundary operator 0: C k (X)-+C k _ 1 (X). Let hECk - 1 (X, A) be an arbitrary cochain, i.e., a homomorphism h: C k (X)-+A. Then the cochain 5h E Ck (X, A) is determined uniquely. It is given by the formula: 5h(a) = h (0 a), i.e., 5h: C k (X) -+A. In other words, the cochain 5h is determined from the diagram.

We will sometimes denote the operator 5: Ck -

1

(X, A)-+Ck (X, A) by 15 k _ I'

DEFINITION 2. The operator 15k _ I: Ck - I-tCk is called the coboundary operator. It is the dual of the operator o.

193


Since iF = 0, 02 = O. Therefore, we obtain a sequence of groups and the homomorphism connecting them, which is of the following form:

where ok Ok _ 1 = O. This sequence is called a cochain complex. Proceeding as in the previous section, consider the groups Ker 0 and 1m 0, and construct the group Hk (X, A) = Ker 0k/lmok - l' DEFINITION 3. The groups Hk (X, A) are called the cohomology groups of the space X with coefficients in the Abelian group A. The elements of the group Bk = 1m Ok _ 1 are called coboundaries, and those of the group Zk = Ker Ok cocycles.

If the space X is path-connected, then HO (X, A) = A. As in the case of homology, the relative cohomology groups are defined in natural fashion. Let Y be a closed subcomplex of the complex X, then Ck (Y) C Ck (X). Let 0 (X, Y) be the group of all those homomorphisms a: C k (X)~A that are equal to zero on Ck (Y). It is clear that 00 (X, Y) C 0 + 1 (X, Y), and this therefore gives rise to the groups Hk (X, Y, A) = Ker o/lmo. They are called the relative cohomology groups. As in the case of cohomology, the exact sequence of the pair and triple arise. Omitting the details of construction, we only give the exact sequence of the pair (verify that it is exact!): ... -+Hk (X, Y, A)-+Hk (X, A)~Hk (Y, A)~ Hk + 1 (X, Y, A)-+... and also of the triple (X, Y, Z): ... ~Hk (X, Y, A)4Hk (X, Z, A)4Hk (Y, Z, A)~Hk

+ 1

(X, Y, A)-+ ...

Singular cohomology groups are homotopy invariant. Proceeding as in §1, we define cellular cohomology groups. To this end, we introduce the cellular cochain groups pk (X, A) defined as the groups Hk (Xk, Xk - 1, A), where Xk is the k-dimensional skeleton of the cell complex X. The exact sequence of the triple (Xk + 1, Xk, Xk - 1) gives rise to the coboundary operator 0: pk (X, A)~ pr- + 1 (X, A). Cellular cohomology groups are defined as the groups Ker O/Im for the cochain complex {Pk (X, A), o}.

°

THEOREM I. For a finite cell complex X, the singular cohomology and cellular cohomology groups are isomorphic, and are finitely generated Abelian groups if the coefficient group is Abelian.

The proof is as given in § 1.

194


2. The problem of the extension of a continuous mapping from a subspace to the whole space. While studying topological variational problems (to which the subsequent sections are devoted), we shall sometimes have to solve the following problem. Let Y be a closed subspace of a topological space X, and let a continuous mapping of this subspace into some space Z be given. The question arises: in which cases can this mapping be extended to a continuous mapping of the whole space X into the space Z? It is clear that such a continuous extension does not always exist. There are topological obstructions which, in certain cases, do not permit us to extend a mapping from a subspace to the whole space. Obstruction theory takes up the study of all the various versions of this question. Here, we only give the results which will be used in the sequel. We already possess all the necessary material to give an account of the required constructions.

3. Obstructions to the extension of mappings. Consider a finite cell complex K and a topological space X. Let a continuous mapping g of the (n - I)-dimensional skeleton Kn - 1 of the cell complex K (Le., union of all its cells of dimensions not exceeding n - 1) into the space X be given. We intend to extend it to a mapping f of the following skeleton Kn in X. For simplicity, we will assume that X is either a simply-connected space or, in case X is one-dimensional, possesses a commutative fundamental group. Consider all n-dimensional an which make up the n-dimensional skeleton of the complex K. Since it is finite, the number of these cells is finite. To construct the required extension, we must be able to extend the original mapping to each ndimensional cell separately. Fix some cell an. Its boundary has already been mapped by the mapping g into the space X. It has to be extended to the whole cell. We will assume that the mapping f: Kn~x under construction coincides with the mapping g on the skeleton Kn - 1 C Kn. Let x: on~K be the characteristic mapping of the cell an, where on is an n-dimensional disc. Then we have the composite mapping sn - 1 = a on_ X-+Kn - 1_ g-+X. Here, we use the fact that the boundary of the cell an is contained in the skeleton Kn - 1 in accordance with the definition of a cell complex. The mapping g x: sn - 1-+ X so obtained determines an element [gX] of the homotopy group 11"n _ 1 (X). This is well-defined in view of the restrictions imposed on the space X. They guarantee that the definition of the elements of the group 1I"n _ 1 (X) does not depend on the choice of a base point. In the general case, we must require that the space X be (n - I)-simple (see the definition in [84]) Le., that the fundamental group act trivially (from the homotopy point of view) on 1I"n _ 1 (X). See this action in Fig. 114. It is clear that, with the above assumptions, X is (n - I)-simple. The mapping gXn: Sn - I-+X can be extended to the mapping of the whole disc on in that and only that case where the element [gX] from 11"n _ 1 (X) equals zero. In the general case, we associated each cell an with a certain element [gx] of the Abelian


195

Figure 114

group G = 'II"n _ 1 (X). Consider the group Pn (K) of the cellular chains of the complex K. It is clear that the constructed correspondence (Jn~[gX] f G can be naturally extended to a certain homomorphism of the group of chains P n (K) into the group G. For this, it suffices to construct the mapping for each cell, and extend it to all chains by linearity. We thereby define a certain cochain c~ from the group of cellular cochains. DEFINITION 4. A cochain c~ f pn (K) with coefficients from the Abelian group G = 'II"n _ 1 (X) is called an obstruction to extending the mapping g from the skeleton Kn - 1 to the skeleton Kn. LEMMA I. A mapping g: Kn - l-+X can be extended to a continuous mapping f: Kn~x if and only if the cochain c~ is identically equal to zero. The proof follows from the definition of an obstruction. LEMMA 2. Let a space X be (n - I)-simple. The cochain c~ E pn (K, 'll"n _ 1 (X» is a cocyle, i.e., oc~ = O. Therefore, the cochain c~ defines a certain cohomology class C~ = [c~] from the group Hn (K, 'll"n _ 1 (X». Since the ideas underlying the proof will not be required by us in the sequel, we omit the proof [84]. In the applications that we shall encounter, the equality oc~ = 0 will follow from an obvious argument. THEOREM 2. Let a space X be (n - I)-simple. A cocycle C~ equals zero (as an element of the cohomology group) if and only if the original mapping g: Kn - l--?X can be extended to a continuous mapping f: Kn~x by preliminarily

196


changing the mapping g on the skeleton Kn skeleton Kn - 2.

I,

and not changing it on the

In contrast with Lemma 2, the plan of the proof of Theorem 2 will be used in applications; hence, we give the proof. Proof. To construct the mapping f, we shall need a new notion closely related to the obstruction c~, viz. the difference cochain. Let two mappings f, g: Kn - I~X coinciding on the skeleton Kn - 2, i.e., f (x) = g (x) for all x € Kn - 2, be given. Let an - 1 C Kn - 1 be an arbitrary cell, and x: Dn - 1--+ K its characteristic mapping. Since the boundary sn - 2 of the disc Dn - 1 is mapped into the skeleton Kn - 2, the two composite mappings fX and gX map the sphere sn - 2 into the complex K in the same way. The cell an - I is mapped differently, generally speaking; however, these, two images have a common boundary, being glued on the image of the sphere sn - 2 in K. Therefore, in the space X, we obtain a spheroid which determines a certain element of the group 7rn _ 1 (X) and measures the deviation of the mapping f from the mapping g on an - I. In this way we have associated each cell an - 1 with an element of the group 7rn _ 1 (X) (here, the (n - I)-simplicity of X is used). We obtain a homomorphism of the group of chains P n _ 1 (K) into the group 7rn _ 1 (X), i.e. an (n - I)-dimensional cochain. This cochain is denoted by dp'g- I, and called the difference cochain of the two mappings f and g (Fig. 115). It follows from its definition that dp'g- 1 = 0 if and only if there exists a homotopy connecting f and g and constant on the skeleton Kn - 2 where f and g coincide.

fJ

Figure 115


197

LEMMA 3. For any mapping f: Kn - I_X and any cochain a f pn - I (K, 1I"n _ I (X», a mapping g:Kn - I~ X can be always chosen, so that it coincides with f on the skeleton K n - 2 and the cochain d is the difference of f and g, i.e., d = dp'g- I.

Proof. Consider an arbitrary cell an - I from K and its image under the mapping f in X. Choose a sufficiently small ball in the centre of the cell, and, considering its image under f in X, cut this image (using the cellular approximation theorem [84]) out of the image of the cell an - I. Then glue to the hole so obtained a spheroid, a representative of that element from the group 11"n _ I (X), which is the value of the cochain d on the cell an - I (Fig. 116). As the new mapping g, take the mapping coinciding with f everywhere except on the distinguished ball, and coinciding on the ball with the mapping realizing the indicated spheroid. Roughly speaking, the distinguished small ball bulges into the spheroid realizing the value of the cochain d on this cell. Performing this operation on each cell, we obtain a certain mapping g. Comparing it with the original, we obviously obtain d = rlP,g= I. The lemma is thus proved.

G' n-i Figure 116

LEMMA 4. The equality holds: Odp'g-

I,

=

q -

C~.

Proof. Due to the definition of a coboundary operator in terms of cellular cochains and chains, we have the equality:

198


To compute the incidence coefficients, we have to consider the following composite mapping

where 11" is the natural factorization, and Pi the projection on the wedge of spheres onto the sphere with number i in this wedge. The degree of the mapping sn - I-,+Sr - 1 so obtained is what is called the incidence coefficient [on: or - 1]. It is easy to prove that the mapping sn - I--+Kn - 1 is homotopic to a mapping under which almost the whole sphere sn - I, with the exception of a finite number of small balls, is mapped into the skeleton Kn - 2, and the small balls are mapped into the skeleton Kn - I, each small ball being mapped onto its cell or - 1 with degree ± 1. The algebraic number of the small balls mapped onto the cell or - 1 (i.e., sum of degrees ± 1) is exactly equal to the required incidence coefficient (Fig. 117). In the figure, the thick line segments on the boundary of the ball schematically represent the small balls of dimension n - 1, which are mapped onto the cells or - I. Let us calculate the value of the cochain cp - c~ on

Figure 117


199

the cell on (Fig. 117), i.e. where the boundary cells or - 1 are mapped. Eventually, we have to determine where the small balls distinguished on sn - 1 are mapped under the composite mapping involving f and g. Since the mappings f and g coincide on the skeleton K n - 2, to compute the value of the cochain on n , i.e., all the the cell on, we have to take the summation over the boundary cells or - I, of the following spheroids from 11"n _ 1 (X). On each cell or - I, we have to take the spheroid coinciding with the value of the difference cochain dp'g- 1 on the cell or - I, and as many times as there are small balls mapped onto the cell or - I. Obviously, this sum gives the incidence coefficient, and the whole cell on will therefore be associated with the sum E[on: or - I] dp'g- 1 (or - I) coinciding with (0 dP,g- I) on, which completes the proof.

ao

Let us return to the proof of Theorem 2. Let the mapping g: Kn - 14 X be extended to the mapping f: Kn_x without being changed on the skeleton Kn - 2, but possibly, being changed on the skeleton Kn - I. Due to Lemma 4, we have: n - 1 = C n - cn But since f is defined on Kn cn = o· therefore cn = odf,g f g. , 'g" g Odp'g- I, so that C~ = O. Let us prove the converse statement. Suppose that C~ = 0, i.e., c~ = Od, where d is a certain cochain. According to Lemma 3, there exists a mapping f:Kn - I-+X coinciding with the mapping g on the skeleton Kn - 2, and such that -d = dp'g- I. Then, due to Lemma 4, we obtain q = C~ + Odp'g- 1 = c~ - od = o. Therefore, the mapping f can be extended to the mapping Kn~x. Thus, we have extended to the mapping g, possibly having a priori changed it on the skeleton Kn - I, but not on Kn - 2, which finishes the proof.

4. The cases of the existence of the retraction of a space onto a subspace which is homeomorphic to the sphere. We prove the retraction theorem which we shall need for the study of variational problems. Consider the Abelian group (additive) U = RI (mod 1), i.e. the circumference, as the coefficients of the cellular homology theory. THEOREM 3. (Hopf). Let an embedding i of the sphere sn - 1 into a finite ndimensional cell complex K be given. Let the homomorphism i*: Hn _ 1 U)-+Hn _ 1 (X, U) induced by this embedding be a monomorphism. Then the sphere S~ - 1 = iS n - 1 is a retract of K, i.e. there exists a continuous mapping f:K-+ S~ - 1 which is an extension of the identity on the sphere S~ - I. Proof. The constructive variant of the proof given here, necessary for the explicit construction of retractions is due to T. N. Fomenk030o • We have to construct a continuous mapping f:K~ sn - 1 such that its restriction to the sphere iS n - 1 embedded into K is the identity mapping of the sphere iS n - 1 onto sn - I. To construct such a mapping, we apply obstruction theory. First of all, choose the simplest cellular decomposition into the sum of two cells for the sphere sn - I:

200


zero-dimensional * and (n - I)-dimensional an - I. Therefore, we can construct the required mapping so that it is cellular and transforms the whole skeleton Kn - 2 into a point on the sphere sn - I. In other words, it suffices to construct a continuous mapping of the quotient complex K/Kn - 2 into the sphere sn - I; here, K = Kn. To apply obstruction theory, we have to take X to be an (n - I)-simple space. Take X = sn - I, then the condition of(n - I)-simplicity is obviously fulfilled. In fact, if n - I > I, then 7r 1 (X) = 0, and if n - I = I, then the group 7r 1 (S I) = Z is Abelian and acts on itself as the identity. Since the skeleton Kn - 2 has been contracted by us to a point, we can assume that the skeleton Kn - 1 coincides with the wedge of(n - I)-dimensional spheres s~

-

1

V sy -

1

V .... V

S~

-

I.

It is clear that the sphere iS n - 1 can be considered to be one of them. Let it be, for definiteness, the sphere S~ - I. Represent each of the spheres Sr - 1 as the simplest cellular decomposition, and denote the corresponding (n - I)-dimensional cells by or - I. We must construct a continuous mapping f:K~sn - 1 such that its restriction to S~ - 1 is the identity mapping onto sn - I. Let oy, ... , o~ be n-dimensional cells of the complex K. Then we may assume that nI 1 n K -- o-vo o

1

U... UOkn - IU0 nu' I_n 1 ••• vu q

Figure 118

(Fig. 118)


201

Preliminarily, we must construct a certain mapping of the skeleton Kn - 1 into sn - I. Now, we construct one such mapping gl which, generally speaking, will not be extended to the mapping of the whole K into sn - I, but can be used for constructing a second and final mapping g2 extendable to the mapping f:K~ sn - I. Since the skeleton Kn - 1 is the wedge of spheres, it suffices to specify the mapping gl on each of them. We specify the identity mapping S~ - 1 ~ sn - 1 on the distinguished sphere S~ - I, and map all the remaining spheres into the base point * f sn - I. This mapping is continuous; however, it is easy to construct examples showing that it cannot be extended to a continuous mapping of the whole complex into the sphere. Let us calculate the obstruction C~I for the mapping gl· Consider an arbitrary n-dimensional cell on. To calculate c~ (on), we have to consider the composite mapping of the (n - I)-dimenslcinal sphere, the boundary of the cell, into the sphere sn - I, and find its degree. Here, we use the fact that 1I"n _ 1 (sn - I) = Z, and that the homotopy classes of a mapping of a sphere into a sphere are classified by the degree of the mapping. This composite mapping is the composite of the characteristic mapping and gl :Kn ~ sn - I. That part of the boundary sphere (Jon, which was mapped into the wedge S? - 1 V .... V S~ - I, i.e. which is not incident with the distinguished sphere S~ - I, will be mapped, finally, into the base point * on the sphere sn - I. Therefore, these portions of the sphere (Jon do not take part in forming the required degree (contribution is zero). The remaining part of the boundary sphere (Jon is mapped onto the distinguished sphere S~ - 1 "twisted" on it as many times as the incidence coefficient [on: o~ - I], and then mapped onto the sphere sn - 1 with the help of the identity mapping. Finally, the boundary sphere (Jon is mapped onto the sphere sn - 1 with the degree equal to the incidence coefficient [on: o~ - I]. Thus, we have calculated that C~I (on) = [on: o~ - I]. We state that this cochain is representable as the coboundary of a certain (n - I)-dimensional cochain d. LEMMA 5. With the assumptions of Theorem 3, the cochain C~I is a coboundary, i.e., C~I = a d, where d f P n - 1 (K,1I"n _ 1 (sn - I) and d (M~ - I) = 0, O~A~ 1. Proof For simplicity, consider the case when, among the n-dimensional cells of the complex K, there is only one cell on with non-zero incidence coefficient with the distinguished cell o~ - I. Roughly, the further argument will be developed as follows. Our purpose is to construct an (n - I)-dimensional cochain d on the wedge of spheres, so that this cochain must be equal to zero on the distinguished cell c~ - 1 (more precisely on the elementary chain 1. o~ - I), and, generally speaking, non-trivial on the remaining spheres of the wedge, i.e., we want to "remove" the cochain from the distinguished sphere S~ - I. We shall use the fact that the cycle S~ - 1 is not homologous to zero in the complex K for the coefficient group U. Recall that the homomorphism induced by an embedding of the sphere has a trivial kernel. We compute the boundary of the cell on. We have:

202


Denoting [an: ar - I] by aj. we can write:

Thus, the chain - ao a~ - I is homologous to the chain Qj a~ - I + ... + Qk a~ - I. Therefore, the cycles defined by them coincide as elements of the homology group Hn _ I (K, Z). It follows from the conditions of the theorem that the cycle }.a~ - I, where 0 1. Then there exists a cell complex denoted by K (11", n) and called the Eilenberg-MacLane space, such that all its homotopy groups are trivial except for the group 11" (K(1I", n», which is isomorphic to the group 11". The following important statement holds: the set n (K, K (11", n» of the homotopy classes of mappings of a finite cell complex K into the space K (11", n) is in one-toone correspondence with the set of elements of the cohomology group Hn (K, 11"). Consider the sphere sn - I, and embed it into the space K (Z, n - 1) as an (n - I)-dimensional skeleton. For this purpose, a certain number of cells should be glued to the sphere in order to "eliminate" all its homotopy groups beginning with dimension n. Meanwhile, we use the fact that 1I"n _ I (sn - I) = Z. The first homotopy group which needs annihilating is the group 1I"n (sn - I). Therefore, we have to glue cells of dimensions n + 1, n + 2, etc., to the sphere sn - I. Eliminating all higher homotopy groups one after another, we obtain a certain complex B, for which 1I"j (B) = 0 when it=n - 1 and 1I"n _ I(B) = Z. It is well

204


known that any two polyhedra which are spaces of type K( 11", n - I) for given n - 1 and 11" are homotopy equivalent. It is clear that Hn - l(K (Z, n - 1), Z) = Z; K(Z, n - 1) = UOOUO n - IUOn + IUO n + 2U... Let a€ Hn - I(K(Z, n - I)Z) be the generator of the group Z. Then (see [84]) the function associating the mapping f:K-+K(Z, n - 1) with the homology class f'l'a € Hn - 1 (K, Z) for any finite complex K determines a one-to-one correspondence between Hn - 1 (K, Z) and the set II (K, K(1I", n» of homotopy classes of mappings of K into K (11", n). The condition of Theorem 3, according to which the homomorphism i.: Hn _ 1 (sn - I,U)-+H n _ I(K,U) has no kernel, can be restated in an equivalent manner in the language of cohomology, viz., the homomorphism i*: Hn - 1 (K, Z)+Hn - 1 (sn - I, Z) induced by the embedding is an epimorphism. Therefore, there exists an element {3€ Hn - 1 (K, Z) such that i* B = a€ Hn - 1 (sn - I,Z). Because of this, there exists a continuous mapping f:K-+K (Z, n - 1) such that f'l'a = {3. Since (fi)*a = a, the restriction of the mapping f to the sphere iS n - 1 embedded into K is homotopic to the identity mapping of the sphere iS n - 1 onto itself. It remains to show that the mapping f can be deformed into a mapping which remains the identity on the sphere sn - I, and will map the whole complex K onto it. But since the complex K is n-dimensional by assumption and since the complex K (Z, n - 1) does not contain cells of dimension n, it follows from the cellular approximation theorem (see, e.g., [84]) that the mapping f can be assumed to transform K into the (n - I)-dimensional skeleton of the space K (Z, n - 1), i.e. the sphere sn - I, which completes the proof of Theorem 3.

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262. Otsuki, T., Minimal hypersurfaces with three principal curvature fields in sn+l. Kodai Math.J., V.l,l (1978), pp. 1-29. 263. Otsuki, T., Minimal submanifolds with M-index 2. J. Dilf Geom., V. 6,2 (1971), pp. 193-211. 264. Otsuki, T., Minimal submanifolds with M-index 2 and generalized Veronese surfaces. J. Math. Soc. Jap., V. 24, 1 (1972), pp. 89-122. 265. Otsuki, T., A construction of closed surfaces of negative curvature in E4. Math. J. of Okayama Univ., V. 3, 2 (1954), pp. 95-108. 266. Pini, M., Minimalflachen fester GauBscher Krummung. Math. Annal.,V. 136, (1958), pp. 34-40. 267. Pini, M., Kuge1bilder ree1ler Minimalfliichen im R4. Math. Z., V. 59, (1953), pp. 290-295. 268. Sabitov, I.K., A minimal surface as the rotation graph of a sphere. Mat. Zametki., V. 2, 6 (1967), pp. 645-656 (Russian). 269. Sabitov, I.K., Minimal surfaces with certain boundary conditions. Mat. Sb., V. 76, 3 (1968), pp. 368-389 (Russian). 270. Sacks, J. and Uhlenbeck, K., Minimal immersions of closed Riemann surfaces. Trans. A mer. Math. Soc., V. 271, 2 (1982), pp. 639-652. 271. Schoen, R. and Yau, S.-T., Compact group actions and the topology of manifolds with non-positive curvature. Topology,V. 18, 4 (1979), pp. 361-380. 272. Seminar on Differential Geometry. Edited by Shing-Tung Yau. Ann. of Math. Studies, V. 102, Princeton University Press, Princeton, N.J., (1982). 273. Serrin, J., A priori estimates for solutions of the minimal surface equation. Arch. R. Mech., V. 14, (1963), pp. 376-383. 274. Serrin, J., On surfaces of constant mean curvature, which span a given space curve. Math. Z., V. 112,2 (1969), pp. 77-88. 275. Sigalov, A.G., Variational problems with admissible surfaces of arbitrary topological types. Uspekhi Mat. Nauk V. 12, 1 (1973), pp. 53-98 (Russian). 276. Simons, J., Minimal cones, Plateau's problem and the Bernstein conjecture. Proc. Nat. Acad. Sci. USA, V. 58,2 (1967), pp. 410-411. 277. Shiffman, M., The Plateau problem for minimal surfaces of arbitrary topological structure. Amer. J. Math., V. 61, (1939), pp .853-882. 278. Thompson, D'A., On Growth and Form. London, 1917. 279. Tomi, F., On the local uniqueness of the problem of least area. Arch. R. Mech.,V. 52, 4 (1973), pp. 312-318. 280. Tomi, F. and Tromba, A., Extreme curves bound an embedded minimal surface of disc type, Math. Z., (1978),158, pp. 137-145. 281. Uhlenbeck, K., Morse theory by perturbation methods with applications to harmonic maps. Trans. Amer. Math. Soc., V. 267, 2 (1981), pp. 569-583. 282. Vikaruk, A.J., Analogs of Nevanlinna theorems for minimal surfaces. Mat. Sb. V. 100,4 (1976), pp. 555-579 (Russian). 283. Yau, S.-T., On almost minimally elliptic singularities. Bull. Amer. Math. Soc., V 83, 3 (1977), pp. 362-364. 284. Le Hong Van. Growth of two-dimensional minimal surface. Uspekhi Mat. Nauk., V. 40, 3 (1985), pp. 209-210 (Russian). 285. Le Hong Van. Calibrated geometry of minimal currents in symmetric spaces. In Trudy seminara po vektornomu i tenzornomu analizu, V. 22, Moscow University Press, Moscow, (1985), pp. 107-118 (Russian). 286. Le Hong Van, A.T. Fomenko. Lagrangian manifolds and Maslov index in the minimal surface theory. Dokl. Akad. Nauk. SSSR, (1988) V. 299, No.1, pp.42-45.


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217

INDEX Absolute maxima 78 Area functional 53, 172

Douglas theorem 54 Drop's geometry 14-15

Barycentric subdivision 180 Bernstein's problem 137 Bernoulli Jacob 2 Bifurcation of the minimal surface 37 Boundary conditions 36 Boundary contour (frame) 8, 9 Boundary operator (in homology theory) 173 Boyle experiments 14 Branch-points of the minimal surface 127, 166

Edge of regression 73 Eilenberg-MacLane space (complex) 203 Enneper minimal surface 36, 49 Equation of the minimal surface 66, 131 Euler-Lagrange equation 35 Euler Leonhard 2 Evolute 73 Exact sequence of the groups 174 Extremal properties of soap films 13 Extremals of the functional 35

Catalan's theorem 27 Catenary line 32, 87 Catenoid 32, 88 Cauchy-Riemann equations 137 Cell-complex 176 Cellular chains 183 Cellular cohomology 193 Cellular homology 180, 184 Chain complex 173 Characteristic of minimal surface 161 Characteristics 68, 69 Coboundary operator 192, 197 Cochain complex 193 Cohomology groups 192 Complete catenoid 90 Complete Riemannian surface 152 Conformal (or isothermal) coordinates 17,52,53 Conformal representation of the minimal surface 169 Convex boundary 51, 132 Critical points 21,36 Critical surface 25 Cross-cup 53

Figures of revolution 86 First fundamental form 16,49 Functional 35 Gaussian curvature 16 Generalized harmonic function 126 Genus of the Riemannian surface 149 Handle 52-54 Harmonic mappings 51,52 Harmonic radius vector 52 Helicoid 27 Homology boundaries 173 Homotopy groups 173 Homology cycles 173 Homotopic mappings 174 Homotopy of chain complexes 173-174 Induced metric 16 Isothermic parameters 133 Jordan curve 127 Jumps of the minimal films 38-40 Klein surface 163

Darboux theorem 136 Developable surface 73 Dirichlet integral 50, 168 Dirichlet functional 36, 51,52, 169 Dirichlet principle 53 Differential equation of the minimal surfaces 66 Double-curvature curve 73

Lagrange 3 Lagrangian 35 Laminar systems 97 Laplace equation 50 Laplace operator 17 Lines of curvature 73 Liquid edges 96 219

220

INDEX

Many-valued functions 35 Mean curvature 16 Membranes 62 Meusnier (Jean-Baptiste-MarieCharles) 6, 7 Minimal radius vector 52 Minimal two-dimensional surfaces 20 Minimal surfaces in animate nature 60 Minimal surface in three-dimensional space 50 Minimal surface of assigned topological type 160 Mobius strip 53 Mobius surface 167 Monge Gaspard 3, 65 Multiplicity of the point of the minimal surface 57 Multiplicity of the self-intersection 57 Non-parametric problem 136 Obstructions to the extention of mappings 194-195 One-sided Riemannian surface 147 One-sided minimal surface 159-160 Orientation of the minimal surface 154-155 Partial catenoid 86 Physical realization of minimal surfaces 27 Plateau Joseph 8, 83 Plateau's physical experiments 7, 22 Plateau problem 56, 125 Poisson Simeon 7, 74 Poisson and Laplace theorem 19 Principal curvatures 20 Problem of the latest area 125 Projective space 189 Ouadruple singular point 59 Radii of curvature 65 Radiolarians 61 Rado T. 123 Reduction of the Riemannian surface 150-151 Regular minimal surface 124 Regular point 124

Relative maxima or minima 78 Representation of the minimal surface 161 Retraction 199 Riemannian surfaces 144, 146 Riemannian volume form 34 Saddle type minimal films 42 Schwarz H.A. 123 Second fundamental form 16 Segner's experiments 15 Semi-Riemannian surface 149 Shape of the drops 15 Singular chain 173 Singular cochain 192 Singular edge of the film 44 Singular points of minimal surface 55 Singular simplex 173 Skeleton figures (of the minimal surfaces) 101 Soap bubbles 9 Soap films 9 Soap films of constant positive curvature 19 Soap films of constant zero curvature 19 Stable columns of liquids 22 Stable minimal surfaces 20, 21 Stationary points 35 Steiner problem 59 Surface of constant mean curvature 16 Surface of equilibrium 16 Surface separating two media 11 Swallow tail 48, 49 Tension energy 11 Thompson D.A.W. 61 Topological genus of the surface 36 Unduloid 84 Uniqueness theorem 127 Unstable surfaces 20, 41, 42 Weierstrass representation of minimal surface 137-\38, 143 Whitney cusp 41 Whitney singularities 49 Wire contour 37 Wire models 97

THE PLATEAU PROBLEM A.T. Fomenko Faculty of Mathematics and Mechanics Moscow State University. USSR Translated from the Russian by Oleg Efimov Containing original research data appearing for the first time in English. the two parts of this book explore the history and current state of the theory of minimal surfaces. Its clear presentation and numerous illustrations make this topic accessible to both students and research workers in the fields of mathematics and physics. Part I: Historical Sun-e)' Charting the historical origins of the Plateau Problem. the author discusses substantial extracts from 18th. 19th and early 20th century works devoted to the investigation of minimal surfaces. including Plateau's famous physical experiments. The theories of homology and cohomology. necessary for an understanding of modern multidimensional variational problems. are elucidated. Part II: The Present State of the Theor)' The author demonstrates the deep rooted connections between the theory of minimal surfaces and modern branches of mathematics. particularly the theory of differential equations. Lie groups and multidimensional variational calculus. About the Author Professor A.T. Fomenko was educated at Moscow State University. He graduated with a DSc in 1972. and won the Award of the Moscow Mathematical Society for his doctoral thesis in 1974. In 19~P he was presented with the Award of the Praesidium of the Academy of Sciences of the USSR. Professor Fomenko has obtained fundamental results in the fields of geometry. topology and multidimensional variational calculus. as well as being involved in scientific methodological work and teaching. Related titles of interest from Gordon and Breach Integrable Systems on Lie Algebras and Symmetric Spaces A.T. Fomenko and V.V. Trofimov Lie Pseudogroups and Mechanics J.F. Pommaret

Space Kinematics and Lie Groups A. Karger and J. Novak ISBN 288124700 8/ Part I ISBN 288124701 6/ Part II ISBN 2881247024 2 part set

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