Global analysis of minimal surfaces

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Series editors M. Berge...

Author: Ulrich Dierkes | Stefan Hildebrandt | Anthony J. Tromba

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Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander M. Kashiwara A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan

341

Minimal surfaces with a free boundary. (a) Tongue; (b) Cusp; (c) Loop. Courtesy of E. Pitts For further volumes: http://www.springer.com/series/138

Ulrich Dierkes Stefan Hildebrandt Anthony J. Tromba

Global Analysis of Minimal Surfaces Revised and enlarged 2nd edition With 43 Figures and 5 Colour Plates

Ulrich Dierkes Faculty of Mathematics University of Duisburg-Essen Campus Duisburg Forsthausweg 2 47057 Duisburg Germany [email protected]

Stefan Hildebrandt Mathematical Institute University of Bonn Endenicher Allee 60 53115 Bonn Germany

Anthony J. Tromba Department of Mathematics University of California at Santa Cruz Baskin 261B CA 95064 Santa Cruz USA [email protected]

This volume is the third part of a treatise on Minimal Surfaces in the series Grundlehren der mathematischen Wissenschaften. Part One is Vol. 339 ISBN 978-3-642-11697-1, Part Two is Vol. 340 ISBN 978-3-642-11700-8. A 1st edition of the treatise appeared as Vols. 295 and 296 of the same series.

ISSN 0072-7830 ISBN 978-3-642-11705-3 e-ISBN 978-3-642-11706-0 DOI 10.1007/978-3-642-11706-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010930924 Mathematics Subject Classification (2010): 49Q05, 53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20, 30C20, 30F60, 58J05 c Springer-Verlag Berlin Heidelberg 1992, 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is the third volume of a treatise on minimal surfaces consisting of altogether three volumes which can be read and studied independently of each other. The central theme is boundary value problems for minimal surfaces, such as Plateau’s problem. The present treatise forms a greatly extended version of the monograph Minimal Surfaces I, II by U. Dierkes, S. Hildebrandt, A. K¨ uster, and O. Wohlrab, published in 1992, which is often cited in the literature as [DHKW]. New coauthors are Friedrich Sauvigny for the first volume and Anthony J. Tromba for the second and third volume. The title of this third volume might be somewhat misleading since many aspects of minimal surface theory treated in the preceding two volumes are also of a global nature. Moreover, the lecture notes collected in [GTMS] deal with problems that are essentially different from those investigated here. Nevertheless we think that the title is justified, because at least Part II of the present book is in an essential way based on notions, methods, and results of general Global Analysis. Furthermore, also Part I sheds light on certain global aspects of minimal surfaces. We are very grateful to Ruben Jakob who read the entire typoscript and pointed out numerous errors and misprints. His assistance was invaluable, and we are greatly indebted to his enthusiasm and tireless energy. We also thank Klaus Bach, Frei Otto, and Eric Pitts for providing us with photographs of various soap film experiments. Many thanks also to Friedrich Tomi for his comments on the index theorems. The continued support of our work by the Sonderforschungsbereich 611 at Bonn University as well as by the Hausdorff Institute for Mathematics in Bonn and its director Matthias Kreck was invaluable. We also thank the Centro di Ricerca Matematica Ennio De Giorgi in Pisa and its director Mariano Giaquinta for generous support of our work. We are especially grateful to Anke Thiedemann and Birgit Dunkel who professionally and with untiring patience typed many versions of the new text. Last but not least we should

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Preface

like to thank our publisher and in particular our very patient editors, Catriona Byrne, Marina Reizakis, and Angela Schulze-Thomin, for their encouragement and support. Duisburg Bonn Santa Cruz

Ulrich Dierkes Stefan Hildebrandt Anthony J. Tromba

Contents

Part I. Free Boundaries and Bernstein Theorems Chapter 1. Minimal Surfaces with Supporting Half-Planes . . . . . . . . . . . . . 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

An Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Minimal Surfaces with Cusps on the Supporting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setup of the Problem. Properties of Stationary Solutions . . . . Classification of the Contact Sets . . . . . . . . . . . . . . . . . . . . . . . . Nonparametric Representation, Uniqueness, and Symmetry of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet’s Integral . . . . . . . . . . . . . . . . . . . . Asymptotic Expansions for Surfaces of the Tongue/LoopType II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Results on the Shape of the Trace. Absence of Cusps. Optimal Boundary Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of the Representation Theorem . . . . . . . . . . . . . . . . . . . . . Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 7 11 13 18 21 23 26 28 34

Chapter 2. Embedded Minimal Surfaces with Partially Free Boundaries . 37 2.1 2.2 2.3 2.4 2.5 2.6 2.7

The Geometric Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inclusion and Monotonicity of the Free Boundary Values . . . . A Modification of the Kneser–Rad´ o Theorem . . . . . . . . . . . . . . Properties of the Gauss Map, and Stable Surfaces . . . . . . . . . . Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniqueness of Freely Stable Minimal Surfaces . . . . . . . . . . . . . Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 44 50 52 60 66 74 vii

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Contents

2.8 2.9 2.10 2.11

Edge Creeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Embedded Minimizers for Nonsmooth Supporting Surfaces . . 96 A Bernstein Theorem for Minimal Surfaces in a Wedge . . . . . 108 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Chapter 3. Bernstein Theorems and Related Results . . . . . . . . . . . . . . . . . . 135 3.1

3.2

3.3

3.4

3.5 3.6 3.7

Entire and Exterior Minimal Graphs of Controlled Growth . . 137 3.1.1 J¨ orgens’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.1.2 Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser’s Bernstein Theorem . . . 140 3.1.3 The Interior Gradient Estimate and Consequences . . . 144 First and Second Variation Formulae . . . . . . . . . . . . . . . . . . . . . 145 3.2.1 First and Second Variation of the Area Integral . . . . . . 146 3.2.2 First and Second Variation Formulae for Singular Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Some Geometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.3.1 Covariant Derivatives of Tensor Fields . . . . . . . . . . . . . . 159 3.3.2 Simons’s Identity and Jacobi’s Field Equation . . . . . . . 161 Nonexistence of Stable Cones and Integral Curvature Estimates. Further Bernstein Theorems . . . . . . . . . . . . . . . . . . . 163 3.4.1 Stability of Minimal Cones . . . . . . . . . . . . . . . . . . . . . . . . 164 3.4.2 Nonexistence of Stable Cones . . . . . . . . . . . . . . . . . . . . . . 172 3.4.3 Integral Curvature Estimates for Minimal and α-Minimal Hypersurfaces. Further Bernstein Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Pointwise Curvature Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.7.1 References to the Literature on Bernstein’s Theorem and Curvature Estimates for n = 2 . . . . . . . . . . . . . . . . . 236 3.7.2 Bernstein Theorems and Curvature Estimates for n ≥ 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3.7.3 Bernstein Theorems in Higher Codimensions . . . . . . . . 242 3.7.4 Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Part II. Global Analysis of Minimal Surfaces Chapter 4. The General Problem of Plateau: Another Approach . . . . . . . 249 4.1 4.2 4.3

The General Problem of Plateau. Formulation and Examples 249 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Symmetric Riemann Surfaces and Their Teichm¨ uller Spaces . 263

Contents

4.4 4.5 4.6 4.7

ix

The Mumford Compactness Theorem . . . . . . . . . . . . . . . . . . . . . 271 The Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Existence Results for the General Problem of Plateau in R3 . 285 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Chapter 5. The Index Theorems for Minimal Surfaces of Zero and Higher Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 The Statement of the Index Theorem of Genus Zero . . . . . . . . 302 Stratification of Harmonic Surfaces by Singularity Type . . . . 304 Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type . . . . . . . . . . . . . . . . . . . . . . . . . 318 The Index Theorem for Classical Minimal Surfaces . . . . . . . . . 324 The Forced Jacobi Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Some Theorems on the Linear Algebra of Fredholm Maps . . . 341 Generic Finiteness, Stability, and the Stratification of the Sets Mλ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Review of Some Basic Results in Riemann Surface Theory . . 354 Vector Bundles over Teichm¨ uller Space . . . . . . . . . . . . . . . . . . . 359 Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators . . . . . . . . . . . . . . . . . . . . . . . . . 365 The Corank of the Partial Conformality Operators . . . . . . . . . 369 The Corank of the Complete Conformality Operators . . . . . . . 377 Manifolds of Harmonic Surfaces of Prescribed Branching Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 The Proof of the Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 385 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Chapter 6. Euler Characteristic and Morse Theory for Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Fredholm Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 The Gradient Vector Field Associated to Plateau’s Problem . 405 The Euler Characteristic χ(Wα ) of Wα . . . . . . . . . . . . . . . . . . . 411 The Sard–Brown Theorem for Functionals . . . . . . . . . . . . . . . . 423 The Morse Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 The Normal Form of Dirichlet’s Energy about a Generic Minimal Surface in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 The Local Winding Number of Wα about a Generically Branched Minimal Surface in R3 . . . . . . . . . . . . . . . . . . . . . . . . . 442 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

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6.8.1 Historical Remarks and References to the Literature . . 447 6.8.2 On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Introduction

The first two chapters deal with local and global properties of minimal surfaces which are minimal or at least stationary points of Dirichlet’s integral in a boundary configuration Γ, S consisting of a surface S, with or without boundary, and a Jordan arc Γ with endpoints on S. Here the supporting surface may only be piecewise smooth, such as the boundary of a wedge. The situation considered in Chapter 1 is a special case of the wedge, as here the surface S is a halfplane, which can be viewed as a wedge with an opening angle of zero degrees. In both chapters we prove unique solvability of certain “partially free” boundary value problems for minimal surfaces. A special feature is the phenomenon of edge creeping at free boundaries which is discussed by several examples. Moreover, in Section 2.10, a Bernstein theorem is proved for minimal surfaces with a free boundary on the faces of a wedge. Chapter 3 provides a fairly comprehensive—although still not complete— presentation of results which lead to Bernstein theorems for solutions of the n-dimensional minimal surface equation as for stationary points of sin as well gular integrals of the type Eα (u) := Ω uα 1 + |Du|2 dx, called α-minimal hypersurfaces. The basic results are integral curvature estimates and pointwise curvature estimates for minimal and α-minimal hypersurfaces. Essential tools are formulae for the first and second variation, Simons’s identity for the second fundamental form and “Jacobi’s field equation”, a discussion about the nonexistence of stable cones, monotonicity formulae, and “Sobolev-type” estimates by Michael and Simon. In particular, a proof of the celebrated curvature estimate of Schoen–Simon–Yau is presented which generalizes Heinz’s estimate from n = 2 to n ≤ 5. In Chapter 4, a general version of the Douglas problem for orientable minimal surfaces is treated, based on results from Teichm¨ uller theory. In Chapter 5, the index theorems by Böhme & Tromba and Tomi & Tromba are presented.

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Introduction

Finally, in the last chapter, global results on Fredholm vector fields are applied to set up local winding numbers and Euler characteristics of vector fields whose zeros are “boundaries” of minimal surfaces. In the Scholia of Chapter 6, an index theorem for minimal surfaces of genus greater than one in an n-dimensional Riemannian manifold (n ≥ 3) with negative curvature is developed. Chapters 4–6 require a number of notions and theorems from Teichm¨ uller theory and Global analysis; we give detailed references or prove results on the spot if they cannot easily be found in the literature. We note that these chapters are profoundly influenced by ideas of S. Smale and his approach to global nonlinear analysis. Smale, and also R. Palais, realized that M. Morse’s Analysis in the Large could be put into the context of an Analysis on infinite dimensional manifolds. Along with R. Thom, Smale developed the handlebody approach to Morse theory, and he discovered that both, in the finite, and infinite dimensional setting, the Morse lemma was unnecessary, a basic observation needed for the work of M. Struwe [4,8] on Morse theory of disktype minimal surfaces in Rn with n ≥ 4. In Chapter 6 it is pointed out why no such theory exists in R3 . In his lectures, Smale called for a truly global approach to problems in nonlinear analysis that sees them all as problems on manifolds, even if they originally are defined on linear or affine spaces. In particular he asked for a reformulation of the Leray–Schauder theory in a truly global way, and he stressed the need to find an approach to the global calculus of variations. The influence of Smale on some of the succeeding work is described in Tromba’s account [29]. Here we note that the approach to Teichm¨ uller theory used in Chapter 4, the index theorems of Chapter 5, and the theory of an Euler characteristic presented in Chapter 6 are all influenced by Smale’s point of view. In particular, the theory of an Euler characteristic of vector fields on Banach manifolds and its application to Plateau’s problem is one answer to Smale’s call for a global Leray–Schauder theory.

Introduction

xiii

Plate I. Four soap films with partially free boundaries on a halfplane. Courtesy of ILF Stuttgart

xiv

Introduction

Plate II. (a) Loop, (b) cusp, (c) tongue. Courtesy of ILF Stuttgart

Introduction

xv

Plate III. Solutions to a general Plateau problem. Courtesy of E. Boix, D. Hoffman, and M. Wohlgemuth

xvi

Introduction

Plate IV. Solution of a Douglas problem which is of higher topological type. Courtesy of D. Hoffman

Part I

Free Boundaries and Bernstein Theorems

Chapter 1 Minimal Surfaces with Supporting Half-Planes

In Chapter 2 of Vol. 2 we have investigated the regularity of stationary minimal surfaces in the class C(Γ, S). Such stationary surfaces had been introduced in Section 4.6 of Vol. 1 (cf. also Chapter 1 of Vol. 2). We have shown that, for a uniformly smooth surface S with a smooth boundary ∂S, the stationary surfaces X belong to the class C 1,1/2 (B ∪ I, R3 ). One of the consequences of results proved in the present chapter will be that this regularity result is optimal. Recall that, according to the results of Chapter 2 of Vol. 2, the nonoriented tangent of the free trace Σ = {X(w) : w ∈ I} of a stationary minimal surface X in C(Γ, S) changes continuously. This, in particular, means that the free trace cannot have corners at points where it attaches to the border of the supporting surface S. On the other hand, since isolated branch points of odd order cannot be excluded, there might exist cusps on the free trace. In fact, experimental evidence suggests that cusps do appear for certain shapes of the boundary configuration Γ, S. In Section 1.1 we shall describe soap film experiments, demonstrating the generation of cusps by a suitable bending process of the arc Γ . Such a physical proof for the existence of cusps is, of course, not conclusive in the mathematical sense although it bears strong evidence for the existence of this phenomenon. In Section 1.2 we therefore present several examples of stationary minimal surfaces with cusps on their traces. In fact, such examples are already well known to us (see, for example, Henneberg’s surface and Catalan’s surface) and have been discussed in Section 3.5 of Vol. 1. The main part of this chapter is devoted to the study of the free trace of a stationary surface X within a boundary configuration Γ, S consisting of a half-plane S and a symmetric curve Γ which has a convex projection onto a plane E orthogonal to ∂S and which connects the two sides of S. After classifying the possible sets of contact of the free trace Σ with the boundary ∂S of the supporting half-plane, we prove a representation theorem for stationary surfaces in C(Γ, S) which is the key to all further results of this

U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften 341, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11706-0 1,

3

4

1 Minimal Surfaces with Supporting Half-Planes

chapter. It essentially states that X can be viewed as a nonparametric surface with respect to the plane E. One of the main consequences drawn from this representation theorem is a uniqueness theorem stating that there can be at most one stationary minimal surface whose trace is touching ∂S, and this surface is area minimizing among all surfaces of C(Γ, S). Furthermore we shall derive asymptotic expansions of a stationary surface along its free boundary I which will imply that C 1,1/2 -regularity is in general the optimal regularity result. Finally, we describe the geometrical shape of the free trace, and we exhibit conditions on Γ which prevent the occurrence of branch points.

1.1 An Experiment Let S be a half-plane and consider some arc Γ that starts in some point P1 on the upper side of S, leads about the edge ∂S, and ends in some point P2 on the lower side of S, as depicted in Fig. 1. It is assumed that Γ has no points

Fig. 1.

in common with S, except for P1 and P2 . We can imagine that Γ is obtained from a circle by cutting it and pulling its ends slightly apart. Suppose that S is the part {x ≥ 0, y = 0} of the x, z-plane and that ∂S coincides with the z-axis. Then we may assume that the projection of Γ onto some plane E orthogonal to the z-axis is nearly circular and certainly convex, and that the z-component of a suitable Jordan representation of Γ is monotonically increasing. In this

1.1 An Experiment

5

Fig. 2. (a) Tongue. (b) Cusp

case, the free trace of a soap film spanned in Γ, S is depicted in Fig. 2. Let us now define the arc Γ in such a way that its endpoints on S are kept fixed and the projection of Γ onto the plane is only slightly altered, whereas the z-component of the representation of Γ changes its signs repeatedly (an odd number of times). During this deformation process the free trace may develop a cusp (see Fig. 2). This can be seen by looking at the free trace in various stages of the bending procedure; cf. Fig. 3. Let us deform the arc Γ by twisting it about some axis in the supporting plane orthogonal to the edge. If the twisting is carried sufficiently far, the originally tongue-shaped free trace narrows more and more, forms for a moment a cusp, which then opens and changes into a loop. This loop as well as the original tongue are attached to the border of S along an interval.

Fig. 3. The free trace during various stages of the bending process

Three different forms of the free trace that were actually observed and photographed during an experiment are reproduced in Plate II. It is interesting to contrast the situation depicted in Fig. 4 with another, related experiment where Γ is a circle, cut at some point, which again has its endpoints on opposite sides of the supporting half-plane S, but this time not spread apart. If the circle is turned about its horizontal diameter, the free

6


Fig. 4. (a) Tongue. (b) Cusp. (c) Loop

Fig. 5. (a, b) Another bending process where no cusps are formed

boundary, originally consisting of two matching segments on either side of S (cf. Fig. 5a), opens and develops a shape, depicted in Fig. 5b, which does not contain a cusp at any stage of the turning process. The symmetry assumptions on S and Γ stated above are essential for the following mathematical discussion, but they are by no means essential for the experiment. The supporting surface S can be an arbitrary smooth surface, planar or not, and Γ can be an arbitrary arc which has no points in common with S except for its endpoints. Of course, the free trace of a soapfilm in the frame Γ, S will then be more complicated and can develop several cusps and selfintersections. A mathematical discussion of this general case has not yet been carried out.

1.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface

7

1.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface In the sequel, B will not denote the unit disk {|w| < 1} but the semidisk B := {w ∈ C : |w| < 1, Im w > 0}, and I denotes the interval I := {u ∈ R : |u| < 1} on the real axis. Finally we introduce the circular arc C := ∂B \ I. Definitions, theorems, etc. concerning surfaces previously defined on the whole disk {|w| < 1} are then carried over to surfaces defined on the semidisk B by means of a conformal map τ : {|w| < 1} → B keeping the three points 1, −1, i fixed. As in (1), we consider the half-plane S = {(x, y, z) ∈ R3 : x ≥ 0, y = 0} as supporting surface. In Sections 3.4 and 3.5 of Vol. 1 we have seen how Schwarz’s formula solving Björling’s problem can be used to construct stationary surfaces X : B → R3 which intersect S perpendicularly in a given curve Σ having a cusp at the origin of the system of coordinates. The surfaces of Henneberg and Catalan are prominent examples of such minimal surfaces. Let us consider the following rescaled version of Henneberg’s surface, a portion of which is pictured in Figs. 1 and 2: x = cosh(2λu) cos(2λv) − 1, (1)

y = − sinh(λu) sin(λv) −

1 sinh(3λu) sin(3λv), 3

z = − sinh(λu) cos(λv) +

1 sinh(3λu) cos(3λv). 3

It follows from 1 X(u, 0) = cosh(2λu) − 1, 0, − sinh(λu) + sinh(3λu) 3 =

4 2 sinh2 (λu), 0, sinh3 (λu) 3

8


that (1) intersects the plane y = 0 in Neil’s parabola 2x3 = 9z 2 ,

(2)

y = 0.

For small values of w we have the expansion x(w) = Re{2λ2 w2 + · · · }, y(w) = Re{2iλ2 w2 + · · · }, 4 z(w) = Re λ3 w3 + · · · . 3 Let us denote by M the portion of (1) which corresponds to the closed semidisk B = {w : |w| ≤ 1, v ≥ 0} in the parameter plane. The surface M is bounded by a configuration Γ, S where S is the half-plane {x ≥ 0, y = 0}, and Γ is the image of the circular arc C under the mapping (1), that is, the arc {X(eiθ ) : 0 ≤ θ ≤ π}. The free boundary of M on S is Neil’s parabola (2); M and S meet at a right angle along this curve. The orthogonal projection of Γ onto the x, y-plane is a smooth closed curve. For a later reference we observe that this curve is convex as long as the ˆ 1.014379 . . . . (It turns out parameter λ remains in the interval 0 < λ ≤ λ0 = that λ0 is the first positive root of the equation tan(2λ) = −2λ.) Certain other algebraic singularities of the free boundary are also possible. For the minimal surface represented by the equations

(3)

x = Re{w2 }, w 4n−2 ω 1+ω dω , y = Re 2i z = Re

0

2 w2n+1 , 2n + 1

Fig. 1. A part of Henneberg’s surface as solution in a configuration Γ, S whose free trace on S has a cusp

1.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface

9

Fig. 2. Another view of Henneberg’s surface in a configuration Γ, S. Courtesy of I. Haubitz

Fig. 3. Two views of two cusps in Henneberg’s surface

the free boundary, i.e., the image of I on the half-plane S = {x ≥ 0, y = 0}, is the curve 4x2n+1 = (2n + 1)2 z 2 , y = 0. We can state even simpler examples if we do not insist on classical curves as free boundaries. One very simple example is furnished by the minimal surface x = Re{w2 − 18λ2 w4 }, (4)

y = Re{iw2 + 18iλ2 w4 },

λ > 0,

z = Re{8λw } 3

which meets the half-plane S orthogonally along the curve

10


Fig. 4. A boundary configuration Γ, S consisting of a disk S and a disjoint Jordan curve Γ . It bounds a stationary minimal surface of annulus type which meets S perpendicularly at an asteroid

Fig. 5. The annulus-type stationary minimal surface within the configuration Γ, S depicted in Fig. 4 is part of the adjoint of Henneberg’s surface. The four cusps correspond to four branch points

x(u) = u2 − 18λ2 u4 ,

y(u) = 0,

z(u) = 8λu3

which has the expansion z = 8λx3/2 + · · · ,

y=0

about the origin. As arc Γ we shall again use the image of the circular arc C, this time under the mapping (4). The orthogonal projection of Γ onto the x, y-plane is the closed curve x = cos θ − 18λ2 cos 2θ, y = sin θ + 18λ2 sin 2θ, This curve is convex if 0 < 18λ2 ≤ 0.117851 . . . .

1 4,

0 ≤ θ ≤ 2π.

that is, if 0 < λ ≤ λ0 :=

√

2/12 =

1.3 Setup of the Problem. Properties of Stationary Solutions

11

It is not at all a priori clear that the above surfaces are solutions of the minimum problem (5)

D(X) → min

in C(Γ, S).

This will, in fact, follow from the uniqueness theorem proved in Section 1.5. In particular Henneberg’s surface (1) provides us with a simple example of a solution of the minimum problem (5) which possesses a cusp on its trace.

1.3 Setup of the Problem. Properties of Stationary Solutions We will now prepare the mathematical discussion to be carried out in the following sections. We begin by fixing the assumptions on the boundary configuration Γ, S which are supposed to hold throughout Sections 1.3–1.9. Assumption A. Let S be the half-plane {(x, y, z) : x ≥ 0, y = 0} in R3 . Moreover, the curve Γ is assumed to be a regular arc of class C 1,α , 0 < α < 1, with the endpoints P1 and P2 , P1 = P2 , which issues from S at right angles and meets S only in its endpoints. Close to P1 , the arc Γ is supposed to lie in the half-space {y ≥ 0}. Assume also that Γ is symmetric with respect to the x-axis, and that the orthogonal projection of Γ onto the x, y-plane is a closed, strictly convex and regular curve γ of class C 1,α . Finally, suppose that the projection of Γ onto γ is one-to-one, except for the endpoints P1 and P2 of Γ which are projected onto the same point of γ. This assumption is satisfied by the examples discussed in Section 1.2. Assume that P (s) = (p1 (s), p2 (s), p3 (s)), 0 ≤ s ≤ L, is a parametrization of Γ by the arc length s such that (1)

P (0) = P1 = (a, 0, −c),

P (L) = P2 = (a, 0, c)

where a > 0 and c > 0. Then P3 := P (L/2) is the uniquely determined intersection point of Γ with the x-axis which must be of the form (2)

P (L/2) = P3 = (−b, 0, 0),

b > 0.

This is illustrated in Fig. 1. Let us now recall that the definition of stationary minimal surfaces was phrased in such a way that these surfaces are precisely the stationary points of Dirichlet’s integral within the class C(Γ, S). In Chapter 2 of Vol. 2 we have formulated the following result: Lemma 1. Every stationary minimal surface in C(Γ, S) is continuous in the closure B of the parameter domain B. From the regularity theory of Chapter 2 in Vol. 2 we can also derive the following result:

12


Fig. 1. Assumption A

Lemma 2. If X is a stationary minimal surface in C(Γ, S) and if Γ, S satisfies Assumption A, then X belongs to C 1 (B, R3 ) and satisfies (3)

y(u) = 0,

zv (u) = 0

for u ∈ I.

Thus y(w) and z(w) can be continued analytically across the interval I = {|u| < 1} of the u-axis, and the extended functions are harmonic in the whole disk {w : |w| < 1}. The set I1 := {u ∈ I : x(u) > 0} is an open subset of R containing the intervals (−1, −1 + 2δ0 ) and (1 − 2δ0 , 1) for some sufficiently small δ0 > 0. Hence the set of contact I2 = {u ∈ I : x(u) = 0} is closed in R. In addition, we have (4)

xv (u) = 0

for u ∈ I1 .

Proof. The regularity theory of Chapter 2 of Vol. 2 yields that X is of class C 1 on B \ {±1}. Since X is stationary in C(Γ, S), it follows that y(u) = 0 for u ∈ I holds as well as xv (u) = 0,

zv (u) = 0 for u ∈ I1 .

1.4 Classification of the Contact Sets

13

The first equation implies yu (u) = 0 for u ∈ I. Furthermore, the relations x(u) ≥ 0 for u ∈ I,

x(u) = 0 for u ∈ I2

imply that xu (u) = 0 for u ∈ I2 , whence Xu (u) = (0, 0, zu (u))

for u ∈ I2

and zu (u) = 0 on I2 except for isolated points. By virtue of Xu , Xv = 0, we infer that zv (u) = 0 for u ∈ I2 , and therefore zv (u) = 0 for all u ∈ I. Thus we have verified (3) and (4), and, in view of the reflection principle, the functions y(w) and z(w) can be continued analytically across the interval I on the u-axis by setting y(u − iv) := −y(u + iv),

z(u − iv) := z(u + iv)

for v ≥ 0. The extended functions y(w) and z(w) are harmonic in the disk {w : |w| < 1}. Since X ∈ C 0 (B, R3 ) and a > 0, the points X(u) lie in the interior of the half-plane S if u is close to ±1. Hence there is a number δ0 > 0 such that the intervals (−1, −1 + 2δ0 ) and (1 − 2δ0 , 1) on the u-axis are contained in I1 . On the part X(I1 ) of the free trace, the surface X meets S perpendicularly. Hence we can continue X(w) analytically across I1 by a reflection with respect ˆ to S, and the extended surface X(w) is a minimal surface on {w : |w| < ˆ 1, w ∈ / I2 }. Moreover, X(w) is continuous on {w : |w| < 1}. Since Γ issues ˆ maps the unit circle {w : |w| = 1} from S perpendicularly, the surface X bijectively onto a closed regular curve of class C 1,α . Then the regularity results ˆ stated in Section 2.12 of Vol. 2 imply that X(w) is of class C 1,α in the strip 1 {1 − δ0 ≤ |w| ≤ 1}. Thus X is of class C on B.

1.4 Classification of the Contact Sets The principal result of this section is the following observation: The free trace of a stationary minimal surface in C(Γ, S) either meets the boundary ∂S of the half-plane S in a single point, or in a single subinterval, or in no point at all. More precisely, we shall prove:

14


Theorem. Let X(w) = (x(w), y(w), z(w)) be a stationary minimal surface in C(Γ, S), and set I1 := {u ∈ I : x(u) > 0}, I2 := {u ∈ I : x(u) = 0}, and x0 := min{x(u) : u ∈ I}. Then only the following three cases can occur: (I) x0 = 0, and I2 consists of a single point u0 ; (II) x0 = 0, and I2 is a closed interval of positive length; (III) x0 > 0, that is, I2 is empty, and there is exactly one point u0 in I such that x0 = x(u0 ). Consequently, we have x(u) > x0 for u ∈ I with u = u0 . Remark. Case I may indeed occur as we see from the examples given in Section 1.2. If we introduce the new supporting surface Sε = {(x, y, z) ∈ R3 : y = 0, x ≥ −ε},

ε > 0,

for some sufficiently small ε > 0 as well as the new coordinates ξ = x + ε,

η = y,

ζ = z,

a surface X(w) = (x(w), y(w), z(w)) of type I is transformed into a surface Ξ(w) = (ξ(w), η(w), ζ(w)) of type III. Hence also the case III may appear. On the other hand, we shall see in Section 1.6 that minima of Dirichlet’s integral are never of type III. As a first step towards the proof of the Theorem we draw some preliminary information from the maximum principle which is formulated as Lemma 1. The trace X(I) is contained in the strip {(x, y, z) : 0 ≤ x < a, y = 0} of the half-plane S whence (1)

0 ≤ x0 < a.

Moreover, we have (2)

−b < x(w) < a

for all w ∈ B.

Proof. In fact, if there were some u ∈ I with x(u) ≥ a, then there would exist some u∗ ∈ I such that x(u∗ ) = max x(u) ≥ a > 0, I

since x(±1) = a. Since x(w) is harmonic and nonconstant in B, the lemma of E. Hopf1 implies that1 xv (u∗ ) < 0. Since u∗ belongs to I1 , this contradicts Lemma 2 of the preceding section. Thus we have proved that x0 := min{x(u) : u ∈ I} satisfies (1). Moreover, the x-component p1 (s) of the representation P (s) of Γ satisfies 1

Cf. Gilbarg and Trudinger [1], p. 33.


−b ≤ p1 (s) ≤ a

15

for 0 ≤ s ≤ L

whence −b ≤ x(w) ≤ a

for w ∈ C.

On account of 0 ≤ x(u) < a

for u ∈ I

we infer relation (2) from the maximum principle.

The next lemma is the crucial step for the proof of the Theorem. We need the following notations: For each value μ ∈ R we define the open (and possibly empty) subsets B(μ), B + (μ), and B − (μ) of B by B(μ) := {w ∈ B : x(w) = μ}, B + (μ) := {w ∈ B : x(w) > μ}, B − (μ) := {w ∈ B : x(w) < μ}. By virtue of Lemma 1, we obtain B + (μ) = ∅, +

B (μ) = B,

B − (μ) = B −

B (μ) = ∅

if μ ≥ a, if μ ≤ −b.

Recall that X(w) provides a topological mapping of the circular arc C onto Γ . By Assumption A there are, for each value μ ∈ (−b, a), exactly two points w1 (μ) = eiθ1 (μ) and w2 (μ) = eiθ2 (μ) on C, 0 < θ1 (μ) < θ2 (μ) < π, with the property that x(w1 (μ)) = x(w2 (μ)) = μ. In addition we set w1 (−b) = w2 (−b) := i, w1 (a) := 1,

w2 (a) := −1,

θ1 (−b) = θ2 (−b) := θ1 (a) := 0,

π , 2

θ2 (a) := π.

For μ ∈ (−b, a), we define the following open subarcs of C: C1+ (μ) := {w = eiθ : 0 < θ < θ1 (μ)}, C − (μ) := {w = eiθ : θ1 (μ) < θ < θ2 (μ)}, C2+ (μ) := {w = eiθ : θ2 (μ) < θ < π}. Lemma 2. For each μ ∈ (−b, a), the set B − (μ) is connected, and the set B + (μ) can have at most two components.

16


Proof. We proceed as follows: (i) First we fix some μ ∈ (−b, a), and denote by Q the component of B − (μ), the boundary of which contains the arc C − (μ). If B − (μ) were not connected, there would exist another nonempty component R of B − (μ). Clearly, ∂R ⊂ B ∪ I ∪ {w1 (μ), w2 (μ)}, x(w) = μ for w ∈ ∂R ∩ (B ∪ C), and x(w) ≤ μ for w ∈ ∂R ∩ I. (Note that ∂R ∩ I is void for μ < 0.) If ∂R ∩ I were empty, the maximum principle would imply that x(w) ≡ μ on R, so that, contrary to the facts, x(w) ≡ μ on B. If ∂R ∩ I is nonempty (this is only possible for μ ≥ 0), let m := inf{x(u) : u ∈ ∂R ∩ I}. We claim that m < μ. Otherwise, if m = μ, we could obtain a contradiction as before. From x(u) ≥ 0 we conclude that 0 ≤ m < μ. Thus, R has to be void if μ ≤ 0. If 0 < μ < a, the value u := sup{u ∈ I ∩ ∂R : x(u) = m} satisfies −1 0 such that (u − ε, u + ε) ⊂ ∂R. By the maximum principle and E. Hopf’s lemma it follows that xv (u) > 0. On the other hand, by the definitions of m and u, a right neighbourhood U of u on I must belong to I1 . By Lemma 2 of Section 1.3, it follows that xv (u) = 0 for u ∈ U, and xv is continuous on I. Thus we arrive at the contradictory conclusion xv (u) = 0. We have proved that B − (μ) is connected for all μ ∈ (−b, a). (ii) Again, we select a value μ ∈ (−b, a). Denote by Q1 and Q2 the two components of B + (μ), the boundary of which contains C1+ (μ) and C2+ (μ) respectively. It is of course possible that Q1 and Q2 are identical. We assert that B + (μ) cannot have further components. Otherwise, if R were such a nonempty component different from Q1 and Q2 , we would have ∂R ⊂ B ∪ I ∪ {w1 (μ), w2 (μ)}, x(w) = μ for w ∈ ∂R ∩ (B ∪ C), and x(w) ≥ μ for w ∈ I ∩ ∂R. If ∂R ∩ I were empty, the maximum principle would lead to a contradiction, as in (i). We may therefore assume that ∂R ∩ I is nonvoid. If −b < μ < 0, the level set l(μ) = {w : x(w) = μ} cannot touch I. In fact, there is a strip sε = {w = u+iv; 0 ≤ v < ε} abutting on I which is not penetrated by l(μ), so that sε ⊂ Q1 ∪ Q2 . But this is incompatible with the assumption ∂R ∩ I = ∅. We turn to the case 0 ≤ μ < a. Neighbourhoods in B of the corner points w = ±1 belong to the components Q1 and Q2 . Hence there is a δ > 0, such that ∂R ∩ I ⊂ {u : |u| < 1 − δ}. Then there exists a point u ∈ ∂R ∩ I in which x(u) attains the maximum value m = max{x(u) : u ∈ I ∩ ∂R}. As in (i) we conclude from the maximum principle that m > μ. Then there is a σ > 0 such that the interval (u − σ, u + σ) on I belongs to the boundary ∂R, whence xv (u) < 0, again on account of E. Hopf’s lemma. On the other hand, 0 ≤ μ < m = x(u) implies that the point u belongs to I1 which leads to the contradicting statement xv (u) = 0. Therefore, B + (μ) has no components other than Q1 and Q2 .


17

Remark. The proof of Lemma 2 yields further information regarding the set B + (μ). We see for instance that B + (μ) is connected for −b < μ < 0. If B + (μ) consists of two different components, then the boundary of one of these components, Q1 , contains all points of the arc C1+ (μ), while C2+ (μ) is part of the boundary of the other component Q2 . Now we turn to the proof of the Theorem. Set u0 := min{u ∈ I : x(u) = x0 },

u0 := max{u ∈ I : x(u) = x0 }.

Clearly we have −1 < u0 ≤ u0 < 1. Then one of the following three, mutually exclusive cases must hold: (α) u0 = u0 ; (β) u0 < u0 ,

x(u) = x0

for all u ∈ [u0 , u0 ];

u0 < u0 ,

x(u) > x0

for some u ∈ (u0 , u0 ).

(γ)

We shall show first that case (γ) cannot occur. In case (γ) we would be able to find two points u1 , u2 ∈ [u0 , u0 ], u1 < u2 , such that x(u1 ) = x(u2 ) = x0 and x(u) > x0 for u1 < u < u2 . Set m := max{x(u) : u1 < u < u2 },

0 ≤ x0 < m < a,

and assume that x(u ) = m, u1 < u < u2 . Then xu (u ) = 0; moreover we have xv (u) = 0 for u1 < u < u2 , since (u1 , u2 ) ⊂ I1 . Therefore, x(w) can be continued analytically as a harmonic function across the segment u1 < u < u2 of the u-axis into the lower half of the w-plane. In a (full) neighbourhood of the point w = u this function has an expansion x(w) = m + Re{κ(w − u )ν + · · · },

κ = 0,

ν ≥ 2,

since ∇x(u ) = 0 and xv (u) = 0 for u1 < u < u2 . From the fact that u = u is a local maximum of x(u) on I we conclude that κ < 0 and ν = 2n, n ≥ 1. A neighbourhood of w = u in B is divided into 2n + 1—at least three—open sectors σ1 , σ2 , . . . , σ2n+1 such that x(w) < m in σ1 , σ3 , . . . , σ2n+1 , and that x(w) > m in σ2 , σ4 , . . . , σ2n . Now consider two points w1 and w2n+1 in σ1 and σ2n+1 respectively. As we know from Lemma 2, the set B − (m) is connected and contains the points w1 and w2n+1 . Thus we can connect w1 and w2n+1 by a path γ˜ contained in B − (m). Connecting w1 and w2n+1 with u in σ1 and σ2n+1 respectively we obtain a closed curve which separates the component Ω2 of B + (m) containing the sector σ2 from the components Q1 and Q2 that were introduced in the proof of the preceding lemma. In other words, the case (γ) would imply that B + (m) has at least three components, which is not true. Having ruled out case (γ), we shall now prove that (β) cannot hold unless x0 = 0. In fact, the inequality x0 > 0 would imply xv (u) ≡ 0 on I, and then the unique continuation principle would yield x(w) ≡ x0 in B if (β) were true. This is again not possible. Therefore the relation x0 > 0 implies that we are in case (α), and the proof is completed.

18


1.5 Nonparametric Representation, Uniqueness, and Symmetry of Solutions the following representation theorem which will be proved in Section 1.9 is the key to all the other results of this chapter. It states that all stationary minimal surfaces X in C(Γ, S) are graphs. Theorem 1 (Representation theorem). Let X be a stationary minimal surface in C(Γ, S), and let x0 be the lowest x-level of the free trace of X, that is, x0 := min{x(u) : u ∈ I}. Moreover, denote by D = D(x0 ) the twodimensional domain in the x, y-plane which is obtained from the interior of the orthogonal projection γ of Γ by slicing this interior along the x-axis from ˆ of the slit domain D, both x = x0 to x = a. In defining the boundary ∂D borders of the slit x0 < x ≤ a will appear, with opposite orientation. ˆ Then the functions x(w), y(w) provide a C 1 -mapping of B onto D ∪ ∂D which is topological, except in case II, where the interval of coincidence I2 = {u ∈ I : x(u) = 0} ˆ corresponds wholly to the point (0, 0) on ∂D. Moreover, the minimal surface M with the position vector X(w) admits a nonparametric representation z = Z(x, y) over the domain D. The function Z(x, y) is real analytic in D, and on both shores of the open segment x0 < x < a, and (1)

∂ ∂ Z(x, y) = lim Z(x, y) = 0 y→+0 ∂y y→−0 ∂y lim

for x0 < x < a.

ˆ in cases I and III. In case II, Z(x, y) is Z(x, y) is continuous on D ∪ ∂D ˆ continuous on D ∪ ∂D \ {(0, 0)} and remains bounded upon approach of the point (0, 0). As we shall immediately see, this result implies the following Theorem 2 (Uniqueness theorem). If X1 and X2 are two stationary minimal surfaces in C(Γ, S) which are normed in the same way, say, X1 , X2 ∈ C∗ (Γ, S), and whose free traces X1 (I) and X2 (I) have the same lowest x-levels, then X1 (w) ≡ X2 (w) on B. In particular, two stationary minimal surfaces in C∗ (Γ, S) coincide on B if both are not of type III. Let X(w) = (x(w), y(w), z(w)), w = u+iv, be a stationary minimal surface in C∗ (Γ, S). Then also ˆ + iv) := (x(−u + iv), −y(−u + iv), −z(−u + iv)) X(u ˆ have the is a stationary minimal surface in C∗ (Γ, S), and the surfaces X and X ˆ same lowest x-levels. Then the uniqueness theorem implies that X(w) ≡ X(w) on B, and we obtain

1.5 Nonparametric Representation, Uniqueness, and Symmetry of Solutions

19

Theorem 3 (Symmetry theorem). Every stationary minimal surface X ∈ C∗ (Γ, S) is symmetric with respect to the x-axis. More precisely, we have (2)

x(u + iv) = x(−u + iv),

(3) (4)

y(u + iv) = −y(−u + iv), z(u + iv) = −z(−u + iv).

In cases I or III we have x0 = x(0)

and

x0 < x(u) < a

for u ∈ I, u = 0.

In case II, I2 is of the form [u1 , u2 ], where 0 < u2 < 1 and u1 = −u2 . Clearly, relations (3) and (4) imply (5)

y(iv) = z(iv) = 0

for all v ∈ [0, 1].

Finally, the nonparametric representation z = Z(x, y) of the minimal surface M, given by X : B → R3 , satisfies Z(x, y) = −Z(x, −y)

for (x, y) ∈ D(x0 ),

and therefore also lim Z(x, y) = − lim Z(x, y),

y→+0

y→−0

x = 0,

in case II. Now we come to the proof of Theorem 2. The domain D introduced in the representation theorem is the same for X1 and X2 , even if the diffeomorphisms B → D given by the first two components differ. Therefore we have the nonparametric representations z = Z1 (x, y) and z = Z2 (x, y) respectively with (x, y) ∈ D, for the two surfaces X1 and X2 . The functions Z1 (x, y) and Z2 (x, y) have the properties stated in Theorem 1 and satisfy (6)

Z1 (x, y) = Z2 (x, y)

for all (x, y) ∈ γ,

where γ is the projection of Γ onto the x, y-plane. We will show that Z1 and Z2 coincide in D. For j = 1, 2, we set ∂ ∂ pj := Zj , qj := Zj , Wj := 1 + p2j + qj2 . ∂x ∂y For fixed (x, y) ∈ D and for t ∈ [0, 1] we introduce the notations p(t) := p1 + t(p2 − p1 ), q(t) := q1 + t(q2 − q1 ), W (t) := {1 + p2 (t) + q 2 (t)}1/2 ,

20


as well as f (t) := (p2 − p1 )

p(t) p1 − W (t) W1

+ (q2 − q1 )

q(t) q1 − . W (t) W1

Note that f (0) = 0. Then, in view of the mean value theorem, there is some t = t(x, y) ∈ (0, 1) such that f (1) = f (t).

(7)

Furthermore, a brief calculation yields f (t) ≥ W −3 (t)[(p2 − p1 )2 + (q2 − q1 )2 ]. Since (W 2 (t)) ≥ 0, we obtain (8)

f (t) ≥ (max{W1 , W2 })−3 [(p2 − p1 )2 + (q2 − q1 )2 ].

For δ > 0 and ε > 0 we now introduce the set Dδ,ε consisting of all points in D the distance of which from (x0 , 0) and (a, 0) exceeds ε, and whose distance ˆ is greater than δ. Let Q be an arbitrary compact subset of Dδ,ε , and from ∂D set m(Q) := max{W1 (x, y), W2 (x, y) : (x, y) ∈ Q},

and

[(p2 − p1 )2 + (q2 − q1 )2 ] dx dy.

I(Q) := Q

Invoking (7) and (8), we arrive at 3 3 I(Q) ≤ m (Q) f (1) dx dy ≤ m (Q) Q

f (1) dx dy. Dδ,ε

Inserting

p2 p1 q2 q1 + (q2 − q1 ) , − − f (1) = (p2 − p1 ) W2 W1 W2 W1

and applying an integration by parts, we obtain that

p1 q1 q2 p2 3 dx + dy . (Z1 − Z2 ) − − − I(Q) ≤ m (Q) W1 W2 W1 W2 ∂Dδ,ε Letting δ decrease to zero, keeping ε fixed, we infer from the boundary conditions (1) and (6) that

p1 q1 q2 p2 dx + dy (Z1 − Z2 ) − − − (9) I(Q) ≤ m3 (Q) W1 W2 W1 W2 Cε

1.6 Asymptotic Expansions for Surfaces of Cusp-Types I and III

21

where Cε denotes the parts of the circles {x2 +y 2 = ε2 } and {(x−a)2 +y 2 = ε2 } ˆ Since the integrand of the right-hand side of which are contained in D ∪ ∂D. (9) is bounded, the line integral tends to zero as ε → 0 whence I(Q) = 0 for every compact subset Q of D. It follows that ∇Z1 (x, y) ≡ ∇Z2 (x, y)

in D,

and therefore also Z1 (x, y) ≡ Z2 (x, y)

in D,

on account of (6). Consequently X1 and X2 are conformal representations of the same nonparametric minimal surface M, with the same parameter domain B and satisfying the same three-point condition. From this we conclude that X1 (w) ≡ X2 (w) because a conformal map of B onto itself has to be the identical map if it leaves three points on ∂B fixed.

1.6 Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet’s Integral The central result of this section is the following Theorem 1. Minima of Dirichlet’s integral in C∗ (Γ, S) are not of type III. In Chapter 4 of Vol. 1 we have proved that there is always a solution of the minimum problem in C∗ (Γ, S). By Theorem 1, this minimum has to be of type I or II. On the other hand, the uniqueness theorem of Section 1.5 states that there is at most one stationary minimal surface in C∗ (Γ, S) if surfaces of type III are excluded. Hence Theorem 1 implies the following result: Theorem 2. (i) Stationary minimal surfaces in C(Γ, S) furnish the absolute minimum of Dirichlet’s integral in C(Γ, S) if and only if they are of type I or II. (ii) There exists one and only one minimum of Dirichlet’s integral in C∗ (Γ, S). Hence the stationary surfaces of type III constructed in Section 1.4 do not minimize Dirichlet’s integral within C(Γ, S). A proof of Theorem 1 can be based on the following asymptotic expansions for surfaces of type I or III: Theorem 3. Let X(w) = (x(w), y(w), z(w)) be of class I or III. Then w = 0 is a first order branch point of X(w), and we have the expansion x(w) = x0 + Re{κw2 + · · · }, (1)

y(w) = Re{iκw2 + · · · }, z(w) = Re{μw2n+1 + · · · },

where κ > 0, μ is real and = 0, and n is an integer ≥ 1.

22


Proof. We note that ∇y(0) = 0, since yu (u) = 0 for all u ∈ I, and yv (0) = 0 by (5) of Section 1.5. In cases I and III we have (2)

x2u (u) + zu2 (u) = yv2 (u) for all u ∈ I.

Combining this identity with yv (0) = 0, we conclude that xu (0) = 0 and zu (0) = 0. Since xv (u) = zv (u) = 0 for all u ∈ I in cases I and III, we see that ∇x(0) = ∇z(0) = 0. Since x(u) > x0 for 0 < |u| ≤ 1, the arguments employed in the proofs of Lemma 2 and the Theorem of Section 1.4 lead, for small w, to an expansion (3)

x(w) = x0 + Re{κw2 + · · · },

κ > 0.

Hence, w = 0 is a branch point of order one for X. From the relation zv (u) = 0, u ∈ I, it follows that z(w) can be extended harmonically across the u-axis and that, in view of Section 1.5, (5), an expansion z(w) = Re{μwm + · · · } is obtained in which μ is real and = 0 and m is an integer ≥ 2. Formula (4) of Section 1.5 shows that this integer must be odd so that, near w = 0, (4)

z(w) = Re{μw2n+1 + · · · },

n ≥ 1.

Recall now that the vector A = 0 appearing in the general expansion formula X(w) = X0 + Re{Awn + · · · } satisfies A, A = 0. Therefore we obtain, in conjunction with the formulas (2), (3) and the relations yv (0) = 0, y(u) = 0 on I, the following local expansion for y(w): y(w) = Re{±iκw2 + · · · }. Here the plus sign must be chosen because yv (u) < 0 for 0 0, v > 0}, so that by virtue of the maximum principle y(w) < 0 for w ∈ Q. Proof of Theorem 1. Because of Section 1.5, (5), y(iv) vanishes for all v ∈ [0, 1]. Since x(0) = x0 ≥ 0, and x(i) = −b < 0, there exists a smallest number v1 in [0, 1) such that x(iv 1 ) = 0. Suppose now that X is a solution of the minimum problem in C∗ (Γ, S) which is of type III. Then, 0 < v1 < 1. Denote by B the slit domain obtained by cutting the semidisk B along the imaginary axis from w = 0 to w = iv 1 . Furthermore, let w = τ (ζ) be the conformal mapping from B onto B , leaving the three points w = +1, −1, i fixed. Then, Y (ζ) = X(τ (ζ)) is again of class C∗ (Γ, S) since y(iv) = 0 for all v ∈ [0, 1]. From the invariance of the Dirichlet integral with respect to conformal mappings we conclude that Y (ζ) is also a solution of the minimum

1.7 Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II

23

problem in C∗ (Γ, S), but of type I, by virtue of the Theorem in Section 1.4. By (1), Y (ζ) = (y 1 (ζ), y 2 (ζ), y 3 (ζ)) possesses an expansion near ζ = 0 of the form y 1 (ζ) = Re{κζ 2 + · · · }, (5)

y 2 (ζ) = Re{iκζ 2 + · · · }, y 3 (ζ) = Re{μζ 2n+1 + · · · },

where κ > 0, μ = 0 and n ≥ 1. Let ζ = α + iβ. We infer from (5) that the images of suitable segments (−ε, 0) and (0, ε), ε > 0, on I under the mapping Y (ζ) are different, that is, y 3 (−α) = y 3 (α ) if 0 < α, α < ε. On the other hand relation (5) in Section 1.5, z(iv) = 0 for 0 ≤ v ≤ 1, implies that y 3 (α) = 0 for 0 ≤ |α| ≤ ε , α ∈ I, if ε is a sufficiently small positive number. Such a discrepancy is not possible, and X cannot be of type III. Finally we shall give another proof of Theorem 1 without using the expansion formula. The symmetry theorem of the previous section shows that the minimum X in C∗ (Γ, S) maps the interval {w = iv : 0 ≤ v ≤ 1} onto the x-axis. If X is of type III, that is, if x0 > 0, then also the value v1 := inf{v ≥ 0 : x(iv) ≤ 0} is positive. Now let τ be the conformal mapping from B onto the slit semidisk B − {iv : 0 ≤ v ≤ v1 } mapping each of the points i, 1, −1 onto itself. Since the Dirichlet integral is conformally invariant, we conclude that X ◦ τ =: Y = (y 1 , y 2 , y 3 ) is another minimum for the Dirichlet integral in C∗ (Γ, S), but Y is of type I. Because of formula (5) in Section 1.5, the third component z(w) of the minimum X vanishes for w = iv, 0 ≤ v ≤ 1. Therefore the third component y 3 (w) of Y (w) satisfies y 3 (u, 0) = 0 and yv3 (u, 0) = 0 on certain intervals (−δ, 0) and (0, δ), δ > 0, which are mapped by τ onto the slit {iv : 0 < v < v1 }. The reflection principle implies that y 3 (w) ≡ 0 on B, which is impossible.

1.7 Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II The aim of this section is the proof of the following Theorem. Let X(w) = (x(w), y(w), z(w)) be a stationary minimal surface in C∗ (Γ, S) which is of type II, and let [u1 , u2 ] be its set of coincidence I2 , −1
0.) Then there are positive numbers κ and μ, and a real number z1 = 0, such that x(w) = Re{iκ(w − u1 )3/2 + · · · } (1)

y(w) = Re{−iμ(w − u1 ) + · · · },

near w = u1 ,

z(w) = Re{z1 − (sign z1 )μ(w − u1 ) + · · · } and x(w) = Re{κ(w − u2 )3/2 + · · · }, (2)

y(w) = Re{iμ(w − u2 ) + · · · },

near w = u2 ,

z(w) = Re{−z1 − (sign z1 )μ(w − u2 ) + · · · }. Moreover, no point on I is a branch point of X(w). Proof. Let h(w) be the holomorphic function in a neighbourhood of w = u1 in B satisfying h(u1 ) = 0 such that x(w) = Re h(w), and g(w) = h (w) = xu (w) − ixv (w). If u ∈ I is close to u1 , we have Re g(u) = 0 for u > u1 , and Im g(u) = 0 for u < u1 . Consider the transformation w = u1 + ζ 2 , and set f (ζ) = g(u1 + ζ 2 ). The function f (ζ) is holomorphic near ζ = 0 in {ζ : Re ζ > 0, Im ζ > 0}, and Re f (ζ) vanishes on the positive real ζ-axis, while Im f (ζ) is zero on the positive imaginary axis. The C 1 -character of x(w) in B allows us to extend f (ζ) by a twofold reflection analytically to a holomorphic function in a full neighbourhood of the point ζ = 0, with an expansion f (ζ) = a0 + a1 ζ + a2 ζ 2 + · · ·

near ζ = 0.

The relations xu (u1 ) = xv (u1 ) = 0 imply that a0 = f (0) = 0. For v ≥ 0 we then get the expansion g(w) = a1 (w − u1 )1/2 + a2 (w − u1 ) + a3 (w − u1 )3/2 + · · · . (We choose the branch of the square root which is positive for large positive values of w.) An integration leads to the expansion x(w) = Re{b0 + b1 (w − u1 )3/2 + b2 (w − u1 )2 + b3 (w − u1 )5/2 + · · · } with complex coefficients bj = pj + iq j . From the relation x(u) = 0 for u > u1 it follows that p0 = p1 = p2 = · · · = 0; we may also assume that q0 = 0. The condition xv (u) = 0 for u < u1 allows us to conclude that q2 = q4 = · · · = 0. Denoting the first non-vanishing coefficient of the remaining ones by iκ, we arrive at x(w) = Re{iκ(w − u1 )n+1/2 + · · · }

1.7 Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II

25

where (−1)n κ < 0, and n ≥ 1. By virtue of formula (2) in Section 1.5 we also have the expansion x(w) = Re{κ(w − u2 )n+1/2 + · · · } for w ∈ B near the value u2 . Arguments similar to those employed in the proofs in Section 1.4 show that we have n = 1 in the above expansions. Thus we obtain x(w) = Re{iκ(w − u1 )3/2 + · · · } near w = u1 ,

(3)

x(w) = Re{κ(w − u2 )3/2 + · · · } near w = u2 .

The harmonic function y(w) vanishes on I as well as for w = iv, 0 ≤ v ≤ 1, while y(eiθ ) < 0 for 0 < θ < π2 and y(eiθ ) > 0 for π2 < θ < π. Consider the two sets Q− = {w : |w| < 1, u > 0, v > 0} and Q+ = {w : |w| < 1, u < 0, v > 0}. Since y(w) ≥ 0 for w ∈ ∂Q+ and y(w) ≤ 0 for w ∈ ∂Q− , the maximum principle implies that y(w) > 0 for w ∈ Q+ and that y(w) < 0 for w ∈ Q− . It then follows from E. Hopf’s lemma that yv (u) > 0 for −1 0, yv (u2 ) < 0. Because y(u) = 0 for all u ∈ I, the function y(w) can be extended as a harmonic function into the lower half of the w-plane. Near w = u1 , the above relations lead to an expansion y(w) = Re{−iμ(w − u1 ) + · · · } with a constant μ > 0. The conformality relation |Xu |2 = |Xv |2 yields zu2 (u1 ) = yv2 (u1 ) so that zu (u1 ) = ±μ, while zv (u1 ) = 0. We set z1 = z(u1 ) and z2 = z(u2 ). Since u1 = −u2 , formula (4) of Section 1.5 implies that z1 = −z2 . Hence, z(w) = Re{z1 ± μ(w − u1 ) + · · · } near w = u1 , z(w) = Re{−z1 ± μ(w − u2 ) + · · · − z1 }

near w = u2 .

The conformality relation |Xu | = |Xv | also implies that 2

2

zu2 (u) = x2v (u) + yv2 (u) for u ∈ I2 , because xu (u) = 0 for u ∈ I2 and yu (u) = zv (u) = 0 for u ∈ I. Assume that xv (u ) = 0 for some u ∈ (u1 , u2 ). Since x(w) can be extended as a harmonic function across I2 , we would then obtain an expansion of the form x(w) = Re{α(w − u )n + · · · }, n ≥ 2, valid in a full neighbourhood of the point w = u . Arguments similar to those employed earlier in conjunction with the properties of the expansions (3) show

26


that this is impossible. Thus, xv (u) = 0 for u1 0 for u ∈ I2 if z1 < 0, and zu (u) < 0 for u ∈ I2 if z1 > 0. This completes the proof of the theorem.

1.8 Final Results on the Shape of the Trace. Absence of Cusps. Optimal Boundary Regularity An inspection of the foregoing proofs shows that the relations yv (u) > 0 for −1 < u < 0, yv (u) < 0

for 0 < u < 1

hold in all three cases I, II, and III. In conjunction with the two expansion theorems of Sections 1.6 and 1.7 we obtain the following result about the shape of the trace of a stationary minimal surface in C(Γ, S). This result exactly corresponds to the experimental observations in Section 1.1. Theorem 1. Let X be a stationary minimal surface in C(Γ, S). In cases I and III, the trace X(u), u ∈ I, is a real analytic curve which is regular except for the branch point w = 0 of order 1. In case II, X has no branch points on I, and the trace curve X(u), u ∈ I, is a regular curve of class C 1,1/2 . From the expansion formulas of Section 1.6, (1), and Section 1.7, (1) and (2), it is apparent that the three generic forms of the trace X(u), u ∈ I, for a solution X of the minimum problem in C∗ (Γ, S) look as depicted in Fig. 1. In conclusion, let us describe a situation in which the trace curve X(u), u ∈ I, is free of cusps. Theorem 2. Suppose that the open subarc of the arc Γ with the end points P1 and P3 lies in the half-space {z < 0}, and that the open subarc of Γ between

Fig. 1. (a) Case II, z1 < 0 (tongue), (b) Case I (cusp), (c) Case II, z1 > 0 (loop)

1.8 Final Results on the Shape of the Trace. Absence of Cusps.

27

P3 and P2 is contained in the half-space {z > 0}. Then there exists exactly one stationary minimal surface X in C∗ (Γ, S). This surface is of type II, and its trace X(u), u ∈ I, on the half-plane S is a regular curve of class C 1,1/2 and has the form of a tongue. Remark 1. The expansions (1) and (2) of Section 1.7 show that the regularity class of a stationary surface of type II is exactly C 1,1/2 (B∪I, R3 ) and no better on I, and Theorem 2 guarantees that there are surfaces of type II. Thus the principal regularity theorem from Chapter 2 of Vol. 2 cannot be improved. Remark 2. The assumptions of Theorem 2 are satisfied if the z-component p3 (s) of the representation P (s) of Γ changes monotonously from z = −c to z = c as s moves from 0 to L; cf. Fig. 2. The situation is altered if Γ is deformed in such a way that p3 (s) changes signs repeatedly (an odd number of times). After such a deformation, the trace may exhibit a cusp; see Fig. 3.

Fig. 2.

Fig. 3.

Proof of Theorem 2. We introduce the two arcs π + iθ , C := w = e : 0 < θ < 2 − iθ π C := w = e : < θ < π . 2 Let X(w) = (x(w), y(w), z(w)) be the minimal surface under consideration. Then z(w) > 0 for w ∈ C + and z(w) < 0 for w ∈ C − . Denote by Q+ and Q− the two components of the open set Q = {w ∈ B : z(w) = 0} for which C + ⊂ ∂Q+ and C − ⊂ ∂Q− respectively. There cannot be further components of Q. In fact, if R were such a component different from Q+ and Q− , then ∂R ⊂ B ∪ I ∪ {i}. Moreover, z(w) = 0 at all boundary points of R in B ∪ {i}. In view of the maximum principle, z(w) cannot vanish everywhere on ∂R.

28


Hence, there is a point on I where z(w) is different from zero, say, positive. Since Q+ is adjacent to C + and Q− is adjacent to C − , the intersection ∂R ∩ I must be contained in a compact subinterval of I. Therefore, there is a point u ∈ I such that z(u ) = max{z(u) : u ∈ ∂R ∩ I} = max{z(w) : w ∈ ∂R} > 0. Clearly, a whole interval on I around u is also contained in ∂R ∩ I. Then, by E. Hopf’s lemma, zv (u ) < 0, in contradiction to the relation zv (u) = 0, which is valid for all u ∈ I. Since Q+ and Q− are the only components of the set Q, we conclude from Section 1.5, (5) that Q+ = {w : |w| < 1, u > 0, v > 0} and

Q− = {w : |w| < 1, u < 0, v > 0}.

By means of arguments familiar from earlier occasions it is seen that zu (u) cannot vanish on the intervals −1 < u < 0 or 0 0 in σ1 , σ3 , . . . , σ2n+1 , and z(w) < 0 in σ2 , σ4 , . . . , σ2n+2 . From Q = Q+ ∪˙ Q− we infer that this is impossible. Thus it follows from the above that the solution X must be of type II. Hence, be the uniqueness theorem of Section 1.5, the surface X is unique, and the description of the sets Q+ and Q− shows that the trace of X on the half-plane S has to be of the form of a tongue. This ends the proof of Theorem 2.

1.9 Proof of the Representation Theorem Now we want to supply the proof of the representation theorem, stated in Section 1.5, which is still missing. It will be based on a detailed discussion of the harmonic components x(w), y(w), z(w) of the stationary minimal surface X ∈ C∗ (Γ, S). For this purpose it is useful to recall the results of Sections 1.3 and 1.4 as well as the definitions of the subsets B(μ), B + (μ), B − (μ) of B and of the arcs C1+ (μ), C2+ (μ), C − (μ) given in Section 1.4. (i) We shall first pursue the discussion of case I assuming that I2 = {u0 }. By Lemma 2 of Section 1.3, the functions x(w), y(w), and z(w) can be continued analytically as harmonic functions across the diameter I into the lower half of the w-plane. Since x(u0 ) = 0 and x(u) > 0 for u = u0 , the function x(w) must have an expansion x(w) = Re{κ(w − u0 )2n + · · · } near w = u0 where κ > 0, n ≥ 1. A neighbourhood of w = u0 in B is divided into 2n+1 (and at least three) open sectors σ1 , σ2 , . . . , σ2n+1 such that

1.9 Proof of the Representation Theorem

29

x(w) > 0 in σ1 , σ3 , . . . , σ2n+1 , and that x(w) < 0 in σ2 , σ4 , . . . , σ2n . Denote by Q1 , Q2 , . . . , Q2n+1 the components of the set B(0) which contain the sectors σ1 , σ2 , . . . , σ2n+1 , respectively. These components are mutually disjoint for topological reasons and because of the maximum principle. Then, by virtue of Lemma 2 of Section 1.4, it follows that n = 1 and that B(0) consists of three different components Q1 , Q2 , Q3 . Clearly, Q2 = B − (0). According to the remark following the same lemma we may assume that C1+ (0) ⊂ ∂Q1 , C2+ (0) ⊂ ∂Q3 . Since x(u) > 0 for u ∈ I, u = u0 , and since x(1) = x(−1) = a > 0, the interval (u0 , 1) belongs to ∂Q1 while the interval (−1, u0 ) is part of ∂Q3 . Then, by our standard reasoning, the gradient of x(u) cannot vanish on I except for u = u0 . On the other hand, xv (u) = 0 on I, so that xu (u) = 0 for u = u0 . Therefore, the function x(u) increases strictly from the value 0 to the value a as u increases from u0 to 1, or decreases from u0 to −1. Furthermore, xu (u) < 0 for −1 0 for u0 0.

We assert that |∇x(w)| > 0 for all w ∈ B. Otherwise we would have ∇x(w0 ) = 0 for some w0 ∈ B. Then, according to Radó’s reasoning (cf. Lemma 2 of Section 4.9 in Vol. 1), the set B(μ) consists of at least four different components. This contradicts Lemma 2 in Section 1.4. Next we consider the harmonic function y(w). We have y(eiθ ) < 0 for 0 < θ < π2 and y(eiθ ) > 0 for π2 < θ < π, as well as y(u) = 0 for −1 ≤ u ≤ 1. As the angle θ increases from zero to π, the function y(eiθ ) decreases from zero to its minimum value ymin , then increases from ymin to its maximum value ymax = −ymin , and finally decreases again to zero. By conformality we have x2u (u) + zu2 (u) = yv2 (u) on I. Since xu (u) = 0 for u = u0 , we see that yv (u) = 0 for all u ∈ I, with the possible exception of u = u0 . It follows from the maximum principle that the open set {w ∈ B : y(w) = 0} has exactly two components, Q+ and Q− , and o’s argument, that y(w) > 0 in Q+ , y(w) < 0 in Q− . Applying once more Rad´ we infer that |∇y(w)| > 0 for all w ∈ B. Therefore, the two components Q+ and Q− are separated in B by an analytic arc A which has points in common with each horizontal line v = Im w = const, 0 < v < 1, considering that y(w) changes signs in B along each such line. We claim that this arc, except for its end points, lies entirely in the domain B − (0), and that it has the end points w = u0 on I and w = i on ∂B. As a first step we shall show that yv (1) < 0 and yv (−1) > 0. For this purpose recall that X(w) can be extended to the full disk {w : |w| ≤ 1} in such a way that X(w) is the position vector of a minimal surface defined on {w : 1 − δ0 < |w| < 1}, for a suitable δ0 > 0.

30


In view of the boundary regularity results stated in Section 2.3 of Vol. 2, the surface X is of class C 1,α in {w ∈ B : 1 − δ0 ≤ |w| ≤ 1}. As the curve X(eiθ ), 0 < θ < 2π, lies on a convex cylinder, the asymptotic expansion at boundary branch points (cf. Section 3.1 of Vol. 2) implies that our minimal surface cannot have branch points on the circular arc C. Hence it follows that |Xu (eiθ )|2 = |Xv (eiθ )|2 > 0 for 0 ≤ θ ≤ π. The arc Γ meets the half-plane S at right angles; therefore Xv (eiθ ) = (0, yv (eiθ ), 0)

for θ = 0

and

θ = π.

Consequently, we have yv (±1) = 0; more precisely, yv (1) < 0 and yv (−1) > 0, since y(eiθ ) < 0 for 0 < θ < π2 and y(eiθ ) > 0 for π2 < θ < π. As we know, yv (u) cannot vanish on I for u = u0 . Therefore, yv (u0 ) = 0, and y(w) has near w = u0 an expansion y(w) = Im{−λ(w − u0 )n + · · · }, where n ≥ 2, and λ is a real number different from zero. Since the set {w ∈ B : y(w) = 0} has exactly two components, we see that n = 2 and λ > 0; that is, near w = u0 , (2)

y(w) = Im{−λ(w − u0 )2 + · · · },

λ > 0.

The above results imply that the arc A which separates the components Q+ and Q− has as its end points (and only points on ∂B) the points w = u0 and w = i. Assume that A, except for its end points, is not contained in B − (0). Then there is a point w1 ∈ B on this arc for which x(w1 ) ≥ 0, y(w1 ) = 0. From the expansion (2) we see that near u = u0 , that is, for small positive values of ρ, the arc A has the representation w = u0 + ρeiθ(ρ) ,

θ(ρ) =

π + O(ρ). 2

It then follows from (1) that x(w) = −κρ2 + O(ρ3 ) for w ∈ A in a neighbourhood of w = u0 . Therefore, if we traverse the arc A from the point w = u0 to the point w = w1 , we shall encounter a negative minimum for the function x(w), restricted to A. Assume that this minimum is attained at the point w2 ∈ B ∩ A. Since y(w) = 0 on A, and A is a regular arc, we have xu yv − xv yu = 0 at w = w2 . Thus, there exist numbers p and q, p2 + q 2 > 0, satisfying the linear equations pxu (w2 ) + qy u (w2 ) = 0,

pxv (w2 ) + qy v (w2 ) = 0.


31

In fact, p = 0 and q = 0, since |∇x(w2 )| > 0 and |∇y(w2 )| > 0. Consider the harmonic function h(w) = p[x(w) − x(w2 )] + q[y(w) − y(w2 )] = px(w) + qy(w) + r, where r = −px(w2 ). This function vanishes at w = w2 , together with its first derivatives. By Radó’s lemma, h(w) must have at least four distinct zeros on the boundary ∂B. On the other hand, since pr = −p2 x(w2 ) > 0, the straight line px + qy + r = 0 in the (x, y)-plane passes through the x-axis ˆ of the slit to the left of the origin and therefore intersects the boundary ∂D domain D = D(0) in at most two points. Moreover, the functions x(w), y(w) ˆ provide a topological mapping of ∂B onto ∂D. Consequently, the function h(w) vanishes on ∂B in at most two points. This is a contradiction to the previous statement. We have proved that the arc A, except for its end points w = u0 , w = i, lies entirely in B − (0). This fact will be used in the following way: Let H(w) be a harmonic function in B of class C 0 (B) such that the open set {w ∈ B : H(w) = 0} consists of exactly four components which are separated in B by four analytic arcs issuing from some point w1 ∈ B. Suppose that two end points of these arcs lie on I, to the left and to the right of w = u0 , and two end points lie on C, to the left and to the right of w = i. Then, regardless of the location of the point w = w1 , the null set of the function H(w) in B must contain two points w and w in which x(w ) = 0,

y(w ) > 0 and

x(w ) = 0,

y(w ) < 0.

It can now be shown that the functions x(w), y(w) provide a topological ˆ We already know that the relation between the mapping from B to D ∪ ∂D. ˆ is a topological one and that interior points of B boundaries ∂B and ∂D are mapped onto interior points of D. The bijectivity of the mapping follows from the monodromy principle once it has been shown that the Jacobian ∂(x, y)/∂(u, v) cannot vanish in B. Assume that ∂(x, y)/∂(u, v) = 0 at some point w1 ∈ B. Then, as before, there exist constants p = 0 and q = 0 satisfying the linear equations pxu (w1 ) + qy u (w1 ) = 0 and pxv (w1 ) + qy v (w1 ) = 0. It follows that the harmonic function H(w) := p[x(w) − x(w1 )] + q[y(w) − y(w1 )] = px(w) + qy(w) + r, r := −px(w1 ) − qy(w1 ), and its first derivatives vanish at w = w1 . Rad´ o’s lemma implies that H(w) must have at least four different zeros on ∂B. On the other hand, any straight ˆ in at most four line px + qy + r = 0, p = 0, in the x, y-plane intersects ∂D points. The case of four distinct points is only possible for pr < 0. Because of ˆ we conclude that H(w) posthe bijectivity of the relation between ∂B and ∂D sesses exactly four different zeros on ∂B if pr < 0. Under the circumstances, the set

32


{w ∈ B : H(w) = 0} consists of exactly four components which are separated in B by four analytic arcs issuing from w = w1 . Two end points of these arcs lie on I, to the left and to the right of w = u0 , and two end points lie on C, to the left and to the right of w = i. The observation formulated earlier implies that there are two points w , w ∈ B for which qy(w ) + r = 0,

qy(w ) + r = 0,

y(w ) > 0,

y(w ) < 0.

These relations are incompatible with the inequality q = 0, and we have proved that the functions x(w), y(w) furnish a topological mapping from B ˆ to D ∪ ∂D. Let w = ω(x, y) be the inverse map, and set Z(x, y) = z(ω(x, y)),

ˆ (x, y) ∈ D ∪ ∂D.

The function Z(x, y) provides a nonparametric representation ˆ {z = Z(x, y) : (x, y) ∈ D ∪ ∂D} of our minimal surface X = X(w), w ∈ B. Z(x, y) is real analytic in D and on both shores of the open segment 0 < x < a of the x-axis (having of course ˆ different limits limy→+0 Z(x, y) and limy→−0 Z(x, y)), continuous in D ∪ ∂D, 1 ˆ and of class C in D ∪ ∂D except at the points (0, 0) and (a, 0). Given that xv (u) = zv (u) = 0 on I, and that xu (u) = 0, yv (u) = 0 for u = u0 , u ∈ I, it also follows from the relation zv xu − zu xv ∂ Z(x, y) = ∂y xu yv − xv yu ω(x,y)=w that lim

y→±0

∂ Z(x, y) = 0 for 0 < x < a. ∂y

Thus, the proof of the theorem is completed for case I. (ii) We turn now to a discussion of case II, assuming that I2 is a closed interval u1 ≤ u ≤ u2 , where −1 < u1 < u2 < 1. We know that y(u) = yu (u) = zv (u) = 0 for |u| < 1 as well as x(u) > 0,

xv (u) = 0 for −1 ≤ u < u1

and

and x(u) = xu (u) = 0 for u1 ≤ u ≤ u2 .

u2 < u ≤ 1


33

The functions y(w) and z(w) can be continued as harmonic functions across the diameter I into the disk {w : |w| < 1}. For the function x(w) such a continuation is possible across the intervals u1 < u < u2 , −1 < u < u1 and u2 < u < 1, but the resulting extended function will have isolated singularities at the points w = u1 and w = u2 . Recall that B − (0) is connected and that B + (0) can have at most two components. From the situation at hand it follows that B + (0) consists of exactly two components and that we have the expansions x(w) = Re{iκ1 (w − u1 )3/2 + · · · },

κ1 > 0,

near w = u1 ,

x(w) = Re{κ2 (w − u2 )3/2 + · · · },

κ2 > 0,

near w = u2 .

and The derivation of these expansions is based on the arguments employed for the proof of Section 1.7, (3), except that we are at the present stage not able to conclude that u1 = −u2 and κ1 = κ2 . From here on, we can follow the line of reasoning used in part (i) of the proof. We find that, as u decreases from u1 to −1 or increases from u2 to 1 the function x(u) increases strictly from the value zero to the value a, and also that xu (u) = 0 for −1 0 for all w ∈ B. As for the function y(w), we see as in (i) that both sets Q+ = {w ∈ B : y(w) > 0} and Q− = {w ∈ B : y(w) < 0} are connected, and that yv (u) > 0 near w = −1, and yv (u) < 0 near w = 1. On (−1, u1 ) ∪ (u2 , 1), we have x2u (u) + zu2 (u) = yv2 (u), and x2u (u) > 0. Hence, yv (u) > 0 for −1 < u < u1 , and yv (u) < 0 for u2 < u < 1. We claim that yv (u1 ) = 0, yv (u2 ) = 0. The assumption yv (u1 ) = 0 leads to Xu (u1 ) = Xv (u1 ) = 0, so that w = u1 would have to be a branch point of X. However, the asymptotic expansion of X(w) near a branch point does not allow for terms containing the power (w −u1 )3/2 . A similar contradiction arises from the assumption yv (u2 ) = 0. It follows that the derivative yv (u) must vanish somewhere in the interval (u1 , u2 ). Since the set {w ∈ B : y(w) = 0} has only two components, our standard reasoning shows that there exists exactly one point u0 ∈ (u1 , u2 ) such that yv (u0 ) = 0. The expansion of y(w) near w = u0 is y(w) = Re{iλ(w − u0 )2 + · · · },

λ > 0.

It follows as in (i) that |∇y(w)| > 0 for w ∈ B and that the Jacobian ∂(x, y)/∂(u, v) cannot vanish in B. The functions x = x(w), y = y(w) provide a mapping between the boundˆ (Here D = D(0) is the slit domain in the (x, y)-plane defined aries ∂B and ∂D. in the statement of the theorem.) This mapping is topological, except on the ˆ From interval [u1 , u2 ] of I which corresponds wholly to the point (0, 0) on ∂D.

34


the non-vanishing of the Jacobian ∂(x, y)/∂(u, v) in B it follows that x(w), y(w) furnish a homeomorphism between B and D. A repetition of the further discussion of part (i) leads to the conclusion that the minimal surface X = X(w), w ∈ B, admits a nonparametric representation z = Z(x, y). The function Z(x, y) has the properties stated in the theorem. (iii) Case III can easily be reduced to case I. For the purpose of this reduction, let x0 = min{x(u) : u ∈ I}, and suppose that x0 = x(u0 ). Then, x(u) > x0 for u = u0 , u ∈ I. We choose a new Cartesian coordinate system with coordinates ξ, η, ζ, defined by the relations ξ = x − x0 , η = y, ζ = z; see Fig. 1 of Section 1.3. Introduce Γ0 := Γ \ (x0 , 0, 0) and the functions ξ(w) := x(w) − x0 ,

η(w) := y(w),

ζ(w) := z(w),

and the surface Y (w) = (ξ(w), η(w), ζ(w)). Furthermore let S be the halfplane {(ξ, η, ζ) : ξ ≥ 0, η = 0}. Then Y (w) is a stationary minimal surface of type I in C∗ (Γ0 , S). Applying part (i) of this proof to Y (w), we may deduce the desired properties of X from those of Y by going back to the old coordinates x, y, z. This completes the proof of the representation theorem.

1.10 Scholia 1. Remarks about Chapter 1 Except for minor modifications and the second proof of Theorem 1 in Section 1.6, the results of this chapter and their proofs are taken from the paper [3] of Hildebrandt and Nitsche. There remains the challenging problem to extend the results of this section to non-planar supporting surfaces S and, more generally, to arbitrary configurations Γ1 , . . . , Γk , S1 , . . . , Sl . Experimental evidence indicates that it should be possible to prove similar results in the general case. A certain generalization is given in the following Chapter 2. 2. Numerical Solutions So far we have not touched upon the problem of numerical solutions of boundary value problems for minimal surfaces. Both the nonparametric minimal surface equation ∇z =0 div 1 + |∇z|2

(1)

and the parametric equations (2)

ΔX = 0,

|Xu |2 = |Xv |2 ,

Xu , Xv = 0

1.10 Scholia

35

have been treated. Here we mention that the partially free boundary problem for (2) with a planar support surface can effectively be solved by means of the finite element method, a comprehensive presentation of which can be found in the treatise of Ciarlet [1]. A numerical approach to partially free problems was given by Wohlrab [2,3]. (However, his formulae are in part faulty. For corrections, see e.g. A. Pape [1].) In addition we want to give some (rather incomplete) references to the literature concerning the numerical treatment of minimal surfaces. The nonparametric equation (1) was dealt with by Concus [1–4] using a finite difference scheme and solving the resulting finite difference equations by a nonlinear successive overrelaxation method. The finite element method was applied to minimal surfaces by many numerical analysts. We only mention the work of Mittelmann [1–6], Jarausch [1], Wohlrab [2,3], and Dziuk [9,10]. Whereas the first three authors used the variational formulation as a point of departure, Dziuk applied an iteration procedure suggested by the mean curvature flow of surfaces. Furthermore we refer to the work of Dziuk and Hutchinson [1–3], Hutchinson [1], Polthier [5,6], D¨ orfler and Siebert [1], and Hinze [1]. We also mention the purely computational work by Wagner [1,2] and Steinmetz [1].

Chapter 2 Embedded Minimal Surfaces with Partially Free Boundaries

In this chapter we consider disk-type minimal surfaces in R3 that span boundary configurations Γ, S consisting of a cylinder surface S and a Jordan arc Γ with endpoints on S, and we ask the following question: When can such a minimal surface be viewed as a nonparametric surface over a planar domain? Any satisfactory answer yields the existence of embedded minimal surfaces that are stationary in Γ, S. Simultaneously, such results lead to uniqueness theorems for the partially free boundary problem which generalize Radó’s uniqueness theorem for Plateau’s problem that was discussed in Section 4.9 of Vol. 1. We begin in Section 2.1 by describing the geometric setup and giving the necessary definitions that are to hold throughout this chapter. In particular we introduce the planar domain G above which a minimal surface X : B → R3 that is stationary in Γ, S is expected to lie as a graph. The first step to verify this expectation is taken in Section 2.2. It consists in verifying that the free trace of X lies above some part of ∂G. To this end we show that, under appropriate assumptions on the configuration Γ, S, the orthogonal projection of the free trace X|I into the plane Π of G is restricted to a certain part Σ of ∂G and does not overshoot ∂G. In Section 2.3 we establish a modification of the Kneser–Radó result (cf. Lemmata 1–3 in Section 4.9 of Vol. 1), which later on is used in proving that X lies as a graph above G, although it might not be a graph above G. For deriving uniqueness we also need various properties of the Gauss map N : B → S 2 ⊂ R3 associated with X. These properties are derived in Section 2.4. In Section 2.5 we first prove a uniqueness theorem for minimal surfaces X that lie on one side of the supporting surface S. This special property of X is not satisfied a priori but has to be verified, what is easy if S is convex and Γ lies on the convex side of S. For nonconvex S this property is fairly difficult to prove; we state a geometric condition on Γ, S when this can be done. Of particular interest in this respect is the main result of Section 2.6 which


37

38

2 Embedded Minimal Surfaces with Partially Free Boundaries

ensures that minimizers and, more generally, freely stable minimal surfaces lie on one side of S. This can be ascertained without any curvature assumption on S, except for smoothness; we only need to know that the orthogonal projection of Γ into the plane Π is convex with respect to G. By a suitable approximation procedure one can carry over this result to nonsmooth S, but the uniqueness result is not as strong as for smooth S; this will be seen in Section 2.9. In Section 2.7 we study the behaviour of solutions X when S is the boundary of a wedge. A “reasonably well behaved” minimal surface X either intersects the edge L of the wedge “transversally”, or a part of the boundary curve X|I adheres to L; we call this behaviour “edge creeping”. If X is in the class T(Γ, S) of transversally intersecting surfaces mapping w = 0 into the edge, we can derive an asymptotic expansion of Xw (w) and Nw (w) at w = 0. In Section 2.8 we use this expansion to show that edge creeping on L must occur if Γ lies on the concave side of S and its height function above the plane Π is monotonic. On the other hand it will be seen in Section 2.9 that there is no edge creeping along the edge of the wedge if Γ lies on the convex side of S. Finally in Section 2.10, we sketch a Bernstein theorem for nonparametric minimal surfaces in a wedge. In the Scholia we describe generalizations of these results to doubly connected minimal surfaces bounded by a polygonal cylinder surface S and a closed Jordan curve Γ winding around S.

2.1 The Geometric Setup In this section we describe the geometric configurations Γ, S to be considered and introduce the notation which is used throughout this chapter. We denote points of R3 by capital letters P, Q, . . . or in coordinates by x = (x1 , x2 , x3 ), y = (y 1 , y 2 , y 3 ) etc. The x1 , x2 -plane Π will be identified with R2 , and the points (x1 , x2 , 0) of this plane are denoted by (x1 , x2 ) or by small letters p, q, . . . . As usual we identify the complex plane C of points w = u + iv with the u, v-plane, and the points w = u + iv with (u, v) ∈ R2 . As parameter domain B of minimal surfaces X(w) = X(u, v) = X 1 (u, v), X 2 (u, v), X 3 (u, v) we choose the semidisk B = {(u, v) ∈ R2 : u2 + v 2 < 1, v > 0} bounded by the closed semicircle C = {(u, v) ∈ R2 : u2 + v 2 = 1, v ≥ 0} and the segment {(u, 0) ∈ R2 : |u| < 1}, which is identified with the open interval I = {u ∈ R : |u| < 1} on the real axis of C. With any surface X : B → R3 in R3 we associate the mapping f := π ◦ X where π denotes the orthogonal projection of R3 onto the plane Π, that is, f (u, v) := X 1 (u, v), X 2 (u, v) .

2.1 The Geometric Setup

39

Let Σ0 := σ(R) be a Jordan arc in Π such that Π \ Σ0 consists of two unbounded, open, connected subsets Ω and Ω of Π = R2 . The parameter ˙ = representation σ : R → R2 be C 1 or at least piecewise C 1 and satisfy |σ(s)| 1. We consider a cylinder surface S := Σ0 × R with the directrix Σ0 which is the orthogonal projection of S into Π. Moreover let Γ be a rectifiable Jordan arc in R3 with the endpoints P1 and P2 . Let p1 := π(P1 ), p2 := π(P2 ), and Γ := π(Γ ) be the orthogonal projections of P1 , P2 , and Γ into the plane Π. We assume that p1 = p2 , and that Γ ist represented by a homeomorphism ξ : [0, 1] → Γ satisfying ξ(0) = p1 and ξ(1) = p2 , as well as ξ(t) ∈ Ω for 0 < t < 1. This means that Γ lies on one side of S and meets S only in its endpoints P1 and P2 . Let p1 = σ(s1 ) and p2 = σ(s2 ), and consider the three subarcs Σ1 := σ (−∞, s1 ) ,

Σ := σ [s1 , s2 ] ,

Σ2 := σ (s2 , ∞)

of Σ0 ; then Σ0 = Σ1 ∪˙ Σ ∪˙ Σ2 . Suppose also that Σ ∪ Γ is the boundary of a simply connected domain G contained in Ω.

Fig. 1. The parameter domain B

Fig. 2. The configuration Γ, S and its projection Γ, Σ

The central idea of the following considerations is to show that the set X(B) in R3 lies as the graph of a function z : G → R above G, i.e. X(B) = graph z, and that there is a 1 − 1 correspondence between the points of B and G. In fact, we intend to show that f is a diffeomorphism from B onto G. Then Z(x1 , x2 ) := (x1 , x2 , z(x1 , x2 )) will be a surface representation equivalent to X; that means, Z is a nonparametric representation of the surface X. To achieve this goal, we shall always assume that Γ is convex with respect to G. Moreover we suppose that Γ is a graph or at least a generalized graph over Γ . This means, the pre-image π −1 (p) of any p ∈ Γ is supposed to contain

40


exactly one point P ∈ Γ , except for at most finitely many points p ∈ Γ for which Γ ∩ π −1 (p) is a whole interval on the vertical axis {p} × R through p. We make the following further assumptions about σ: If σ is not smooth, then there are finitely many points t1 , . . . , tl ∈ (s1 , s2 ) such that σ|Ij with I1 := (−∞, t1 ], Il+1 := [tl , ∞), Ij := [tj−1 , tj ] for 2 ≤ j ≤ l − 1 is an immersed curve of class C 3 , i.e. the vertices gj := σ(tj ) of Σ0 lie in the interior of Σ. Secondly we suppose that σ (s) and σ (s) tend to limits as s → ∞ and s → −∞ (whence σ (s) → 0 as s → ±∞). If σ is of the class C 3 then S is a complete C∗3 -submanifold of R3 , that is, S is of class C 3 and satisfies a uniformity condition (B) at infinity in the sense of Definition 2, Section 2.6 in Vol. 2. If σ is only piecewise smooth we use the one-sided tangents σ (s + 0) and σ (s − 0) instead of σ (s) at s = t1 , . . . , tl . We suppose that the tangents σ (tj + 0) and σ (tj − 0) enclose an angle βj = 0, π, 2π at any vertex qj = σ(tj ) of Σ, and that Γ meets Σ0 nontangentially at p1 and p2 . Let t : Σ0 → S 1 and ν : Σ0 → S 1 be two piecewise continuous fields of unit vectors along Σ0 such that t(σ(s)) = σ (s) and ν(σ(s)) · σ (s) = 0 provided that σ is differentiable at the value s. We assume that ν(σ(s)) points into Ω , i.e. into the exterior of ∂G for all s ∈ [s1 , s2 ] where σ is differentiable. We extend t and ν to piecewise continuous and piecewise smooth vector fields τ (x) and n(x) on R3 satisfying τ (x1 , x2 , x3 ) = t(x1 , x2 ), 0) , n(x1 , x2 , x3 ) = ν(x1 , x2 ), 0 for all points x = (x1 , x2 , x3 ) ∈ S = Σ0 × R which do not lie on an edge, i.e. which satisfy π(x) := (x1 , x2 ) = qj for 1 ≤ j ≤ l. Similarly we can define a piecewise continuous function κ : R3 → R with κ(x) = κ(x1 , x2 , 0) such that κ(p, 0) is the curvature of ∂G with respect to the inner normal −n(p) of ∂G at any point p ∈ ∂G where ∂G is smooth. The sign of the curvature κ(p, 0) is chosen in such a way that κ(p, 0) ≥ 0 (or ≤ 0) if the boundary ∂G is convex (or concave respectively) with respect to G in a sufficiently small neighbourhood of a regular point p ∈ ∂G. We may assume that τ , n, κ are smooth close to any regular point of the support surface S. ˚ and Σ ˚ we denote the interior parts C \ {1, −1} and Σ \ {p1 , p2 } of By C the closed arcs C and Σ respectively. If Γ is a true graph above Γ we can write it in the form Γ = x1 , x2 , γ(x1 , x2 ) : (x1 , x2 ) ∈ Γ where γ is the height function of Γ above Γ . If ξ(t) = (ξ 1 (t), ξ 2 (t)), 0 ≤ t ≤ 1, and ξ 3 := γ ◦ ξ, then ξ(t) = (ξ 1 (t), ξ 2 (t), ξ 3 (t)) is a representation of Γ providing a homeomorphism of [0, 1] onto Γ . Actually we may assume that ˙ ξ : [0, 1] → R3 is at least Lipschitz continuous, and that |ξ(t)| ≡ const > 0 a.e. on [0, 1]. For mappings X : B → R3 of the Sobolev class H21 (B, R3 ) the Dirichlet integral 1 |∇X|2 du dv D(X) := 2 B

2.1 The Geometric Setup

41

is finite. As in Section 4.6 of Vol. 1 we call such a map X a surface of the class C(Γ, S) if its Sobolev trace X|I ∈ L2 (I, R3 ) satisfies X(u, 0) ∈ S for almost all u ∈ (−1, 1) and if the Sobolev trace X|C furnishes a continuous, monotonic mapping of C onto Γ satisfying X(1, 0) = P1 and X(−1, 0) = P2 . (For the sake of brevity the notation “monotonic” is used in the sense of “weakly monotonic”.) As usual we say that X ∈ C(Γ, S) is stationary in the boundary configuration Γ, S if d D(X ) =0 d =0 holds true for every admissible variation {X }|| 0 such that X ∈ C 1,1/2 (B ∪ (uj − , uj + )). (ii) If X ∈ M(Γ, S) and Γ ∈ C 2,α for some α ∈ (0, 1), then X is of class ˚ R3 ). C 2,α (B ∪ C, Proof. Because of the monotonicity of f = π ◦ X on I for a surface X ∈ M∗ (Γ, S), X(u, 0) is restricted to a C 3 -part of S if uj − < u < uj , 0 < 1, and so the results of Vol. 2, Section 2.7 yield X ∈ C 1,1/2 (B ∪ (uj − , uj )). An analogous reasoning leads to X ∈ C 1,1/2 (B ∪ (uj , uj + )) for 0 < 1 whence X ∈ C 1,1/2 (B ∪ (uj − , uj + )). The other assertions follow from Vol. 2, Sections 2.3 and 2.7, as in the proof of Proposition 1. Remark 1. If X ∈ M(Γ, S) lies on the same side of S as Γ and if σ ∈ C 3 , then there exist no boundary branch points of X on I. Similarly, if σ is merely piecewise C 3 and X ∈ M∗ (Γ, S), then neither on the regular set I ∗ (X) nor on any E-interval Ij (X) there is a branch point of X. This follows immediately from the asymptotic expansions of Section 2.10 in Vol. 2. Remark 2. If X ∈ M∗ (Γ, S) then f (u, 0) = qj on Ij (X) for any E-vertex qj of X whence fu (u, 0) = 0 on Ij , and therefore (5) Xu (u, 0) = 0, 0, Xu3 (u, 0) for u ∈ Ij . Therefore X has no branch points on Ij if and only if Xu3 (u, 0) = 0 on Ij . Moreover we have Xv3 (u, 0) = 0 on I whence (6)

Xu ∧ Xv = (−Xu3 Xv2 , Xu3 Xv1 , 0) = Xu3 · (−Xv2 , Xv1 , 0) on Ij

for any E-interval Ij of X.

2.2 Inclusion and Monotonicity of the Free Boundary Values If X ∈ C(Γ, S) ∩ C 0 (B, R3 ) then X|I is a continuous curve connecting P1 and P2 on S. Consequently f |I with f := π ◦ X is a continuous curve in Π connecting p1 and p2 in such a way that f (I) ⊂ Σ0 . Therefore we have Σ ⊂ f (I). Now we describe two conditions which guarantee that, in fact, f (I) = Σ. Let us first assume that σ ∈ C 3 . By L(p) we denote the straight line in Π perpendicular to Σ0 at p.

2.2 Inclusion and Monotonicity of the Free Boundary Values

45

Condition (B). For any p ∈ Σ1 ∪ Σ2 = Σ0 \ Σ, the normal line L(p) meets the set G ∪ Σ0 only at the point p.

Fig. 1. Condition (B) is satisfied by Σ0 , G where Σ0 = Σ1 ∪ Σ ∪ Σ2

Proposition 1. If σ ∈ C 3 satisfies Condition (B) and X is of class M(Γ, S) then f (I) = Σ. Proof. Since X ∈ C 0 (B, R3 ), the curves X|I and f |I are continuous, f = π ◦ X. Hence there are two numbers s , s with s ≤ s1 < s2 ≤ s such that ˜ = {σ(s) : s ≤ s ≤ s }. Set p := σ(s ), p := σ(s ). ˜ := f (I) is given by Σ Σ We claim that p = p1 and p = p2 . In fact, suppose that for instance p = p1 ; then p ∈ Σ1 . With the tangent vector t(p ) of Σ0 at p we form the function ϕ : B → R by (1)

ϕ(w) := t(p ) · f (w),

w ∈ B.

Clearly ϕ is continuous on B, harmonic in B, and nonconstant. On account of Condition (B) and the geometric assumptions on G and Σ0 , it follows that ˜ \ {p }) ∪ G is contained in the halfspace the set (Σ H+ (p ) := p ∈ Π : (p − p ) · t(p ) > 0 , whereas p lies on L(p ) = ∂H+ (p ). Consequently ϕ assumes its minimum m on B at some point u ∈ I, and f (u , 0) = p . Then fv (u , 0) = λ(u )ν f (u , 0) = λ(u )ν(p ), whence ϕv (u , 0) = t(p ) · fv (u , 0) = 0. However, E. Hopf’s lemma implies ϕv (u , 0) > 0, and so we have arrived at a contradiction. Thus we obtain s = s1 .

46


Similarly, if s > s2 we form ψ : B → R by ψ(w) := t(p ) · f (w),

w ∈ B,

where p := σ(s ). Then ψ assumes its maximum M on B at some point u ∈ I, and f (u , 0) = p . Again we infer from (1) and fv (u , 0) being parallel to ν(p ) that ψv (u , 0) = 0, whereas E. Hopf’s lemma yields ψv (u , 0) < 0, a contradiction; thus s = s2 . If σ0 is merely piecewise C 3 we replace condition (B) by another assumption on the boundary configuration. To formulate this assumption we use the following notation: If p = σ(s), p = σ(s) and s < s, we write p < p. If p , p ∈ Σ0 and p < p then [p , p ] := {p ∈ Σ0 : p ≤ p ≤ p }. Similarly we define (p , p ), [p , p ), and (p , p ]. Condition (M1), (i) G ∩ L(p) is empty for any p ∈ Σ1 ∪ Σ2 . (ii) L(p) ∩ Σ1 = {p} for p ∈ Σ1 ; L(q) ∩ Σ2 = {q} for q ∈ Σ2 . (iii1 ) If p ∈ Σ1 then L(p) ∩ Σ2 is either empty or consists of a single point q; in the second case we have for any q ≥ q: The set L(q) ∩ Σ1 is either empty or consists of exactly one point p that satisfies p < p. If we exchange the roles of Σ1 and Σ2 in (iii), we say that Γ, S satisfies Condition (M2), i.e. we assume in this case (i) and (ii), but replace (iii1 ) by (iii2 ) If q ∈ Σ2 then L(q) ∩ Σ1 is either empty or consists of a single point p; in the second case we have for any p ≤ p: The set L(p) ∩ Σ2 is either empty or consists of exactly one point q satisfying q < q. Note that Condition (B) implies Condition (M1) and Condition (M2). Therefore the next result is a generalization of Proposition 1; its proof uses the same ideas. Proposition 2. Let X ∈ M(Γ, S) ∩ C 0 (B, R3 ), and suppose that Γ, S satisfies Condition (M1) (or (M2) respectively). Then we have f (I) = Σ, and therefore f (∂B) = ∂G. Proof. Since f |I is a continuous curve contained in Σ0 with f (I) ⊂ Σ0 and f (1) = p1 , f (−1) = p2 , there are two points p ∈ Σ 1 and p ∈ Σ 2 such that f (I) = [p , p ]. Let us assume Condition (M1) and suppose that p < p1 . Then we have either (I) L(p ) ∩ [p2 , p ] = ∅, or (II)

L(p ) ∩ [p2 , p ] = ∅.

In case (I) we again form ϕ : B → R by ϕ(w) := t(p ) · f (w)

for w ∈ B,


47

Fig. 2. Four cases where Σ0 , G satisfies Condition (M1 )

which is continuous on B, harmonic in B, and nonconstant. Then m := minB ϕ is assumed on ∂B, and (i) implies that ϕ assumes m neither on C nor on f˜−1 (Q) where f˜ := f |I and Q := {q1 , q2 , . . . , ql } is the set of vertices of Σ0 which is contained in Σ. Hence there is some u0 ∈ I \ f˜−1 (Q) such that ϕ(u0 , 0) = minB ϕ = t(p ) · p . We know that f (w) is of class C 1 in a sufficiently small neighbourhood of w0 = (u0 , 0) in B and that fv (w0 ) · t(p ) = 0 whence ϕv (w0 ) = 0, whereas E. Hopf’s lemma leads to ϕv (w0 ) > 0, a contradiction, and so case (I) is ruled out. In case (II) we have L(p ) ∩ Σ2 = {q} with p2 < q. By the same reasoning as in case (I) we can rule out that p < q. So we can assume p2 < q ≤ p . By (iii) we infer that either L(p ) does not meet Σ1 , or it meets Σ1 in exactly one point p satisfying p < p . Thus L(p ) does not intersect [p , p1 ], and so we are in the same situation with respect to p as before with regard to p , and we again

48


obtain a contradiction using now the auxiliary function ψ(w) := t(p ) · f (w). Hence it is impossible that p < p, and we obtain p = p1 , and therefore f (I) = [p1 , p ]. Suppose now that p2 < p . By virtue of (i) and (ii) it follows that L(p ) does not intersect f (I) except for p . Thus we can apply the above reasoning with the auxiliary function ψ(w) := t(p )·f (w), and we obtain a contradiction. It follows that p2 = p , and so we have f (I) = [p1 , p2 ], i.e. f (I) = Σ. Assuming (M2) instead of (M1) we can proceed in an analogous manner. Note that Condition (B) is never satisfied if Σ0 is a wedge consisting of two rays meeting at a corner q1 under an angle different from 180◦ , whereas (M1) or (M2) is satisfied provided that ∂G is reasonably well-behaved. Now we want to state a condition that implies monotonicity of the boundary values f |I of f = π ◦ X. Condition (C). The set G is contained in the union of all normal lines L(p) with p ∈ Σ, and no two lines L(p) and L(p ) with p, p ∈ Σ and p = p intersect in G.

Fig. 3. Condition (C): The lines L(p) with p ∈ Σ from a field on G

This condition states that the lines L(p) with p ∈ Σ form a “field” on G. Proposition 3. Let σ ∈ C 3 and suppose that Condition (C) is satisfied. Moreover, let X ∈ M(Γ, S), f = π◦X, and suppose that f (B) ⊂ G. Then f |I yields a weakly monotonic mapping of I onto Σ, and so X lies in M∗ (Γ, S).


49

Proof. Since X ∈ C 0 (B, R3 ), it is easy to see that f (I) = Σ. Consider now the continuous function ω : G → R defined by ω(x, y) := s if (x, y) ∈ G ∩ L(σ(s)), s1 ≤ s ≤ s2 . Clearly, ω(G) = [s1 , s2 ]. We introduce the continuous function ψ : B → R by ψ := ω ◦ f ; then ψ(B) = [s1 , s2 ] = ψ(I) and ψ(−1, 0) = s1 , ψ(1, 0) = s2 . It remains to show that ψ|I is monotonic as ω ist strictly increasing, and therefore f is monotonic if and only if ψ is monotonic. If ψ were not monotonic, we could find points u1 and u2 such that −1 < u1 < u2 < 1 and ψ(u1 ) > ψ(u2 ). Then we choose some s∗ ∈ (ψ(u2 ), ψ(u1 )) and consider the connected components Z1 and Z2 of {w ∈ B : ψ(w) > s∗ } and {w ∈ B : ψ(w) < s∗ } respectively with u1 ∈ Z1 and u2 ∈ Z2 . We claim that both Z1 and Z2 have a nonempty intersection with the circular arc C. ˚ were disjoint. Then Z 2 ⊂ B ∪I In fact, suppose for instance that Z2 and C ∗ ∗ and ψ(w) ≤ s on Z 2 , ψ(w) = s on ∂Z2 ∩ B. Set s0 := inf Z 2 ψ. Since ψ is nonconstant on I ∩ Z2 we have s0 < s∗ . Thus ψ|Z assumes its minimum at some point w0 ∈ Z2 , s0 = ψ(w0 ) = ω(f (w0 )), and p0 := f (w0 ) lies on the straight line L(p0 ) given by L(p0 ) := p ∈ Π : p − σ(s0 ) · σ (s0 ) = 0 . Consider the function ϕ : Z 2 → R defined by ϕ(w) := f (w) · σ (s0 ), which is continuous on Z 2 , harmonic in Z˚2 , and nonconstant. On account of Condition (C) it follows that f (Z 2 ) is contained in the closed halfspace {p ∈ Π : (p − σ(s0 )) · σ (s0 ) ≥ 0}, and so ϕ assumes its minimum on Z 2 in w0 . By the maximum principle w0 cannot be an interior point of Z2 since ˚ is supposed to be empty we have w0 ∈ ˚ For / C. ϕ is nonconstant. As Z2 ∩ C ∗ ∗ w ∈ B ∩ ∂Z2 we have ψ(w) = s , that is, f (w) ∈ L(s ) ∩ G. Because of s0 < s∗ it follows that ϕ(w) > ϕ(w0 ) for w ∈ B ∩ ∂Z2 . Therefore w0 has to be contained in I ∩ Z2 , i.e. w0 = u0 ∈ I. We then infer that there is an > 0 such that (u0 − , u0 + ) ⊂ ∂Z2 , and so we can apply E. Hopf’s lemma, arriving at ϕv (u0 , 0) > 0. However, we have ϕv (u0 , 0) = fv (u0 , 0) · σ (s0 ) = λ(u0 )ν f (u0 , 0) · t f (u0 , 0) = 0, a contradiction. Hence the intersection Z2 ∩ C is nonempty, and similarly we can prove that Z1 reaches C. Hence there exist two points w1 = eiϕ1 and w2 = eiϕ2 with 0 ≤ ϕ1 < ϕ2 ≤ π such that w1 ∈ ∂Z1 and w2 ∈ ∂Z2 . The definition of Z1 and Z2 implies ψ(w1 ) > s∗ > ψ(w2 ) which is impossible since ψ(eiϕ ) = ω(f (eiϕ )) is a decreasing function of ϕ ∈ [0, 2π]. Hence we have arrived at a contradiction, and therefore ψ(u, 0) has to be increasing. Finally we formulate conditions which guarantee that f (B) ⊂ G. Proposition 4. Suppose that G lies between the two lines L(p1 ) and L(p2 ) normal to Σ0 at p1 and p2 respectively. Moreover, let X ∈ M(Γ, S) and suppose that f := π ◦ X ∈ C 0 (B, R2 ) and satisfies f (I) = Σ and f (B) ∩ Ω = ∅ (i.e. f (B) lies on the same side of Σ0 as Γ ). Then we have f (B) ⊂ G.

50


Proof. Since f is harmonic in B and continuous on B, the maximum principle implies that f (B) lies in the convex hull H(G) of G, which in turn lies between the lines L(p1 ) and L(p2 ). As Γ is convex with respect to G, it follows that ˜ which is contained in the boundary of H(G) consists of Γ and a second arc Σ Ω ; this arc meets Γ only in the two points p1 , p2 and lies between L(p1 ) and L(p2 ). Since we have assumed that f (B) ∩ Ω is empty, we obtain f (B) ⊂ G. Remark 1. Note that G lies between L(p1 ) and L(p2 ) if one of the Conditions (B), (M1), or (M2) is satisfied.

2.3 A Modification of the Kneser–Radó Theorem Another essential tool in proving uniqueness of solutions in M(Γ, S) is the following generalization of the Kneser–Rad´ o lemma (see Vol. 1, Section 4.9, Lemma 3). Proposition 1. Let f ∈ C 0 (B, R2 ) be harmonic in the semidisk B and suppose that there is a Jordan domain G in R2 such that f (B) ⊂ G, and that f |∂B maps ∂B monotonically onto ∂G. Then f |B maps B diffeomorphically onto G. If f |∂B is strictly monotonic, then f maps B homeomorphically onto G. Proof. Recall that a Jordan domain G in R2 is the interior of a closed Jordan curve in R2 . Let X(w) = (x(w), y(w), z(w)). We begin by proving that the Jacobian Jf of the mapping f (w) = x(w), y(w) , w ∈ B, does not vanish in B. In fact, if Jf (w0 ) = 0 for some w0 ∈ B, then there are numbers a, b ∈ R with a2 + b2 = 1 such that (1)

a∇x(w0 ) + b∇y(w0 ) = 0.

Set ν := (a, b), c := −ν · f (w0 ), g(p) := ν · p + c for p ∈ R2 , and consider the straight line L := {p ∈ R2 : g(p) = 0}. Since p0 := f (w0 ) ∈ L ∩ G, the set L ∩ G is nonempty. Define h : B → R by h := g ◦ f , i.e. h(w) = ν · f (w) + c

for w ∈ B.

The function h is continuous on B and harmonic in B. By the definition of g we have g(p0 ) = 0 whence h(w0 ) = 0, and (1) implies hu (w0 ) = 0 and hv (w0 ) = 0. Because of f (∂B) = ∂G, the set f (∂B) is not contained in L, and therefore h(w) ≡ const. Let F : B → C be the holomorphic function that satisfies F (w0 ) = 0 and Re F = h. Then we have F (w0 ) = 0 and F (w) ≡ 0. Hence there is an integer m ≥ 2 and a complex number am = 0 such that the Taylor expansion of F (w) at w = w0 can be written as

2.3 A Modification of the Kneser–Rad´ o Theorem

51

F (w) = am (w − w0 )m + · · · . We infer that in a sufficiently small neighbourhood U of w0 the zeros of h lie on m different, real analytic arcs through w0 which divide U into 2m sectors, and 2m ≥ 4. Let B + and B − be the open subsets of B where h(w) is positive or negative respectively, and set B0 := {w ∈ B : h(w) = 0}. By the definition of h we have f (B0 ) ⊂ L ∩ G. Invoking the maximum principle, we see that no connected component B ∗ of B + or B − has a boundary ∂B ∗ containing no more than one point of ∂B; otherwise we would have h(w) = 0 on ∂B ∗ and therefore h(w) ≡ 0 on B ∗ , a contradiction. Hence the connected component Z of B0 containing w0 has the property that Z ∩ ∂B contains at least four (different) consecutive points w1 , w2 , w3 , w4 such that on the subarcs β1 , β2 , β3 , β4 of ∂B between those points there exist four other points ζj ∈ βj , j = 1, 2, 3, 4, satisfying either h(ζ1 ) > 0, h(ζ2 ) < 0, h(ζ3 ) > 0, h(ζ4 ) < 0 or the opposite set of inequalities. Let C be the closed connected component of L ∩ G such that p0 ∈ C. Then the inclusion f (B0 ) ⊂ L ∩ G implies that f (Z) ⊂ C whence f (Z) ⊂ C. Since the boundary map f |∂B : ∂B → ∂G is weakly monotonic, we obtain four different consecutive points qj := f (wj ) on ∂G, 1 ≤ j ≤ 4. By construction, these points lie in C, and therefore the connected component C of L ∩ G contains at least four different points of ∂G, the points q1 , q2 , q3 , q4 . We can assume that the straight segment [q1 , q4 ] := {p ∈ R2 : p = (1 − t)q1 + tq4 , 0 ≤ t ≤ 1} lies in C, and that q2 , q3 ∈ [q1 , q4 ]. We then infer that the subarc Γ ∗ of ∂G containing q1 , q2 , q3 , q4 must either be a subset of the halfspace H+ := {p ∈ R2 : g(p) ≥ 0} or of the halfspace H− := {p ∈ R2 : g(p) ≤ 0}. However, this contradicts the fact that each of the open halfspaces int H+ and int H− contains at least one of the points f (ζj ), j = 1, 2, 3, which lie on Γ ∗ between q1 and q4 . Thus we infer that (2)

Jf (w) = 0

for all w ∈ B.

Hence the mapping f |B is open and locally one-to-one. In conjunction with f (B) ⊂ G we infer that f (B) is an open set contained in G. Moreover, the set G := G \ f (B) is either empty and therefore G = f (B), or G is nonempty and open. In fact, if G were nonempty and nonopen, we could find a point p ∈ G and a sequence of points pj ∈ f (B) with pj → p. Then there are points wj ∈ B satisfying f (wj ) = pj . We can assume that wj → w; then w ∈ B and f (w) = p. Because of f (∂B) ⊂ ∂G and p ∈ G ⊂ G it follows that w ∈ B, and therefore p ∈ f (B) = G \ G , a contradiction. Consequently G would be the disjoint union of two nonempty open sets, which is impossible. Hence we have G = f (B). Invoking now the monodromy principle, we infer that f |B maps B bijectively onto G, and therefore f |B is a real analytic diffeomorphism of B onto G. If f |∂B is one-to-one then f yields a continuous bijective map of B onto G. It follows that f is a homeomorphism of B onto G.

52


Let us now apply this general result on harmonic mappings to the projection f := π ◦ X of a minimal surface X ∈ M∗ (Γ, S) and to a domain G in R2 as described in Section 2.1. Proposition 2. Suppose that X ∈ M∗ (Γ, S) and that f (B) ⊂ G. Then f |B is a diffeomorphism of B onto G. If f |∂B is strictly monotonic then f maps B homeomorphically onto G.

2.4 Properties of the Gauss Map, and Stable Surfaces So far in this treatise we have not systematically used the Gauss map N of a minimal surface X : B → R3 although it plays a prominent role in the Weierstrass representation formula of a minimal surface as well as in other places. Therefore we take now the opportunity to derive some general properties of the Gauss map; then we turn to special properties of N which are relevant for surfaces X in the classes M(Γ, S) and M∗ (Γ, S) described in Section 2.1. In the sequel we always assume that X : B → R3 is a minimal surface which may have branch points in B. As it was shown in Vol. 1, Section 3.2, one can extend N to a continuous function B → S 2 ⊂ R3 . Using stereographic projection, it follows that N is real analytic in B. If X can be extended to a smooth mapping, say, on B, we can derive from the asymptotic expansion of Xw at boundary branch points that N possesses a continuous extension to B, using the same reasoning as in the interior; moreover, N is smooth in B \ {branch points of X}. Similarly we can argue if X is merely smooth in B ∪ Γ0 where Γ0 is an open subarc of ∂B. Let S be the set of branch points of X in B, and U be the set of umbilical points w0 ∈ B where the Gauss curvature K of X vanishes. By formula (6) of Section 1.4 of Vol. 1 we have

|Nu |2 |Xu |2 Nu · N v Xu · X v = −K Nu · N v |Nv |2 Xu · X v |Xv |2 on B \ (S ∪ U) whence the equations (1)

|Nu |2 = |Nv |2 ,

N u · Nv = 0

and (2)

|∇N |2 = −K|∇X|2 = −2K|Xu |2

are satisfied on B \ {S ∪ U}. If K(w) ≡ 0 in B \ S then X(w) is a planar surface and therefore N (w) ≡ const; thus (1) and (2) are trivially satisfied on B. If K(w) ≡ 0 then the zeros of K(w) in B \ S are isolated, and so we obtain (1) and (2) in B \ S by a continuity argument. Moreover, the branch points of X

2.4 Properties of the Gauss Map, and Stable Surfaces

53

are isolated in B, and so we obtain (1) in B, whereas (2) holds only in B \ S since K is singular at the branch points. From (1) we infer (3)

2|Nu ∧ Nv | = |∇N |2

in B,

and by formula (44) from Vol. 1, Section 1.2, we have (4)

Nu ∧ Nv = KXu ∧ Xv

in B \ S.

Since K ≤ 0, it follows that Nu ∧ Nv = −|Nu ∧ Nv |N , whence by (3) we have 2Nu ∧ Nv = −N |∇N |2 .

(5)

Moreover, the mapping N : B → S 2 ⊂ R3 is regular on B \ (S ∪ U) and has mean curvature 1 in direction of the inner normal of S 2 , i.e. in the direction of −N . If we apply Corollary 2 of Theorem 1 in Section 2.5 of Vol. 1 to N , taking (1) into account, we obtain ΔN = 2Nu ∧ Nv , and by (5) it follows that (6)

ΔN + N |∇N |2 = 0

holds in B \ (S ∪ U). A continuity argument yields that equation (6) is satisfied in B. Thus N is a harmonic mapping from B into the two-dimensional sphere S 2 . From Xu ∧ Xv = EN with E := |Xu |2 we infer that (7)

Xu ∧ N = −Xv ,

Xv ∧ N = Xu ,

and (4) together with K ≤ 0 implies that (8)

N u ∧ N = Nv ,

Nv ∧ N = −Nu .

Furthermore from H = 0 and the Weingarten equations (see Vol. 1, Section 1.3, (30)) it follows that (9)

Xu ∧ Nv + Nu ∧ Xv = 0.

The Dirichlet integral 1 D(N ) = 2

|∇N |2 du dv B

of N is just the total curvature of the minimal surface X. In fact, the area element dA of X is given by

54


dA = W du dv,

W=

EG − F2 = E = |Xu |2 ,

and so (2) implies (10)

|K| dA

D(N ) = X

since K ≤ 0. Thus the minimal surfaces X of finite total curvature are exactly those with D(N ) < ∞. For these surfaceswe can define the second variation δ 2 A(X, Y ) of the area functional A(X) = B |Xu ∧ Xv | du dv at X in a normal direction Y = λN with λ ∈ C 1 (B) as in the case of a regular surface X. In fact, let us first assume that X is regular. Then we have (see Vol. 1, Section 2.9, formula (17)): |∇X λ|2 + 2Kλ2 dA. δ 2 A(X, Y ) = B

By Section 1.5, (13 ) of Vol. 1 we also have |∇X λ|2 = (1/E)|∇λ|2 ,

whence

|∇λ|2 + 2K|Xu |2 λ2 du dv.

δ 2 A(X, Y ) = B

On account of (2) it follows that 2 (|∇λ|2 − |∇N |2 λ2 ) du dv (11) δ A(X, Y ) =

for Y = λN.

B

Moreover, Y = λN yields |Yu |2 = λ2u + λ2 |Nu |2 , and therefore

|Yv |2 = λ2v + λ2 |Nv |2

|∇Y | du dv =

(|∇λ|2 + λ2 |∇N |2 ) du dv

2

(12) B

B

whence (13)

1 2 δ A(X, Y ) ≤ D(Y ) 2

for Y = λN.

Note that (11) is well defined for λ ∈ C 1 (B) if D(N ) < ∞, and so we can use (11) as a definition of δ 2 A(X, λN ) even if S is nonempty. To simplify notation we write 2 (14) δ A(X, λ) := (|∇λ|2 − |∇N |2 λ2 ) du dv for λ ∈ C 1 (B) B

instead of δ 2 A(X, λN ).


55

Let Γ be a closed rectifiable Jordan curve in R3 and define C(Γ ) as in Vol. 1, Section 4.2, Definition 3. If Γ is in C 2,α , 0 < α < 1, then the total curvature X |K|dA of any minimal surface X ∈ C(Γ ) is finite, and its set S of branch points in B is finite (see Vol. 2, Section 2.11, Theorem 2); hence the Dirichlet integral D(N ) of the Gauss map N of X is finite. Suppose that X is a minimal surface in C(Γ ) which is a local minimizer of the area functional A among all surfaces Z ∈ C(Γ ) having the same Dirichlet boundary values as X, i.e. with Z(w) = X(w) on ∂B. If X is regular, i.e. S is empty, we have for any λ ∈ Cc1 (B) that A(X + ελN ) ≥ A(X)

if |ε| 1,

whence d A(X + ελN ) = 0 and dε ε=0

d2 A(X + ελN ) ≥ 0. 2 dε ε=0

The computations above as well as in Section 2.8 of Vol. 1 (see formula (16)) show that d2 A(X + ελN ) = δ 2 A(X, λ) dε2 ε=0 where δ 2 A(X, λ) is given by (14), and so we have 2 (15) δ A(X, λ) = (|∇λ|2 − |∇N |2 λ2 ) du dv ≥ 0 B

for any λ ∈ Cc1 (B). If S is nonvoid, we obtain in the same way that (15) is satisfied for any λ ∈ Cc1 (B \ S). Since the capacity of a single point is zero and S consists of only finitely many points, it follows by a standard argument that (15) holds for any λ ∈ C 1 (B) with λ|∂B = 0, and then for any ˚21 (B) ∩ L∞ (B). Thus the following definition of stability makes sense as λ∈H it is satisfied by the (local) minimizers of area: Definition 1. A minimal surface X ∈ C(Γ ) is said to be stable if (15) is ˚1 (B) ∩ L∞ (B). satisfied for every λ ∈ H 2 Note that the functional (16)

(|∇λ|2 − |∇N |2 λ2 ) du dv

Q(λ) := B

is a quadratic form in λ. Its Euler equation (17)

Δλ + |∇N |2 λ = 0 in B

is the Jacobi equation for the minimal surface X, and the vector fields Y = λN with λ being a solution of (17) are the Jacobi fields corresponding to X.

56


Equation (6) yields the interesting fact that every component N j of the Gauss map N = (N 1 , N 2 , N 3 ) is a solution of the Jacobi equation: (18)

ΔN j + |∇N |2 N j = 0 in B.

We remark that in the equations (17) and (18) it plays no role whatever boundary conditions on X we have required; it suffices to know that X : B → R3 is a nonconstant minimal surface. Moreover, we can consider (16) for λ ∈ H 1,2 (B) ∩ L∞ (B) if D(N ) < ∞. Let us now consider the special boundary configuration Γ, S, described in Section 2.1, and the class M(Γ, S) of minimal surfaces that are stationary in Γ, S. Proposition 1. If Γ ∈ C 2,α , 0 < α < 1, σ ∈ C 3 , and X ∈ M(Γ, S), then we have X |K| dA < ∞ whence D(N ) < ∞. Proof. Using the asymptotic expansions developed in Sections 2.10 and 3.4 as well as the Gauss–Bonnet formula, we obtain similarly as in Remark 3 of Vol. 2, Section 2.11 that X K dA < ∞, and (7) implies D(N ) < ∞. The Gauss map N of X is related to the “projected map” f := π ◦ X in the following way: (19)

WN 3 = Jf = det(fu , fv ) = (Xu ∧ Xv )3 .

Proposition 2. Let X be of class M(Γ, S) ∩ C 2 (B ∪ I∗ , R3 ) satisfying the boundary condition (20)

Xv (u, 0) = λ(u) · n(X(u, 0))

for u ∈ I∗ ,

where I∗ is an open subset of I, and suppose that f := π◦X satisfies Jf (w) ≥ 0 on B, f (I∗ ) ⊂ Σ, and E := |Xu |2 > 0 on I∗ . Then we obtain Jf (w) > 0 on I∗ . Proof. Note that W = E > 0 on I∗ , and so N = W−1 Xu ∧ Xv is of class C ∞ (B, R3 ) ∩ C 1 (B ∪ I∗ , R3 ). The assumption Jf ≥ 0 on B implies N 3 ≥ 0 on B ∪ I. Then we infer from (18) that ΔN 3 ≤ 0 in B, i.e. N 3 is superharmonic in B. Suppose now that Jf (u0 , 0) = 0 for some u0 ∈ I∗ . Then either N 3 (w) ≡ 0 in B, which is impossible (see the following Proposition), or Nv3 (u0 , 0) > 0 on account of E. Hopf’s lemma. Since n(x1 , x2 , x3 ) = (ν(x1 , x2 ), 0) for x ∈ S, equation (20) implies that Xv (u, 0) = (fv (u, 0), 0) for u ∈ I∗ and (21)

fv = λ · ν(f ) on I∗ ,

λ ∈ C 1 (I∗ ).

By differentiating both sides with respect to u we see that fuv = λu ν(f ) + λν (f )fu

on I∗ ,


57

where ν denotes the tangential derivative of ν(p) along Σ. On the other hand we infer from N 3 W = det(fu , fv ) that Nv3 W + N 3 Wv = det(fuv , fv ) + det(fu , fvv )

on B ∪ I∗ ,

whence by (21) Nv3 W = −N 3 Wv + det(λν (f )fu , fv ) + det(fu , fvv ) on I∗ . Since Nv3 (u0 , 0)W(u0 , 0) > 0 and N 3 (u0 , 0) = 0, it follows that fu (u0 , 0) = 0. Moreover, we have |fv | = |Xv | = |Xu | > 0 on I∗ . Finally, f (I) ⊂ Σ and (21) imply that fu · fv = 0 on I∗ whence Jf = |fu | · |fv | and therefore Jf (u0 , 0) > 0, contrary to our assumption Jf (u0 , 0) = 0. Thus we obtain Jf = 0 on I∗ , and because of Jf ≥ 0 on B it follows that Jf > 0 on I∗ , as we have claimed. Proposition 3. Suppose that Γ is a regular Jordan arc of class C 2,α , and that Γ is given as a graph of a height function γ ∈ C 2,α above Γ . Secondly assume that one of the Conditions (B), (M1 ), or (M2 ) holds true, and that ˚ contains no branch points of X, and we X ∈ M(Γ, S) ∩ C 0 (B, R3 ). Then C ˚ have N 3 > 0 on C. Proof. By the maximum principle we infer from Propositions 1 and 2 of Section 2.2 that f (B) is contained in the convex hull H(G) of G and Γ ⊂ ∂H(G), using Remark 1 and the reasoning of the proof of Proposition 4 in Section 2.2. ˚ and its image p0 := f (w0 ) on Γ ˚. Consider now an arbitrary point w0 ∈ C Then there is a linear function g : Π → R, g(p) := a · p + b,

p ∈ Π,

with a ∈ R2 , |a| = 1, and b ∈ R, such that L(p) := {p ∈ Π : g(p) = 0} is a support line of the convex set H(G) at p0 , and we can assume that H(G) lies in the halfspace {p ∈ Π : g(p) ≤ 0}. Then the function ϕ := g ◦ f is continuous in B, harmonic in B, and satisfies ϕ(w0 ) = 0, ϕ(w) ≤ 0 for all w ∈ B as well as ϕ(w) ≡ const. From E. Hopf’s lemma we infer that the ˚ satisfies normal derivative (∂/∂r)ϕ(w0 ) of ϕ at w0 ∈ C (22)

∂ ϕ(w0 ) > 0. ∂r

Here we use polar coordinates r, θ about the origin such that w = u+iv = reiθ . Since ϕr = a · fr , it follows that |fr (w0 )| > 0 whence

58


|Xθ (w0 )| = |Xr (w0 )| ≥ |fr (w0 )| > 0,

(23)

˚ and therefore |Xu (w0 )| > 0. Thus there are no branch points of X on C. Now we set ξ(θ) := X 1 (cos θ, sin θ), η(θ) := X 2 (cos θ, sin θ), ζ(θ) := X 3 (cos θ, sin θ) = γ(ξ(θ), η(θ)),

0 ≤ θ ≤ π.

Then we have ζθ = γx (ξ, η)ξθ + γy (ξ, η)ηθ , and for some suitable constant c it follows that ζθ2 ≤ c[ξθ2 + ηθ2 ], whence |Xθ |2 ≤ (1 + c)[ξθ2 + ηθ2 ].

(24)

From (23) and (24) we infer that |fθ (w0 )| > 0.

(25)

Since ϕ|C assumes its maximum at w = w0 we have ϕθ (w0 ) = 0 and therefore a · fθ (w0 ) = 0,

(26) whereas (22) implies (27)

a · fr (w0 ) > 0.

From (23) and (25)–(27) we conclude that fr (w0 ) and fθ (w0 ) are linearly in˚ This implies dependent, and so we obtain Jf (w0 ) = 0 at any point w0 ∈ C. ˚ since f (cos θ, sin θ) is a parametrization of Γ which is positively Jf > 0 on C oriented with respect to G. Finally we want to derive a mixed boundary condition for the third coordinate N 3 of the Gauss map N of a minimal surface X on the free boundary of X. Proposition 4. Suppose that σ ∈ C 3 and X ∈ M(Γ, S), and let I be the open set on R obtained by removing the boundary branch points from I. Then N ∈ C 1,β (B ∪ I , R3 ) for any β ∈ (0, 1) and (28)

Nv3 = κ(X)[N · τ (X)]2 {Xv · n(X)}N 3

on I .


59

Proof. It suffices to prove (28). Let e3 := (0, 0, 1). On account of (8) it follows that Nv3 = Nv · e3 = −(N ∧ Nu ) · e3 = −(Nu ∧ e3 ) · N, and so we have on B ∪ I .

Nv3 = −(N ∧ e3 )u · N

(29)

Since Xv (u, 0) is perpendicular to S at X(u, 0), the vectors N (u, 0) and e3 are tangent to S, and so N (u, 0) ∧ e3 is perpendicular to S. Hence there is a function ρ : I → R such that N (u, 0) ∧ e3 = ρ(u)n(X(u, 0))

(30)

on I.

It follows on I that ρ = (N ∧ e3 ) · n(X) = N · (e3 ∧ n(X)), whence ρ = N · τ (X)

(31)

on I.

From (29) and (30) we obtain on I .

Nv3 = −[ρn(X)]u · N

Moreover, we have Xv = λn(X) with λ = 0 on I whence n(X) · N = 0 on I and therefore Nv3 = −ρN · [n(X)]u on I. Consequently, Nv3 = −ρN · ∇n(X)Xu

(32)

on I.

Now we interpret ∇n(X(u, 0)) as Weingarten map of the surface S at X(u, 0). It is a symmetric linear operator on the tangent space TX S to S with the eigendirections τ (X), e3 and the eigenvalues κ(X) and 0. Because of (31) it follows that N · ∇n(X) · Xu = ρτ (X) · ∇n(X) · τ (X)[Xu · τ (X)] and

on I

τ (X) · ∇n(X) · τ (X) = κ(X) on I ,

whence (33)

N · ∇n(X) · Xu = ρκ(X)[Xu · τ (X)]

on I .

Furthermore, τ = e3 ∧ n implies Xu · τ (X) = Xu · (e3 ∧ n(X))

on I,

and because of |Xu | = |Xv |, Xu · Xv = 0 and the free boundary condition Xv = −|Xv |n(X)

on I

we obtain (34)

Xu · τ (X) = (N · e3 )[Xv · (−n(X))]

Relation (28) is now a consequence of (32)–(34).

on I.

60


Note that ∇n(x) at x ∈ S is the Weingarten map of S with respect to −n(x) = (−ν(x1 , x2 ), 0) where −ν(p) is the inner normal of ∂G at p = (x1 , x2 ).

2.5 Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface In this section we shall formulate conditions which guarantee that there is at most one stationary minimal surface within the boundary configuration Γ, S. Clearly such a uniqueness result can only be true up to conformal equivalence since the equation ΔX = 0 and the conformality relations |Xu |2 = |Xv |2 , Xu · Xv = 0 are conformally invariant. Therefore we fix an arbitrary point P3 ˚ and consider the class on Γ M (Γ, S) := {X ∈ M(Γ, S) : X(i) = P3 }. The elements X of M (Γ, S) satisfy the 3-point condition X(wj ) = Pj with j = 1, 2, 3 and w1 = 1, w2 = −1, w3 = i; thus there are no two conformally equivalent surfaces in M (Γ, S). Hence “uniqueness in M(Γ, S)” actually means “uniqueness in M (Γ, S)”. More general, “uniqueness in a subclass of M(Γ, S)” means “uniqueness up to conformal equivalence”. The key idea in the following proofs of uniqueness is to find for any X ∈ M(Γ, S) an equivalent nonparametric representation Z(x, y) = (x, y, z(x, y)), (x, y) ∈ G, satisfying a mixed boundary value problem for the minimal surface equation (1 + zy2 )zxx − 2zx zy zxy + (1 + zx2 )zyy = 0 in G

(1)

and then to apply the maximum principle for comparing any two solutions of this boundary value problem. The first step in this approach is the following result: Proposition 1. Let X be of class M(Γ, S), σ ∈ C 3 , and then suppose that f := π ◦ X satisfies (A1) f (B) ⊂ G, (A2) f (∂B) = ∂G, (A3) f |∂B is weakly monotonic. Then we have: (i) f |B is a real analytic diffeomorphism from B onto G with Jf > 0 on B. (ii) If h := (f |B )−1 , then the function z : G → R defined by (2)

z(x, y) := X 3 (h(x, y)),

(x, y) ∈ G,

is a real analytic solution of the minimal surface equation (1), and the surface

2.5 Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface

(3)

Z(x, y) := (x, y, z(x, y)),

61

(x, y) ∈ G,

is equivalent to X|B ; in particular, X(B) = graph z = Z(G). We call Z the nonparametric representation of X above G. (iii) If f |∂B is a homeomorphism from ∂B onto ∂G, then f maps B homeomorphically onto G, and z as well as Z can be extended continuously onto G such that X(B) = Z(G). (iv) If Γ = graph γ then z ∈ C 0 (G ∪ Γ ) and (4)

z=γ

on Γ .

(v) Let I = I (X) := I \ {branch points of X on I}. Then we obtain Jf > 0 on I , z ∈ C 2 (G ∪ I ) with Σ := f (I ), and (5)

∂z =0 ∂ν

on Σ .

Proof. For the sake of convenience we have written x for x1 and y for x2 ; so Π is now the x, y-plane. Assertions (i) and (iii) follow from Proposition 1 in Section 2.3. Thus the surface X|B = (f |B , X 3 |B ) is equivalent to X ◦ h = (f ◦ h, X 3 ◦ h) = (idG , z) = Z, and so Z is a regular surface of zero mean curvature. This implies equation (1) (see Chapter 2), and we have proved (ii). Since X|C is a homeomorphism of C onto Γ and Γ = graph γ, the map f |C is a homeomorphism of C onto Γ . It follows that z is continuous on G ∪ Γ and satisfies (4), i.e. we have (iv). Finally, on account of Section 2.4, Proposition 2 we obtain Jf > 0 on I (X) since we have Jf > 0 on B and |Xu |2 > 0 on I (X). Furthermore we recall that X ∈ C 2,β (B ∪ I, R3 ). It follows that f maps B ∪ I diffeomorphically onto G ∪ Σ , and we have f ∈ C 2 (B ∪ I , R2 ), h ∈ C 2 (G ∪ Σ , R2 ) whence z ∈ C 2 (G ∪ Σ ). The normal N of X is of class C 1 (B ∪ I , R3 ) ∩ C 0 (B ∪ I, R3 ) and perpendicular to n(X) = (ν(f ), 0) along I. On the other hand, we have N ◦ h = ±{1 + zx2 + zy2 }−1/2 · (zx , zy , −1)

on Σ ,

and so it follows that ν · ∇z = 0 on Σ , i.e. we have proved (5).

Let us note that any minimal surface X as in Proposition 1 is an embedded regular minimal surface. Now we can state the first main result. Theorem 1. Let σ ∈ C 3 , and suppose that Condition (B) (or (M1 ), or (M2 ) respectively) and Condition (C) are fulfilled. Secondly, let X be a minimal surface of class M(Γ, S) which lies on the same side of S as Γ . Then we have:

62


(i) There are no branch points of X on B ∪ I. (ii) The Gauss map N of X satisfies N 3 > 0 on B ∪ I. (iii) f |B is a real analytic diffeomorphism from B onto G with Jf > 0 on B, and Z := (idG , z) with z := X 3 ◦ h, h := (f |B )−1 is an equivalent ˚ as well as ∂z/∂ν = 0 on Σ. ˚ representation of X satisfying (1) in G ∪ Σ 2,α 2,α and γ ∈ C (Γ ), with (iv) If Γ = graph γ then z = γ on Γ . If also Γ ∈ C ˚ contains no branch points of X, and N 3 > 0 on C. ˚ 0 < α < 1, then C Proof. (i) By virtue of the asymptotic expansion of Xw at I (cf. Section 2.10 of Vol. 2) there cannot be any branch point of X in I since we have assumed that X lies on the same side of S as Γ . (iii) Proposition 1 (or 2 respectively) of Section 2.2 implies f (I) = Σ, and Proposition 3 of Section 2.2 ensures that f |I is weakly monotonic. Furthermore we infer from Section 2.2, Proposition 4 and Remark 1 that f (B) ⊂ G. Thus assumptions (A1)–(A3) of the preceding Proposition are satisfied, and so we obtain the statements of (iii). (ii) In particular, Proposition 1 in conjunction with (i) yields Jf > 0 on B ∪ I whence N 3 > 0 on B ∪ I. (iv) Proposition 1 implies z = γ on Γ , and the other assertions of (iv) follow from Section 2.4, Proposition 3. Remark 1. If Σ is convex with respect to G, the domain G is convex, and Condition (B) (or (M1 ), or (M2 ) respectively) imply that f (∂B) = ∂G. Using the maximum principle it follows that f (B) ⊂ G, and therefore any X ∈ M(Γ, S) lies automatically on the same side of S as Γ . Hence this assumption in Theorem 1 is superfluous if Σ is convex with respect to G. Remark 2. If Σ is concave with respect to G then Condition (C) is fulfilled for all admissible arcs Γ . However, Γ has to be sufficiently close to a convex arc Σ in order that Condition (C) be satisfied. Remark 3. The inequality (6)

N 3 (w) > 0 for w ∈ B

means that the spherical image N (B) of the set B lies in the upper hemisphere of S 2 , and that the surface X : B → R3 nowhere has a vertical tangent plane. Similarly, (7)

N 3 (w) > 0 on I

implies that the free trace X|I is never vertical on the supporting surface S. Remark 4. In Section 2.6 we shall see that both Condition (C) and the assumption that X lies on one side of S can be omitted if we assume that X is a minimizer of area in the class C(Γ, S). More generally, this is true for local minimizers or even for “freely stable” surfaces X ∈ M(Γ, S) (see Section 2.6).


63

We now use Theorem 1 to prove a uniqueness result for surfaces X of class M(Γ, S) which lie on the same side of S as Γ . Theorem 2. Let σ be of class C 3 , and suppose that Condition (B) (or (M1 ), or (M2 ) respectively) and Condition (C) are fulfilled. Then (up to conformal equivalence) there exists at most one surface X ∈ M(Γ, S); it has to lie on one side of S. Proof. Suppose that X1 and X2 are surfaces in M(Γ, S) which lie on the same side of S as Γ . Let Z1 (x, y) = (x, y, z1 (x, y)),

Z2 (x, y) = (x, y, z2 (x, y)),

(x, y) ∈ G,

be the nonparametric representations of X1 and X2 respectively above G, which exist according to Theorem 1. Then the difference z := z1 − z2 satisfies a linear elliptic equation (8)

azxx + 2bzxy + czyy + dzx + ezy = 0 in G,

˚ and satisfies which is uniformly elliptic on every G G ∪ Σ (9)

z = 0 on Γ ,

∂z ˚ = 0 on Σ, ∂ν

provided that Γ = graph γ. We claim that z(x, y) = 0 on G. Otherwise, applying the maximum principle together with a limit process, we would obtain ˚ that Z assumes both its maximum and its minimum at the free boundary Σ. By means of E. Hopf’s lemma we would then arrive at a contradiction to ˚ ∂z/∂ν = 0 on Σ. If Γ is merely a generalized graph, the function z is discontinuous at points p ∈ Γ corresponding to vertical pieces of Γ , and so the above reasoning is not valid. We replace it by an argument that was already used in Chapter 1, and earlier in Section 7.3 of Vol. 1. Set Wj := 1 + |∇zj |2 where ∇zj = (∂zj /∂x, ∂zj /∂y), j = 1, 2. Since both z1 and z2 satisfy (1), we have div(W1−1 ∇z1 − W2−1 ∇z2 ) = 0

in G.

Let G G be a subdomain of G with a smooth boundary ∂G . Then we obtain ϕ div(W1−1 ∇z1 − W2−1 ∇z2 ) dx dy = 0 G

for any ϕ ∈ C 1 (G), and an integration by parts yields

−1 −1 −1 ∂z1 −1 ∂z2 dH1 ∇ϕ · (W1 ∇z1 − W2 ∇z2 ) dx dy = ϕ · W1 − W2 ∂ν ∂ν G ∂G where ν is the exterior normal to ∂G , and H1 is the 1-dimensional Hausdorff measure. Inserting ϕ = z1 − z2 we arrive at

64


(10)

(∇z1 − ∇z2 ) · (W1−1 ∇z1 − W2−1 ∇z2 ) dx dy

−1 ∂z1 −1 ∂z2 dH1 . (z1 − z2 ) · W1 − W2 = ∂ν ∂ν ∂G

G

By the reasoning used in Section 1.5 we infer that (11)

(max{W1 , W2 })−3 |∇z1 − ∇z2 | ≤ (∇z1 − ∇z2 ) · (W1−1 ∇z1 − W2−1 ∇z2 )

holds true everywhere on G. Fix some set G0 G G, and set m(G0 ) := sup max{W1 , W2 }. G0

Then we obtain from (10) and (11) that −3 m (G0 ) (12) |∇z1 − ∇z2 | dx dy G0

−1 ∂z1 −1 ∂z2 dH1 , (z1 − z2 ) · W1 − W2 ≤ ∂ν ∂ν ∂G where ν = ν (∂G ) is the exterior normal to ∂G . Note that the functions ∂ ˚ zj and Wj−1 ∂ν zj are uniformly bounded on G, and that zj ∈ C 1 (G ∪ Σ), 0 ˚ ∂zj /∂ν = 0 on Σ as well as zj ∈ C (G ∪ Γ ) and z1 − z2 = 0 on Γ , where Γ is the set Γ minus finitely many points. Therefore we can choose a sequence {Gn } of domains Gn G with the exterior normals νn to ∂Gn such that G1 G2 · · · Gn · · · and

−1 ∂z1 −1 ∂z2 dH1 = 0. (z1 − z2 ) · W1 − W2 lim n→∞ ∂G ∂νn ∂νn n Then (12) implies

|∇z1 − ∇z2 | dx dy = 0 G0

whence ∇z1 = ∇z2 on G0 , and therefore also on G since G0 can be chosen as an arbitrary open set with G0 G. This implies z1 (x, y) ≡ z2 (x, y) on G because of z1 = z2 on Γ . Remark 5. (i) If Σ is convex with respect to G, every X ∈ M(Γ, S) lies on the same side of S as Γ , provided that Condition (B) (or (M1 ), or (M2 ) respectively) is fulfilled; see Remark 1. (ii) If Σ is concave with respect to G, then Condition (C) is fulfilled for any admissible arc Γ ; see Remark 2. (iii) If one restricts X to the class of freely stable surfaces in M(Γ, S) then one can omit Condition (C) as well as the assumption that X is to lie on one side of S; see Remark 4.


65

(iv) Are there other cases than those described in (i) and (iii) where one can omit the assumption that X ∈ M(Γ, S) a priori lies on one side of S? This is, in fact, possible if we strengthen Condition (C), which requires that the normal lines L(p), p ∈ Σ form a field on G. Instead we now demand that these lines form a field on K(G), where K(G) is the convex hull of G: Condition (C∗ ). The family of lines L(p), p ∈ Σ, furnishes a foliation of the set K(G), that is, each point q ∈ K(G) lies on exactly one line L(p) with p ∈ Σ.

Fig. 1. The family {L(p)}p∈Σ furnishes a foliation on K(G)

Then we obtain the following result: Theorem 3. Let σ be of class C 3 , Γ = graph γ with γ ∈ C 0 , and suppose that Conditions (B) and (C∗ ) are satisfied. Then the class M(Γ, S) contains (up to conformal reparametrization) exactly one element X. This surface possesses a nonparametric representation above G, Z(x, y) = (x, y, z(x, y)),

(x, y) ∈ G,

˚ of the minimal surface equation by means of a solution z ∈ C 0 (G)∩C 2 (G∪ Σ) (1) satisfying ∂z ˚ = 0 on Σ. z = γ on Γ and ∂ν The proof of this result uses similar ideas as before, but it is quite involved and somewhat tedious. For details we refer the reader to Hildebrandt and Sauvigny [1]. The essential step is a degree argument, by which it is shown that f is a homeomorphism of B onto G and f |B∪I yields a diffeomorphism ˚ (see Proposition 4 of [2]). The degree argument uses from B ∪ I onto G ∪ Σ the fact that |fu | > 0 on I, which is proved by using Condition (C∗ ). The proof resembles that of Proposition 3 in Section 2.2, but it relies on a more general maximum principle and on the Hartman–Wintner lemma.

66


2.6 Uniqueness of Freely Stable Minimal Surfaces As we deal with a partially free boundary value problem, it is reasonable to consider variations X ε of X ∈ M(Γ, S) that vary X along the free boundary I. In this case, however, one can no longer operate with purely normal variations X ε = X + λN , since in general they will not satisfy X ε (I) ⊂ S. In order to produce an admissible variation X ε ∈ C(Γ, S) we have to add correction terms of higher order. Therefore we consider now variations of the following kind: (1)

X ε (w) = X(w) + ελ(w)N (w) +

ε2 Z(w) + o(ε2 ) 2

as ε → 0.

We assume X ε to be admissible, i.e. X ε ∈ C(Γ, S), and compute the second derivative (d2 /dε2 )A(X ε )|ε=0 without using the results of Section 2.9 in Vol. 1. Let us first determine how the second order term Z(w) is to be chosen so that X ε becomes admissible for |ε| 1. Clearly we have to add a second-order correction term in direction of n(X) if we want to correct a possible lift-off of the trace of (X + ελN )|I from the supporting surface S since n|S is the surface normal of S. Thus we assume Z to be of the form (2)

Z(w) = μ(w)n(X(w))

with a scalar factor μ(w) whose values on I are to be determined. Let us denote differentiation with respect to ε by a superscript dot: ∂/∂ε = · . For fixed u ∈ I the mapping ε → X ε (u, 0), |ε| 1 has to describe a curve on S, and so we necessarily have X˙ ε (u, 0) · n(X ε (u, 0)) = 0

for |ε| 1.

Differentiation with respect to ε yields ¨ ε (u, 0) · n(X ε (u, 0)) + X˙ ε (u, 0) · ∇n(X ε (u, 0)) · X˙ ε (u, 0) = 0, X and for ε = 0 it follows that Z · n(X) + λN · ∇n(X) · λN = 0 on I. Because of (2) this relation is equivalent to μ = −λ2 [N · ∇n(X) · N ]

on I.

Here ∇n(x) denotes the Weingarten operator on the tangent space Tx S of S at x ∈ S, and the special structure of S yields N · ∇n(X) · N = κ(X)[N · τ (X)]2 (cf. also Section 2.4), and so we infer that μ = −λ2 κ(X)[N · τ (X)]2

on I.

on I

2.6 Uniqueness of Freely Stable Minimal Surfaces

67

Therefore the correction term Z is of the form (3)

Z = −λ2 κ(X)[N · τ (X)]2 n(X)

on I.

Conversely, if we define Z : B → R3 by (4)

Z := −λ2 κ(X)[N · τ (X)]2 n(X),

a simple flow argument shows that we can construct an admissible variation X ε of X, |ε| 1, which is of the form (1). Summarizing, we have found: Lemma 1. For any X ∈ M(Γ, S) and any λ ∈ C 1 (B) satisfying λ = 0 on C there exists a twice differentiable one-parameter family {X ε }|ε| 0

(11)

for all w ∈ B.

Furthermore X has no branch points in B \ {±1}, and f := π ◦ X satisfies Jf > 0 in B \ {±1}. It follows that f yields a homeomorphism of B onto G and a diffeomorphism of B \ {±1} onto G \ {p1 , p2 }. Proof. By assumption we have (10). A suitable approximation yields (12)

2

δ A(X, λ) ≥ 0 for all λ ∈ H21 (B) ∩ L∞ (B) with λ|C = 0.

Set ω := N 3 and define ω − by ω − (u, v) :=

ω(u, v) if ω(u, v) < 0, 0 if ω(u, v) ≥ 0.

70


We have ω ∈ H21 (B) ∩ C 0 (B) as well as ω(±1) = 1 (see Vol. 2, Section 2.4), and Proposition 3 of Section 2.4 yields ω > 0 on C. This implies ω − ∈ H21 (B) ∩ C 0 (B) and supp ω − ⊂ B ∪ I,

(13)

in particular ω − = 0 on C. Denote by B + and B − the open subsets of B where ω is positive or negative respectively. Because of ω > 0 on C the set B + is nonempty. We now want to show that B − is void. To this end we note that B − B ∪ I. We claim that 2

δ A(X, ω − ) = 0.

(14)

We first prove this equation in case that X has no branch points on I. Then N and therefore also ω are of class C 1 on B ∪ I, and so we infer from (|∇ω − |2 + 2EK|ω − |2 ) du dv = (|∇ω|2 + 2EK|ω − |2 ) du dv B−

B

via an integration by parts that (|∇ω − |2 + 2EK|ω − |2 ) du dv B = ω · (−Δω + 2EKω) du dv − B−

I∩∂B −

ωωv ω du,

using a suitable approximation device. From (2) and (6) of Section 2.4 we infer that Δω − 2EKω = 0 in B whence (|∇ω − |2 + 2EK|ω − |2 ) du dv = − ωωv du. (15) I∩∂B −

B

Furthermore, E(X, ω − ) =

[(Xu · Z)u + (Xv · Z)v ] du dv B

with Z := −|ω − |2 κ(X)[N · τ (X)]2 n(X) and therefore E(X, ω − ) = [(Xu · Z)u + (Xv · Z)v ] du dv. B−

An integration by parts yields − ω 2 κ(X)[N · τ (X)]2 {Xv · n(X)} du (16) E(X, ω ) = I∩∂B −

since ω − = ω on B − . By adding (15) and (16) we arrive at

2.6 Uniqueness of Freely Stable Minimal Surfaces 2

δ A(X, ω − ) = −

71

I∩∂B −

ω · {ωv − κ(X)[N · τ (X)]2 [Xv · n(X)]ω} du.

On account of formula (28) in Section 2.4, we finally obtain (14). If there exist branch points wj on I, we choose radii εj,n > 0 with εj,n → 0 as n → ∞ such that Z(ρ, θ) := N (wj + ρeiθ ) satisfies π |Zθ (ρ, θ)| dθ → 0 as n → ∞. 0

ρ=εj,n

(This can be achieved by the same reasoning as in the Courant–Lebesgue lemma, since D(N ) < ∞.) From Section 2.4, (1) we infer |Zρ | = ρ−1 |Zθ | whence π |Zρ (ρ, θ)|ρ dθ → 0 as n → ∞. 0

ρ=εj,n

Note that ds = ρ dθ is the element of the arc length on the semicircle Cj,n := {w ∈ C : |w − wj | = εj,n , Im w > 0}, and −Zρ is the normal derivative of N with respect to the normal to Cj,n pointing to wj . Thus we obtain

(17) ωωρ ds → 0 as n → ∞. Cj,n

j

Furthermore we also have

ω 2 |κ(X)|[N · τ (X)]2 |Xv · n(X)| du → 0 as n → ∞ (18) j

Ij,n

for Ij,n := {u ∈ R : |u − uj | ≤ εj,n } since the integrand of these integrals is bounded. If we now replace the domain of integration, B, by Bn := B \ j Sj,n , Sj,n := {w ∈ B : |w − wj | < εj,n } in the above reasoning, we obtain 2

δ A(X, ω − ) = 0 + Rn

with Rn → 0,

taking (17) and (18) into account. Let now ϕ be an arbitrary function of class Cc∞ (B) and set λ := ω − + εϕ. Then λ ∈ H21 (B) ∩ L∞ (B) and λ = 0 on C, and by (12) it follows that 2

Φ(ε) := δ A(X, ω − + εϕ) ≥ 0 for all ε ∈ R, whereas (14) implies that Φ(0) = 0. Thus we obtain Φ (0) = 0 which means that (∇ω − · ∇ϕ + 2EKω − ϕ) du dv = 0. B

Since 2EK = −|∇N |2 is real analytic in B, it follows that also ω − is real analytic in B. On the other hand ω − vanishes on the nonempty open set B + ,

72


and so ω − is zero on all of B. Therefore B − is empty, and we have N 3 = ω ≥ 0. Because of ΔN 3 = −|∇N |2 N 3 ≤ 0 the function N 3 is superharmonic in B, and so the maximum principle implies N 3 > 0 on B (otherwise we would have N 3 (w) ≡ 0 on B which is impossible by virtue of N 3 > 0 on C). Moreover, a sufficiently small disk Bε (w0 ) in B is in first order an m-fold copy of a disk, m ≥ 1; then N 3 (w0 ) > 0 implies that f (Bε (w0 )) is in first order an m-fold copy of a solid ellipse centered at f (w0 ). Thus f (w0 ) is an inner point of f (B), and so the mapping f |B is open. Furthermore, by Proposition 1 (or 2 respectively) of Section 2.2 we have f (∂B) = ∂G. We then conclude that f (B) = G, and so X(B) lies on the same side of S as Γ . Consequently X has no branch points on I, that is, W = E > 0 on I. Furthermore, Jf = WN 3 and N 3 > 0 in B imply that Jf ≥ 0 in B. By Section 2.4, Proposition 2 it follows that Jf > 0 on I, and so in particular fu (u, 0) = 0 on I. In addition, ˚ and so f |∂B furnishes a Proposition 3 of Section 2.4 implies fθ = 0 on C, homeomorphism of ∂B onto ∂G. Invoking now Proposition 1 of Section 2.3 we infer that f yields a homeomorphism of B onto G and a diffeomorphism from B \ {±1} onto G \ {p1 , p2 }. From Proposition 2 we immediately derive the following result: Theorem 1. Suppose that Γ ∈ C 2,α , 0 < α < 1, and σ ∈ C 3 satisfy Condition (B) or (M1 ) (or M2 respectively). Then (up to conformal equivalence) there exists exactly one freely stable minimal surface X ∈ M(Γ, S). This surface is the unique minimizer of Dirichlet’s integral as well as of the area functional in the class C(Γ, S). It can be represented as a graph of a scalar function z ∈ C 0 (G) ∩ C 2 (G \ {p1 , p2 }) which satisfies the minimal surface equation (19)

(1 + zy2 )zxx − 2zx zy zxy + (1 + zx2 )zyy = 0

in G

as well as the boundary conditions (20)

z=γ

on Γ

and

∂z =0 ∂ν

˚ on Σ.

Proof. The statements of this theorem are proved in the same way as Theorems 1 and 2 of Section 2.5. The only difference is that we need assume Condition (B), since Proposition 2 above provides us with all the information that we earlier had drawn from Condition (C). If we merely aim for existence without obtaining uniqueness, we can replace Condition (B) by a considerably weaker requirement. ˜1 ∪˙ Σ ˜ ∪˙ Σ ˜2 in Π with the same ˜0 = Σ Condition (R). There is an arc Σ ˜ = Σ and Γ, Σ ˜0 ˙ ˙ properties as Σ0 = Σ1 ∪ Σ ∪ Σ2 stated in 2.1 such that Σ satisfy Condition (B).

2.6 Uniqueness of Freely Stable Minimal Surfaces

73

Then we obtain Theorem 2. Suppose that Γ ∈ C 2,α , 0 < α < 1, and σ ∈ C 3 satisfy Condition (R). Then there exists a freely stable minimal surface X ∈ M∗ (Γ, S) with N 3 > 0 on B which can be represented as a graph of a scalar function z of class C 0 (G) ∩ C 2 (G \ {p1 , p2 }) satisfying (19) and (20). ˜ we obtain a minimizer X for the configProof. Applying Theorem 1 to Γ, Σ ˜ ˜ ˜ uration Γ, S where S := Σ0 × R. Such a minimizer is a freely stable surface in M∗ (Γ, S). The other assertions on X follow from Theorem 1. For example, Condition (R) follows from the next condition, which is easy to check. Condition (A). The lines L(p1 ) and L(p2 ) normal to Σ0 at p1 and p2 respectively do not meet G and are parallel to and different from each other. Remark 1. Theorem 2 is a much stronger existence result than the existence statements of Theorem 1 or of Section 2.5, Theorem 3. However, the uniqueness assertions of those theorems might be false, even if G is a convex domain. This can be seen by means of the following Example 1. Let Γ, S be a boundary configuration whose orthogonal projection into the x1 , x2 -plane looks as in Fig. 1, while the orthogonal projection Γ ∗ of Γ into the x2 , x3 -plane has a shape as depicted in Fig. 2. Inspecting

Fig. 1. The projection of Γ, S on the x1 , x2 -plane

Fig. 2. The projection Γ ∗ of Γ on the x2 , x3 -plane

74


Fig. 1 it is evident that Γ, Σ satisfy Condition (A) but not (B) or (M1 ). Choosing a, h, and ε appropriately we obtain a disk-type minimal surface X ∈ M(Γ, S) which is shaped like a “collar” and whose area is less than 2ah; so its area is also less than the area of the domain G∗ in the x2 , x3 plane whose boundary consists of the arc Γ ∗ and the segment of the x3 -axis between the endpoints P1∗ and P2∗ of Γ ∗ . Yet the number meas G∗ is certainly a lower bound for the area of the nonparametric minimal surface which according to Theorem 2 exists as a graph of a solution z to the boundary value problem (19), (20) above the domain G. This surface is also a freely stable surface in M(Γ, S), and so there exist at least two freely stable minimal surfaces in M(Γ, S). By assuming that Γ, S is symmetric with respect to the x2 -axis we even obtain that there exist at least three freely stable minimal surfaces that are stationary in Γ, S. Note also that in our example the domain G is convex. The reasoning of Section 2.5 yields that the solution of (19), (20) is unique, whereas M(Γ, S) contains at least 2 (or 3) surfaces.

2.7 Asymptotic Expansions In this section we assume that S is the boundary of a wedge, i.e. of a sectorial domain Wα given as Wα := x ∈ R3 : x1 = r cos ϕ, x2 = r sin ϕ, r > 0, 0 < ϕ < π + απ where α is a real parameter satisfying −1 < α ≤ 1. The case α = 1 was treated in Chapter 1; so we shall mainly deal with the case |α| < 1. The line L := {x ∈ R3 : x1 = 0, x2 = 0} is the edge of Wα . For arbitrary β ∈ [0, 2π] we define the halfplane Hβ := x ∈ R3 : x1 = r cos β, x2 = r sin β, r > 0 and the plane Πβ := x ∈ R3 : x1 = r cos β, x2 = r sin β, r ∈ R . Then we have S = ∂Wα = H0 ∪ L ∪ Hπ+απ . We assume that the endpoints P1 and P2 of Γ lie on H0 and Hπ+απ respectively. Then Σ is the polygonal arc p1 0p2 with the endpoints p1 = π(P1 ), p2 = π(P2 ) and the vertex 0. For X ∈ M∗ (Γ, S) the vertex 0 is either of T-type or E-type (see Section 2.1, Definition 3). Normalizing X by the condition (1)

f (0) = 0

2.7 Asymptotic Expansions

75

where f := π ◦ X, we either have (f |I )−1 (0) = 0, or else (f |I )−1 (0) = [u , u ] with −1 < u < u < 1. We say that X is of class T(Γ, S) in the first case and of class E(Γ, S) in the second one. Set I1 := (−1, 0) and I2 := (0, 1). Then I = I1 ∪ {0} ∪ I2 , and we have Proposition 1. Any X ∈ T(Γ, S) has the following properties: (i) X(I1 ) ∈ Hπ+απ , X(I2 ) ∈ H0 , X(0) ∈ L. (ii) X is real analytic on B ∪ I1 ∪ I2 , and Xv (u, 0) is perpendicular to Ππ+απ on I1 and to Π0 on I2 . (iii) X 3 possesses a harmonic extension to D = {w ∈ C : |w| < 1} with X 3 (w) = X 3 (w). Proof. (i) follows immediately from the definition of T(Γ, S), and both (ii) and (iii) are proved in the same way as Proposition 2 in Section 2.1. To formulate the next result, we introduce the standard orthonormal base ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 e1 := ⎝ 0 ⎠ , e2 := ⎝ 1 ⎠ , e3 := ⎝ 0 ⎠ 0 0 1 of R3 as well as the rotated orthonormal base ⎛ ⎞ ⎛ ⎞ cos θπ − sin θπ e1 (θ) := ⎝ sin θπ ⎠ , e2 (θ) := ⎝ cos θπ ⎠ , 0 0

⎛ ⎞ 0 e3 := ⎝ 0 ⎠ 1

for 0 ≤ θ ≤ 2. Furthermore we define the complex vectors (2) and

1 j := √ (e1 − ie2 ), 2 1 j(θ) := √ e1 (θ) − ie2 (θ) , 2

1 j := √ (e1 + ie2 ) 2 1 j(θ) := √ e1 (θ) + ie2 (θ) . 2

As usual we write a · b = a1 b1 + a2 b2 + a3 b3 for any a, b ∈ C3 . Then we obtain j · j = 0, j · e3 = 0, j ∧ j = ie3 ,

j · j = 0, j · e3 = 0,

j · j = 1, e3 · e3 = 1,

j ∧ e3 = −ij,

j ∧ e3 = ij;

therefore {j, j, e3 } is an orthonormal base of C3 equipped with the scalar product a, b := a · b. We also have ⎛ ⎞ ⎛ ⎞ 1 1 1 ⎝ 1 −i ⎠ , j = j(0) = √ ⎝ i ⎠ j = j(0) = √ 2 2 0 0

76


⎛

⎛ ⎞ ⎞ cos θπ + i sin θπ 1 iθπ 1 e j(θ) = √ ⎝ sin θπ − i cos θπ ⎠ = √ ⎝ −i ⎠ 2 2 0 0

and

whence j(θ) = eiθπ j,

(3)

j(θ) = e−iθπ j.

It follows that also {j(θ), j(θ), e3 } is an orthonormal base of C3 . Note that Πθπ = span{e1 (θ), e3 } = span{e1 (1 + θ), e3 }, and so the orthog⊥ = span{e2 (θ)} = span{e2 (1 + θ)}. onal complement is given by Πθπ ⊥ Lemma 1. For any a, b ∈ R3 with a ∈ Πθπ , b ∈ Πθπ , θ ∈ R, there are uniquely determined numbers λ, μ, ν ∈ R such that

a − ib = λeiθπ j + μe−iθπ j + νe3 .

(4)

Proof. By assumption there are real numbers κ1 , κ2 , κ3 such that a = κ1 e1 (θ) + κ3 e3 ,

b = κ2 e 2 ,

whence a − ib = κ1 e1 (θ) − iκ2 e2 (θ) + κ3 e3 . Since

1 e1 (θ) = √ j(θ) + j(θ) , 2

1 −ie2 (θ) = √ j(θ) − j(θ) 2

it follows that κ2 κ1 a − ib = √ j(θ) + j(θ) + √ j(θ) − j(θ) + κ3 e3 2 2 =

κ1 − κ 2 κ 1 + κ2 √ j(θ) + √ j(θ) + κ3 e3 . 2 2

This implies (4), taking (3) into account.

Set I := {u ∈ R : 0 < |u| < 1} = I1 ∪ I2 , and let α be a real parameter with −1 < α ≤ 1. We obtain Lemma 2. Suppose that X ∈ C 1 (B ∪ I , R3 ) is a minimal surface in B. Then there are uniquely determined functions h1 (w), h2 (w), h3 (w) which are holomorphic in B and continuous on B ∪ I such that (5)

Xw (w) = h1 (w)wα j + h2 (w)w−α j + h3 (w)e3

for w ∈ B ∪ I ,

(6)

0 = Xw (w) · Xw (w) = 2h1 (w)h2 (w) + h23 (w),

(7)

|Xw (w)|2 = |h1 (w)|2 |w|2α + |h2 (w)|2 |w|−2α + |h3 (w)|2 .


77

Proof. Since {j, j, e3 } is a Hermitian orthonormal base in C3 , we can write φ(w) := Xw (w) as (8)

φ(w) = k1 (w)j + k2 (w)j + k3 (w)e3 .

From ΔX = 0 in B we infer φw (w) = 0, and so the functions k1 (w) = φ(w), j, k2 (w) = φ(w), j, k3 (w) = φ(w), e3 are holomorphic in B and continuous on B ∪ I . Setting h1 (w) := w−α k1 (w),

h2 (w) := wα k2 (w),

h3 (w) := k3 (w)

we obtain three functions hl (w) that are holomorphic in B and continuous on B ∪ I , and from (8) we infer (5). The conformality relations imply the equation Xw · Xw = 0, and the other assertion follows from a straight-forward computation. Lemma 3. For X ∈ T(Γ, S) the associated functions hl (w) of its representation formula (5) are real valued on I = I1 ∪ I2 . Proof. By virtue of Proposition 1 we know that Xu (u, 0) ∈ Π0 ,

Xv (u, 0) ∈ Π0⊥

Xu (u, 0) ∈ Ππ+απ ,

for u ∈ I2 ,

⊥ Xv (u, 0) ∈ Ππ+απ

for u ∈ I1 .

Applying Lemma 1 to Xw (u) = 12 [Xu (u, 0) − iXv (u, 0)] we infer that for any u ∈ I2 there are real numbers λ, μ, ν such that Xw (u) = λj + μj + νe3 . Comparing this formula with (5) we obtain μ = h2 (u)u−α ,

λ = h1 (u)uα ,

ν = h3 (u)

for u ∈ I2 .

Therefore the functions hl (u) are real for u ∈ I2 . Similarly Lemma 1 implies that, for any r ∈ (0, 1), there are real numbers λ, μ, ν such that Xw (reiπ ) = λeiαπ j + μe−iαπ j + νe3 , and (5) yields Xw (reiπ ) = h1 (w)rα eiαπ j + h2 (w)r−α e−iαπ j + h3 (w)e3 for w = reiπ ∈ I1 , and so h1 (w) = λr−α ,

h2 (w) = μrα ,

h3 (w) = ν

for w = reiπ ∈ I1 .

Proposition 2. Any X ∈ T(Γ, S) can be written in the form (9)

Xw (w) = h1 (w)wα j + h2 (w)w−α j + h3 (w)e3

for w ∈ B ∪ I

where h3 (w) is holomorphic on the disk D = {w ∈ C : |w| < 1}, whereas h1 (w) and h2 (w) are holomorphic on D := D\{0}. Moreover we have hl (w) = hl (w) for w ∈ D as well as:

78


(i) If 0 < α ≤ 1 then h2 (w) is holomorphic in D while h1 (w) has at most a pole of first order at w = 0. (ii) If −1 < α < 0 then h1 (w) is holomorphic in D while w = 0 is at most a pole of first order for h2 (w). (iii) If α = 0 then both h1 (w) and h2 (w) are holomorphic in D. (iv) If α = 1 then h2 (0) = 0. Proof. The representation formula (9) is just formula (5) of Lemma 1, and Lemma 4 yields hl (w) ∈ R for w ∈ I . By Schwarz’s reflection principle we can extend h1 , h2 and h3 to holomorphic functions on D satisfying hl (w) = hl (w) for l = 1, 2, 3. Since X ∈ H21 (B, R3 ) we have Xw ∈ L2 (B, C3 ), (w), h3 (w) and so formula (6) implies that the functions wα h1 (w), w−α h2 ∞ ∞ n n (D, C). Let h (w) = a w , h (w) = are of class L 1 2 −∞ n −∞ bn w , ∞ 2 n h3 (w) = −∞ cn w be the Laurent series of h1 , h2 , and h3 respectively. Then 1 1 ∞ ∞

|an |2 r2n+2α+1 dr < ∞, |bn |2 r2n−2α+1 dr < ∞, 0

n=−∞ ∞

n=−∞

|cn |

0

n=−∞

1

r2n+1 dr < ∞.

2 0

The last relation yields cn = 0 for n ≤ −1, and so h3 is holomorphic in D. (This follows also from Proposition 1.) If 0 ≤ α ≤ 1 we infer that an = 0 for n ≤ −2 and bn = 0 for n ≤ −1. On the other hand, −1 < α ≤ 0 implies that an = 0 for n ≤ −1 and bn = 0 for n ≤ −2. Thus we obtain (i)–(iii), and (iv) 1 means b0 = 0 which follows from 0 r2n−2α+1 dr = ∞ for n = 0 and α = 1. An immediate consequence of the above result is Proposition 3. If X ∈ T(Γ, S) then there is a real number ν > −1 and a vector a ∈ C3 \ {0} with a · a = 0 such that (10)

Xw (w) = awν + o(|w|ν )

as w → 0.

Proof. Formula (10) is a direct consequence of Proposition 2. Moreover, X ∈ T(Γ, S) implies that X(w) ≡ 0 whence the leading coefficient a in (10) can be chosen different from zero. Finally Xw · Xw = 0 yields a · a = 0. Now we shall investigate the asymptotic behaviour of the Gauss map N : B → R3 of a surface X ∈ T(Γ, S) with f (0) = 0 at the singular point w = 0. We have N = W−1 Xu ∧ Xv if W = |Xu ∧ Xv | = 0, and Xw ∧ X w =

i Xu ∧ Xv . 2


79

The conformality relations imply 1 1 1 |∇X|2 = |Xu |2 = W, 4 2 2

|Xw |2 = and so we have

N = −i

(11)

Xw ∧ Xw |Xw |2

away from the branch points and from w = 0. We also know that N (w) possesses a continuous extension to B ∪ I on account of (9) and of the asymptotic expansions at branch points, which is real analytic on B ∪ I and satisfies ΔN + N |∇N |2 = 0 and |Nu |2 = |Nv |2 ,

Nu · N v = 0

on B ∪ I . By Proposition 1 it follows that N (w) is perpendicular to e2 (0) for w ∈ I2 and to e2 (α) for w ∈ I1 . Therefore N (0) is perpendicular to span{e2 (0), e2 (α)}. If 0 < |α| < 1 it follows that N (0) ∈ span{e3 }, that is, N (0) = (0, 0, ±1). Thus we have proved: Proposition 4. The Gauss map N : B ∪ I → S 2 ⊂ R3 of X ∈ T(Γ, S) is real analytic on B ∪ I , continuous on B ∪ I, and satisfies N (0) = ±e3 provided that 0 < |α| < 1. We have N (0) = e3 if N 3 (w) ≥ 0 for w ∈ B close to zero. Proposition 5. If X ∈ T(Γ, S) then the third component N 3 of its Gauss map N = (N 1 , N 2 , N 3 ) satisfies (12)

N 3 (w) =

|wα h1 (w)| − |w−α h2 (w)| |wα h1 (w)| + |w−α h2 (w)|

where hl (w) are given by the representation formulas (9) or (5). Proof. From (6) we infer (13)

|h3 |2 = 2|h1 ||h2 |.

Then we can write formula (7) as 2 (14) |Xw (w)|2 = |wα h1 (w)| + |w−α h2 (w)| . On the other hand, formula (9) implies Xw (w) ∧ Xw (w) = i |wα h1 (w)|2 − |w−α h2 (w)|2 e3 + (. . . )j + (. . . )j, taking j ∧ j = ie3 , j ∧ e3 = −ij, j ∧ e3 = ij into account. Because of Xw = Xw it follows that (15) Xw (w) ∧ Xw (w) · e3 = i |wα h1 (w)|2 − |w−α h2 (w)|2 . By virtue of (11), (14), and (15) we obtain (12).

80


Now we use (12) to prove that the term h1 (w)wα j is the dominating term in the representation formulas (9) or (5), provided that N 3 (0) = 1, which follows from N 3 (w) ≥ 0 for w ∈ B close to 0. To this end we introduce the function (16)

ρ(w) :=

1 − N 3 (w) 1 + N 3 (w)

for |w| 1,

by means of which we can write (12) in the equivalent form |w−α h2 (w)| = ρ(w)|wα h1 (w)|.

(17) Then (13) implies (18)

|h3 (w)| =

2ρ(w)|wα h1 (w)|.

If N 3 (0) = 1 we have ρ(w) → 0 as w → 0, and so |wα h1 (w)| is the dominating term in (9). We can summarize these results as follows, taking Proposition 4 into account: Proposition 6. Let X ∈ T(Γ, S), and suppose that either α = 1 and N 3 (0) = 1, or 0 < |α| < 1 and N 3 (w) ≥ 0 for w ∈ B close to 0. Then the coefficients w−α h2 (w) and h3 (w) in the representation formula (9) are estimated by (17) and (18) where ρ(w) → 0 as w → 0. Thus we have (19) Xw (w) = wα h1 (w) j + o(1)j + o(1)e3 as w → 0. Now we are prepared to prove the main results of this section. Theorem 1. Let X be a minimal surface of class T(Γ, S), and suppose that either α = 1 and N 3 (0) = 1 or |α| < 1 and N 3 (w) ≥ 0 for w ∈ B close to 0. Then there are uniquely determined functions hl (w), l = 1, 2, 3, which are holomorphic in D = {w ∈ C : |w| < 1} and describe Xw by means of the representation (20)

Xw (w) = wα h1 (w)j + w−α h2 (w)j + h3 (w)e3

for w ∈ B ∪ I .

Proof. Formula (20) is just the representation formula (9) or (5) respectively. We know already that h3 (w) is holomorphic; therefore it remains to show that h1 (w) and h2 (w) are holomorphic in D. If α = 0 this follows from Proposition 2, (iii). Suppose now that −1 < α < 0. By Proposition 2, (ii) we know that h1 is holomorphic in D, while h2 has at most a first-order pole at w = 0. From (17) and (18) we infer that |h2 (w)| = ρ(w)/2 |w|α |h3 (w)| for 0 < |w| 1 where ρ(w) → 0 as w → 0. Then w = 0 cannot be a first-order pole of h2 (w), and so h2 is holomorphic in D.


81

If 0 < α ≤ 1, we know from Proposition 2, (i) that h2 is holomorphic in D, while h1 can have at most a pole of first order at w = 0. Thus we have the Laurent expansions h1 (w) =

a−1 + a0 + a1 w + · · · , w

h2 (w) = b0 + b1 w + · · · ,

where aν , bν ∈ R since the hl are real on I . Let X0 := X(0) = (0, 0, z0 ). If b0 = h2 (0) = 0, formula (6) implies that h1 is holomorphic. So we merely have to consider the case b0 = 0. Note that Xw is holomorphic in B ∪ I , and we have Xw (w) = O(|w|α−1 ) as w → 0 on account of (19). Therefore the line integral w

Xw dw,

F (w) :=

w ∈ B ∪ I ,

0

exists and defines a holomorphic function on B ∪ I satisfying F (0) = 0. Since F = Xw it follows that X(w) = X0 + 2 Re F (w). Suppose now that a−1 = 0. On account of Proposition 6 we then have F (w) = Xw (w) = a−1 wα−1 j + · · · where + · · · denotes terms of higher order in w. This implies F (w) = and therefore X(w) = X0 +

1 a−1 wα j + · · · α 2a−1 Re(wα j) + · · · . α

In particular we obtain √ a−1 2 α X(u) = X0 + u e1 + · · · α

for 0 ≤ u < 1

as a−1 ∈ R. Since X(I2 ) ⊂ H0 it follows that a−1 > 0; thus X(w) = X0 + c Re(wα j) + · · · with c := 2a−1 /α > 0. For −1 ≤ u < 0 we have u = reiπ , r = |u|, 0 ≤ r < 1, and so X(u) = X0 + crα Re(eiπα j) + · · · = X0 + crα Re j(α) + · · · ,

82


which implies 1 X(u) = X0 + √ c|u|α e1 (α) + · · · for u ∈ I1 2 √ with X0 = (0, 0, z0 ). Since X0 + (1/ 2)c|u|α e1 (α) ∈ Hαπ we see that X(u) lies in Hαπ up to the first order if u ∈ I1 , which contradicts the fact that X(I1 ) ∈ Hπ+απ (see Proposition 1) since e1 (α) = −e1 (1 + α) and therefore Hαπ ∩ Hπ+απ = ∅. Thus a−1 has to be zero, and so also h1 is holomorphic in D. For α = 1 the statement of Theorem 1 can be strengthened as follows: Theorem 2. If X ∈ T(Γ, S), α = 1, and N 3 (0) = 1, then there exist holomorphic functions H1 , H2 , H3 on D such that (21)

Xw (w) = wH1 (w)j + w3 H2 (w)j + w2 H3 (w)e3

on B ∪ I.

In particular, Xw (w) is holomorphic in the disk D and X has a branch point at w = 0. Proof. For α = 1 formula (20) reads as Xw (w) = wh1 (w)j + w−1 h2 (w)j + h3 (w)e3

on B ∪ I ,

where h1 , h2 , h3 are holomorphic in D. Moreover, N 3 (0) = 1 implies ρ(w) → 0 as w → 0. According to (18) we have |h3 (w)| = 2ρ(w)|wh1 (w)|, and so w = 0 is a zero of h3 of at least second order, and a zero of h23 of at least fourth order. Suppose that h1 (0) = 0. Then h23 = 2h1 h2 implies that h2 vanishes of at least fourth order at w = 0. On the other hand, (17) yields |h2 (w)| = ρ(w)|w2 h1 (w)|. Hence w = 0 is also a zero of fourth order for h2 if h1 (0) = 0. So we can write h2 and h3 as h2 (w) = w4 H2 (w), h3 (w) = w2 H3 (w) where H2 and H3 are holomorphic in D. Setting H1 := h1 we obtain (21). Remark 1. For α = 1 the polygon is degenerate; either its edge [0, p1 ] is contained in [0, p2 ], or vice versa. Monotonicity of the mapping f |I : I → Σ means now that f maps [−1, 0] and [0, 1] monotonically onto the edges [p2 , 0] and [0, p1 ] respectively. Remark 2. The degenerate case α = 1 corresponds exactly to the “half-plane case” investigated in the preceding Chapter, and the surfaces X ∈ T(Γ, S) are the surfaces of “cusp type I”.


83

Finally we want to show that the Gauss map N of a surface X ∈ M∗ (Γ, S) has a finite Dirichlet integral 1 |∇N |2 du dv 2 B for every B B ∪ I. Lemma 4. If X ∈ T(Γ, S) then its Gauss map satisfies (22)

Nv · [N − N (0)] = 0

for all u ∈ I := I \ {0}.

Proof. By the reflection principle, X can be extended as a minimal surface across I1 = (−1, 0) and I2 = (0, 1); thus N is real analytic on B ∪ I . On I2 the surface X intersects the plane Π0 perpendicularly; so Xv ∈ span{e2 }, and therefore N (u) ⊥ e2 for u ∈ I2 . Thus N |I2 describes a circular arc in Π0 whence Nu (u) ⊥ e2 for u ∈ I2 . If Nv (u) = 0 we trivially have (23) Nv (u) · N (u) − N (0) = 0. For u ∈ I2 with Nv (u) = 0 it follows that Nu (u) = 0 as we have |Nu | = |Nv |. Moreover, |N |2 = 1 implies N · Nu = 0 and N · Nv = 0; thus Π0 is spanned by N (u) and Nu (u). Since Nu · Nv = 0 it follows that Nv ⊥ Π0 whence Nv (u) ⊥ N (u) − N (0), i.e. we have proved (23) for u ∈ I2 . For u ∈ I1 , equation (23) is proved similarly by noting that on I1 the surface X intersects the plane Ππ+απ perpendicularly. Lemma 5. If X ∈ E(Γ, S) and f˜−1 (0) = [u , u ] where f = π ◦ X, f˜ = f |I and −1 < u < u < 1, then the Gauss map N of X satisfies Nv (u) · N (u) − N (u ) = 0 for u ∈ (−1, u ) ∪ (u , u ), (24) Nv (u) · N (u) − N (u ) = 0 for u ∈ (u , u ) ∪ (u , 1). Proof. By the reflection principle, N is real analytic on I \ {u , u }. Similarly as in the proof of Lemma 4 we obtain Nv · N − N (u ) = 0 on I1 and Nv · N − N (u ) = 0 on I2 . Suppose now that u ∈ I0 := (u , u ). Then Xu (u) lies in span{e3 } and therefore N ⊥ e3 for u ∈ I0 . Hence N |I0 is a circular arc in the equator plane Π, and so Nu (u) ⊥ e3 for u ∈ I0 . Since Nv ⊥ N, Nu we have Nv (u) ⊥ Π for u ∈ I0 , and therefore Nv (u) ⊥ N (u) − N (u ) , Nv (u) ⊥ N (u) − N (u ) for u ∈ I0 as N |I is continuous.

84


Theorem 3. If X ∈ T(Γ, S) then its Gauss map N satisfies |∇N |2 du dv < ∞ for all B B ∪ I. B

Proof. It suffices to prove

B

|∇N |2 du dv < ∞ for

B := w ∈ C : |w| < ρ, Im w > 0 with 0 < ρ 1. For this purpose we choose numbers r and R such that 0 < r < R < 2R < 1, and consider a Lipschitz continuous function ζ(t) on {0 ≤ t < ∞} which is zero for 0 ≤ t ≤ r/2 and t ≥ 2R, one for r ≤ t ≤ R, and linear on [r/2, r] and [R, 2R]. Denote by η(w) the cut-off function η(w) = ζ(|w|) which is uniformly Lipschitz continuous on C and whose support is contained in the disk B 2R (0). Now we multiply the equation −ΔN = N |∇N |2 by the test function φ := η 2 · (N − N0 )

(25)

with N0 := N (0)

and integrate over B. Then we obtain − ΔN · (N − N0 )η 2 du dv = η 2 (N − N0 ) · N |∇N |2 du dv. B

B

An integration by parts leads to Nv (u) · N (u) − N0 η 2 (u) du + ∇N · ∇ (N − N0 )η 2 du dv I B 2 2 = η (N − N0 ) · N |∇N | du dv. B

The boundary integral vanishes on account of Lemma 4, and ∇N · ∇ (N − N0 )η 2 du dv B = η 2 |∇N |2 du dv + 2η∇η∇N · (N − N0 ) du dv. B

B

Thus we obtain

η 2 |∇N |2 du dv = J1 + J2 B

where

J1 :=

η 2 (N − N0 ) · N |∇N |2 du dv, J2 := − 2η∇η∇N · (N − N0 ) du dv. B

B


Set

85

δ(R) := sup |N (w) − N0 | : w ∈ B ∪ I, |w| ≤ 2R .

Since N (w) is continuous on B ∪ I we have δ(R) → 0 as R → 0. Furthermore we note that 0 ≤ η(w) ≤ 1, ∇η(w) = 0 on BR (0) \ B r (0), |∇η(w)| = 2/r on Br (0)\B r/2 (0), |∇η(w)| = 1/R on B2R (0)\B R (0) and η(w) = 0 on C\B2R (0). Then we infer that η 2 |∇N |2 du dv, |J1 | ≤ δ(R) B

and Schwarz’s inequality leads to 1 2 2 2 η |∇N | du dv + δ (R) |∇η|2 du dv |J2 | ≤

B B where is an arbitrary positive number and |∇η|2 du dv = 4r−2 π(r2 − r2 /4) + R−2 · π(4R2 − R2 ) = 6π; B

hence

η 2 |∇N |2 du dv + 6π −1 δ 2 (R).

|J2 | ≤ B

Then we obtain

1 − − δ(R)

η 2 |∇N |2 du dv ≤ 6π −1 δ 2 (R).

B

Now we set = 1/4 and choose R ∈ (0, 1) so small that δ(R) ≤ 1/4. Then 1 − − δ(R) ≥ 1/2, and we arrive at η 2 |∇N |2 du dv ≤ 3π B

whence

|∇N |2 du dv ≤ 3π B∩[BR (0)\Br (0)]

if 0 < r < R 1. By letting r → +0 it follows that |∇N |2 du dv ≤ 3π if 0 < R 1. B∩BR (0)

Theorem 4. If X ∈ E(Γ, S) then its Gauss map N satisfies |∇N |2 du dv < ∞ for all B B ∪ I. B

86


Proof. It suffices to prove |∇N |2 du dv < ∞

for u = u , u and 0 < R 1.

B∩BR (u)

This is proved in a similar way as Theorem 3 if we replace the test function (25) by φ = η 2 (N − N1 ) or φ = η 2 (N − N2 ) respectively where N1 := N (u ), N2 := N (u ) and η(w) := ζ(|w − u |) or ζ(|w − u |) respectively, taking Lemma 5 into account.

2.8 Edge Creeping In this section we first show that edge creeping must occur for a surface X ∈ M(Γ, S) if S is the boundary of a wedge Wα with an angle β = π+απ ∈ (π, 2π) such that Γ lies on the concave side of S and has a monotonic height function. Then we describe an explicit example of an embedded surface X ∈ M(Γ ∗ , S ∗ ) in a boundary configuration Γ ∗ , S ∗ where X creeps along the edge L∗ of S ∗ . 1. A sufficient condition for edge creeping. Here we use the notation Wα , L, Hβ introduced in Section 2.7. So Wα is a wedge in R3 with the boundary S = H0 ∪ L ∪ Hπ+απ , and Γ is an arc which lies on the side of S with the angle β := (1 + α)π and whose end points P1 and P2 lie on H0 and Hβ respectively. As in Section 2.7 we decompose the class of surfaces M∗ (Γ, S) normalized by the condition f (0) = 0 (with f := π ◦ X) into the two disjoint classes T(Γ, S) and E(Γ, S) where the latter one comprises the surfaces with edge creeping along L. Then we obtain Theorem 1. Suppose that Γ = {(ξ 1 (t), ξ 2 (t), ξ 3 (t)) : 0 ≤ t ≤ 1} faces the concave side of Wα , i.e. 0 < α ≤ 1, and possesses a nonconstant, monotonically increasing height function ξ 3 : [0, 1] → R. Then X is of class E(Γ, S). Proof. Suppose that X ∈ T(Γ, S). Then X is oriented in such a way that N 3 (0) = (0, 0, 1), and by Section 2.7, Theorems 1 and 2, we have the representation formula Xw (w) = wα h1 (w)j + w−α h2 (w)j + h3 (w)e3 , where h1 , h2 , h3 are holomorphic in D = {w ∈ C : |w| < 1}. Moreover, by Section 2.7, (16) and (18), we also have |h3 (w)| ≤ 2ρ(w)|wα h1 (w)| for w ∈ D with |w| 1 and ρ(w) → 0 as w → 0. We infer h3 (0) = 0 and obtain the Taylor expansion 3 (w) = cw + · · · Xw

whence

for w ∈ D,

2.8 Edge Creeping

X 3 (w) = a0 + Re{c2 w2 + c3 w3 + · · · }

87

for w ∈ D,

where a0 ∈ R and c2 , c3 , . . . ∈ C. By virtue of Section 2.1, Proposition 2, we also have X 3 (w) = X 3 (w) for w ∈ D and Xv3 (u, 0) = 0 for u ∈ I. Let P1 = (p1 , a1 ) = ξ(0), P2 = (p2 , a2 ) = ξ(1), and suppose for instance that ξ 3 is increasing; then a1 < a2 . By the maximum principle, X 3 assumes its maximum and minimum on D at the boundary, and so we obtain a1 < X 3 (w) < a2 for all w ∈ D and in particular a1 < a0 = X 3 (0) < a2 . By a standard reasoning the set A0 := {w ∈ D : X 3 (w) = a0 } has to meet ∂D in at least four points, and because of the mirror symmetry of X 3 , the set A0 ∩ C contains as many points as its mirror image. Consequently A0 ∩ C contains at least two points, and ±1 ∈ / A0 . Thus there exist two numbers θ1 and θ2 with 0 < θ1 < θ2 < π such that X 3 (eiθ1 ) = a0 = X 3 (eiθ2 ). Since X(eiθ ) is weakly monotonic, it follows that X 3 (eiθ ) ≡ a0 for all θ ∈ [θ1 , θ2 ]. Then there is a connected, open subset Ω0 of B such that {eiθ : θ1 ≤ θ ≤ θ2 } ⊂ Ω 0 and X 3 (w) = a0 for all w ∈ ∂Ω0 . The maximum principle implies X 3 (w) ≡ a0 on Ω 0 whence X 3 (w) ≡ a0 on D. This is a contradiction to X 3 (w) ≡ const. Therefore X cannot be of class T(Γ, S), and thus we obtain X ∈ E(Γ, S). Remark 1. In the next section we shall generalize Theorem 1 to the situation where Σ is a polygon with several corners. Remark 2. The result of Theorem 1 is optimal in the sense that X ∈ M∗ (Γ, S) can be of class T(Γ, S) if the height function ξ 3 of Γ has at least one maximum or minimum between P1 and P2 . To see this, we choose a wedge W and an arc Γ with its endpoints on different faces of W such that Γ faces the convex side of ∂W with an angle β at the edge L of W , satisfying π/2 < β < π. As we shall see in Section 2.9 there is an area minimizing surface X in Γ , S contained in W , and this surface is of type T(Γ , S ) since the wedge W is convex with respect to Γ . Reflecting Γ , S and X at one of the faces of S we get a congruent configuration Γ , S and a congruent surface X , sitting in the wedge W , with X ∈ T(Γ , S ). Then we can ˜ which is either put X and X piecewise together to a minimal surface X defined on the slit domain D \ I1 , or on D \ I2 . Choosing a suitable conformal mapping τ of B onto this slit domain which maps I1 and I2 onto either one ˜ ◦ τ a minimal surface of of the two edges of this slit, we obtain by X := X class T(Γ, S) where S is the boundary ∂W of the wedge W := int(W ∪ W ) and Γ is the arc Γ ∪ Γ , which faces the concave side of W with the opening angle 2β ∈ (π, 2π). If we assume that the height function of Γ is nonconstant and strictly monotonic, then the height function of Γ possesses exactly one extremum between its endpoints.

88


2. An example for edge creeping. Now we shall construct an explicit example showing the phenomenon of edge creeping. In the sequel, Γ and S will not have the meaning fixed in Section 2.1; instead Γ ∗ and S ∗ to be defined later on will take their place. Let E and E0 be two planes in R3 intersecting orthogonally in a straight line L∗ (see Fig. 1). This line divides E into two closed halfplanes S and S . Furthermore E decomposes R3 into two closed halfspaces H and H . We

Fig. 1.

consider a rectifiable Jordan arc Γ contained in H which meets E only in its endpoints P0 and P1 , and we assume that P0 lies on the boundary line L∗ of the halfplane S whereas P1 is contained in S \ L∗ . Then the mirror image Γ of Γ at the plane E lies in the halfspace H , and Γ0 := Γ ∪ Γ is a closed rectifiable Jordan curve. We assume that the orthogonal projection of R3 onto the plane E0 maps Γ0 one-to-one onto a closed convex Jordan curve γ0 in E0 . By Radó’s theorem, Γ0 bounds exactly one disk-type minimal surface X0 : D → R3 defined on D := {w ∈ C : |w| < 1} which maps ∂D homeomorphically onto Γ0 and 1, i, −1 onto P1 , P, P0 . Here P denotes a preassigned point on the arc Γ between its endpoints P1 and P0 . Let B be a semidisk as in Section 2.1 with the boundary ∂B = C ∪ I consisting of the circular arc C and the open interval I in R. There is a ˜ of Dirichlet’s integral in C(Γ, E) which is of class M(Γ, E); in minimizer X particular, ˜ = 0, ΔX

˜ v |2 , ˜ u |2 = |X |X

˜u · X ˜ v = 0 in B. X

˜ is real analytic on B ∪ I and meets E perpendicularly. By the maxThen X ˜ has no branch points on I. ˜ ⊂ H, and therefore X imum principle X(B)

2.8 Edge Creeping

89

˜ can be extended to a minimal surface By Schwarz’s reflection principle, X 3 ˆ o’s theorem X : D → R which maps ∂D homeomorphically onto Γ0 . By Rad´ ˜ by X(i) ˜ ˆ : D → R3 is an ˆ = X0 if we normalize X = P , and so X we have X embedded surface in R3 that lies as a graph above Ω 0 where Ω0 is the bounded ˜ maps I homeomorphically domain in E0 with ∂Ω0 = γ0 . It follows that X| I ˜ ˜ ˜ ˜ on E. onto clos Σ, the closure of the free trace Σ := X(I) of X Let Min(Γ, S) be the set of minimizers of Dirichlet’s integral in C(Γ, S). By Sections 2.5 and 2.7 of Vol. 2, any Y ∈ Min(Γ, S) is a minimal surface of class C 1,1/2 (B ∪ I, R3 ) such that Y ∈ C 0,α (B, R3 ) for some α ∈ [0, 1) if Γ, S satisfies a chord-arc condition, what we tacitly shall assume. Moreover Yv (w) is perpendicular to E for any w ∈ Y −1 (S \ L∗ ). We say that Y ∈ Min1 (Γ, S) if there is at least one interval [u1 , u2 ] in I with u1 < u2 such that Y (u) ∈ L∗ for all u ∈ [u1 , u2 ]; otherwise Y is said to be of class Min2 (Γ, S). For Y ∈ Min2 (Γ, S) it follows that Yv (w) is perpendicular to E for any w ∈ I, and therefore Y ∈ M(Γ, E). By our preceding remark, Y is conformally equivalent to X0 |B where X0 is the uniquely determined minimal surface bounded by Γ0 . We set (1) (2)

a(Γ, S) := inf{A(Z) : Z ∈ C(Γ, S)} = inf{D(Z) : Z ∈ C(Γ, S)}, a(Γ, E) := inf{A(Z) : Z ∈ C(Γ, E)} = inf{D(Z) : Z ∈ C(Γ, E)}.

Since C(Γ, S) ⊂ C(Γ, E), it follows that (3)

a(Γ, E) ≤ a(Γ, S).

The uniqueness result above yields that any Y ∈ Min2 (Γ, S) has to be area minimizing in C(Γ, E), i.e. a(Γ, S) = A(Y ) = a(Γ, E). Hence we obtain the following result: Lemma 1. If a(Γ, E) < a(Γ, S) then Min2 (Γ, S) is empty, and so for every Y ∈ Min(Γ, S) there exists at least one interval [u1 , u2 ] in I with u1 < u2 such that Y (w) ∈ L∗ for all w ∈ [u1 , u2 ]. In order to achieve a(Γ, E) < a(Γ, S) we assume that Γ consists of a straight arc L with the endpoints P0 , P and a smooth arc Γ1 with the endpoints P, P1 such that L meets L∗ perpendicularly at P0 (see Fig. 3). As depicted in Fig. 2, let l and γ1 be the orthogonal projections of L and Γ1 into the plane E and let P ∈ E be the image of P under this projection. We choose Γ such that the following holds: (i) γ := γ1 ∪ l is a rectifiable Jordan arc with the endpoints P1 and P0 ; (ii) Γ lies as a graph above the curve γ ⊂ E; (iii) the orthogonal projection P of P into E lies in the interior of the halfplane S ; (iv) γ1 meets L∗ in exactly one point Q∗ .

90


Let γ˜ be the subarc of γ which is bounded by Q∗ and P0 , and denote by c the interval on L∗ between P0 and Q∗ . Then the Jordan curve c ∪ γ˜ bounds a bounded domain Ω in E. We can assume that γ is convex with respect to Ω and meets L∗ perpendicularly in Q∗ (and, of course, in P0 ). Now we consider the collar surface K consisting of the straight segments perpendicular to E between γ and Γ . If the inclination of Γ against E is sufficiently small, the area of the collar becomes as small as we please. Thus we can achieve that (4)

area Ω > area K.

Let Z be an arbitrary minimizer of D (and therefore also of A) in C(Γ, S). Since Z is continuous on B, we conclude that Ω is contained in the orthogonal projection of Z(B) into E. This implies A(Z) ≥ area Ω, and consequently (5)

a(Γ, S) ≥ area Ω.

Moreover we can find a parametrization Z˜ : B → R3 of K by some surface ˜ whence Z˜ ∈ C(Γ, E) such that area K = A(Z) (6)

area K ≥ a(Γ, E).

Then it follows from (4)–(6) that (7)

a(Γ, S) > a(Γ, E).

Summarizing the above results and applying Lemma 1 we obtain

Fig. 2. The domain Ω

Lemma 2. Let us choose Γ as described above, so that in particular (4) holds. Then there is an area–minimizing minimal surface Y of class C(Γ, S) such that Y ([u1 , u2 ]) ⊂ L∗ for some pair of points u1 , u2 ∈ I with u1 < u2 .

2.8 Edge Creeping

91

Fig. 3. The boundary configuration Γ ∗ , S ∗ where S ∗ = S1 ∪ S2

In the following discussion we consider a minimal surface Y ∈ C(Γ, S) as in Lemma 2. Let Q be the point on Γ that is projected onto Q∗ ∈ γ by the orthogonal projection of R3 onto E. Then Q decomposes Γ into two subarcs Γ + and Γ − such that the interior of Γ + lies above E0 , whereas the interior of Γ − lies below E0 (see Fig. 1). We can assume that Y (1) = P1 ,

Y (i) = P,

Y (−1) = P0 .

Next we introduce the open set I + := {w ∈ I : Y (w) ∈ S \ L∗ } and its complement I0 := I \ I + = {w ∈ I : Y (w) ∈ L∗ }. We have the free boundary condition (8)

Yv (w) ⊥ E

for any w ∈ I + ∩ I.

Choose Cartesian coordinates such that E0 is the x, y-plane, E the y, z-plane, L∗ the y-axis, H = {x ≥ 0}, and denote by e1 = (1, 0, 0) the normal to E that points into the halfspace H. Then Y has the representation (9)

Y (w) = (x(w), y(w), z(w)),

w ∈ B,

and we obtain (10)

I0 = {w ∈ I : z(w) = 0}.

92


The condition Yv (w) ⊥ E means that Yv (w) = λ(w)e1 for some scalar λ(w) ∈ R. The maximum principle implies Y (B) ⊂ H; hence Y has no branch points on I, and we have Yv (w) = 0 on I. We then infer that λ(w) > 0 for all w ∈ I + ∩ I as well as λ ∈ C 0 on I + ∩ I since Y ∈ C 1,1/2 (B ∪ I, R3 ). Since Y (w) ∈ E for w ∈ I and Y (w) ∈ L∗ for w ∈ I0 , it follows that x(w) = 0 on I and z(w) = 0 on I0 whence xu (w) = 0 on I and zu (w) = 0 on I0 . Then |Yu |2 = |Yv |2 implies yu2 + zu2 = λ2 on I + ∩ I and yu2 = |Yv |2 on I0 . Thus we have proved: Lemma 3. Any Y (w) = (x(w), y(w), z(w)) as in Lemma 2 satisfies on I + ∩ I;

(11)

xv (w) = λ(w) > 0,

(12)

yu (w)2 + zu (w)2 = λ2 (w) > 0

(13)

x(w) = 0,

xu (w) = 0

on I;

(14)

z(w) = 0,

zu (w) = 0,

yu (w)2 = |Yv (w)|2 > 0

yv (w) = zv (w) = 0 on I + ∩ I;

on I0 .

Now we claim: Lemma 4. There is a number u0 ∈ I such that I0 = (−1, u0 ] and I + = (u0 , 1). + − − Proof. Denote by G+ 1 , G2 , . . . and G1 , G2 , . . . the (nonvoid) connected components of the set B ∗ := {w ∈ B : z(w) = 0} where z(w) is positive or − ∗ negative respectively. We call G+ j the positive components of B and Gj the negative components. Note that Y |C furnishes a homeomorphism of C onto Γ . Let w0 = eiθ0 with 0 < θ0 < π be the pre-image of Q ∈ Γ under this mapping; then w0 decomposes C into two subarcs C − and C + with the endpoints −1, w0 and w0 , 1 respectively, and

Γ − = Y (C − ),

Γ + = Y (C + ).

It follows that z(eiθ ) > 0 for 0 ≤ θ < θ0 , z(eiθ ) < 0 for θ0 < θ < π and z(w0 ) = z(−1) = 0. By numbering the components of B ∗ appropriately we can assume that − − C + ⊂ ∂G+ 1 and C ⊂ ∂G1 . Then we have (15)

G− k ⊂ B ∪ I 0 ∪ {w0 }

(16)

G+ j ⊂ B ∪ [−1, 1 − ε) ∪ {w0 }

for k ≥ 2, for j ≥ 2 and some ε ∈ (0, 1),

since z(w) > 0 on I + and for w ∈ B with |w − 1| 1. We infer from (15) that the harmonic function z(w) vanishes on ∂G− k for , and therefore z(w) ≡ 0 on B, a contradiction. k ≥ 2 whence z(w) ≡ 0 on G− k ∗ Thus G− 1 is the only negative component of B . If j ≥ 2, relation (16) implies that z(w) = 0 for w ∈ ∂G+ j \ I. A similar + + reasoning as before shows that ∂Gj ∩ I is nonempty (otherwise, z(w) ≡ 0

2.8 Edge Creeping

93

on G+ j ). Then hj := z|G+ assumes its positive maximum at some point j

+ wj ∈ G+ j ∩ I , and so z(w) has a local maximum at wj ∈ I. Since z(w) is nonconstant, E. Hopf’s lemma implies zv (wj ) < 0 which contradicts (11). ∗ Consequently G+ 1 is the only positive component of B . + − + − ∗ Setting G := G1 and G := G1 , we have B = G+ ∪ G− . Hence there is a number u0 with −1 ≤ u0 ≤ 1 − ε such that

∂G− ∩ I = [−1, u0 ],

(17)

∂G+ ∩ I = [u0 , 1].

Consider now the subset I0+ of R defined by I0+ := I0 ∩ ∂G+ .

(18)

We claim that I0+ has no interior points. Otherwise there would exist an open interval (u , u ) contained in I0+ with u < u . Then it would follow that zv (w) ≥ 0 for w ∈ (u , u ). On the other hand, we will show that the minimum property of Y implies zv (w) ≤ 0

(19)

on I.

Thus we would obtain zv (w) = 0 on (u , u ) ⊂ I0 . By (10) we have z(w) = 0 on I0 , and so we are in the situation to apply E. Hopf’s lemma to any point w ∈ (u , u ), whence zv (w) < 0 for w ∈ (u , u ), a contradiction. So the interior of I0+ is empty. It remains to prove (19). To this end we consider admissible vector fields Φ in the sense that Φ ∈ Cc∞ (B ∪ I, R3 ) and Y + εΦ ∈ C(Γ, S) for all ε ∈ [0, ε0 ) with some ε0 = ε0 (Φ) > 0. The minimum property of Y yields D(Y + εΦ) ≥ D(Y )

for all ε ∈ [0, ε0 ),

whence

(Yu · Φu + Yv · Φv ) du dv ≥ 0. B

Green’s formula yields (Yu · Φu + Yv · Φv ) du dv = − ΔY · Φ du dv + B

B

I

∂ Y · Φ du ∂n

where n is the exterior normal to ∂B. Since ΔY = 0 and ∂Y /∂n = −Yv on I, it follows that Yv · Φ du ≤ 0. (20) I

Suppose that ϕ ∈ Cc∞ (B ∪ I) satisfies ϕ ≥ 0 on I. Then Φ := (0, 0, ϕ) is an admissible vector field in the sense described above, and so (20) implies

94


zv ϕ du ≤ 0. I

On account of the fundamental lemma of the calculus of variations this leads to (19). Therefore the set I0+ defined by (18) has no interior points in R, and so every point of I0+ is a limit of points from I + . Since zv = 0 on I + , it follows that zv (w) = 0 on I0+ = I0 ∩ ∂G+ .

(21)

Recall now that ∂G+ ∩ I = [u0 , 1] where −1 ≤ u0 ≤ 1 − ε and 0 < ε < 1, + and note that z(w) ≥ 0 on G . Thus the harmonic function z|G+ assumes its + minimum on I0 , and z = 0 = z(u0 ). min + G

Suppose that = {u0 }. Then there is a point w∗ ∈ (u0 , 1) such that z(w∗ ) = 0, and by E. Hopf’s lemma it follows that zv (w∗ ) > 0. Since w∗ ∈ I0+ , this is a contradiction to (21). Thus we have proved that z(w) > 0 on (u0 , 1]. Since there is an interval [u1 , u2 ] with −1 < u1 < u2 < 1 such that z(w) = 0 on [u1 , u2 ], it follows that [u1 , u2 ] ⊂ (−1, u0 ), and therefore u0 > 0. Moreover, ∂G− ∩ I ⊂ I0 , and consequently z(w) = 0 on [−1, u0 ]. This completes the proof of the lemma. I0+

Because of (14) we have |yu (w)| > 0 for all w ∈ (−1, u0 ]. Hence Y furnishes a homeomorphism of I 0 = [−1, u0 ] onto its image Y (I 0 ), and therefore Y maps I 0 ∪ C homeomorphically onto the piecewise smooth Jordan arc Y (I 0 ) ∪ Γ . Furthermore the minimum property of Y implies that Y is freely stable (with [u0 , 1] as free boundary). Furthermore it is not difficult to verify that the Gauss mapping N of Y has a finite Dirichlet integral. Then the reasoning of Section 2.6 implies: Lemma 5. Any Y : B → R3 as in Lemma 2 can be represented as a graph over some domain in E0 , and in particular Y |I yields a homeomorphism of I onto the trace Y (I) in S. Let L be the straight line containing the segment L, and denote by T the reflection of R3 in L. We set S1 := S, S2 := T S1 , Γ2 := T Γ1 , P2 := T P1 , S ∗ := S1 ∪ S2 , Γ ∗ := Γ1 ∪ Γ2 . Then S ∗ is the boundary of a wedge W ∗ with the edge L∗ and a small inclination angle θ between the two faces S1 and S2 of S ∗ , and certainly θ < π. Moreover, Γ ∗ = Γ1 ∪ Γ2 is a smooth Jordan arc facing the concave side of S ∗ , and P1 , P2 ∈ S ∗ are the endpoints of Γ ∗ . Clearly Γ ∗ meets S ∗ only in P1 and P2 . Now we want to reflect Y ∈ M(Γ, S) across the line L. In order to obtain a surface parametrized on the semidisk B we reparametrize Y over the quarterdisk B0 := {w = u + iv : |w| < 1, u > 0, v > 0}.

2.8 Edge Creeping

95

For this purpose we consider the conformal mapping τ : B0 → B of B0 onto B whose continuous extension to B 0 maps 0, 1, i onto −1, 1, i respectively, and define the reparametrization Y0 : B 0 → R3 of Y by Y0 := Y ◦ τ . The surface Y0 maps 0, 1, i onto P0 , P1 , P and the interval J0 := {ti : 0 ≤ t ≤ 1} onto the segment L in L. Then we extend Y0 by reflection to a minimal surface X : B → R3 : for u + iv ∈ B 0 , Y0 (u + iv) X(u + iv) := T Y0 (−u + iv) for − u + iv ∈ B 0 , and we define u∗ > 0 by u∗ := τ −1 (u0 ). By construction, X is continuous and satisfies X(I) ⊂ S ∗ and X(1) = P1 , X(−1) = P2 , X(0) = P0 , X(i) = P . Moreover, X maps C and I homeomorphically onto Γ ∗ and Σ ∗ respectively, where Σ ∗ := X(I), and the interval I ∗ := [−u∗ , u∗ ] is mapped into L, whereas X(I \ I ∗ ) ⊂ S ∗ \ L. It follows from the construction that X is a stationary minimal surface in the configuration Γ ∗ , S ∗ , i.e. X ∈ M(Γ ∗ , S ∗ ). Since f (w) := (x(w), y(w)) maps C homeomorphically onto the orthogonal projection Γ ∗ of Γ ∗ into the plane E0 , we have X ∈ M∗ (Γ ∗ , S ∗ ), and f (0) = (0, 0). Therefore X is of class E(Γ ∗ , S ∗ ), and we have proved: Theorem 2. Let Γ ∗ , S ∗ be a boundary configuration as described above, satisfying (7). Then there exists an embedded minimal surface X ∈ M∗ (Γ ∗ , S ∗ ) which creeps on the edge L of S ∗ ; in fact, X ∈ E(Γ ∗ , S ∗ ).

Fig. 4. Side-view of the configuration Γ, S ∗ from a plane perpendicular to L∗

Remark 3. Note that Γ, S can be chosen in such a way that Γ ∗ , S ∗ satisfies Condition (B) of Section 2.2; see Fig. 4. Then it turns out that the surface X constructed in Theorem 2 is unique in M∗ (Γ ∗ , S ∗ ) and minimizes both A and D in C(Γ ∗ , S ∗ ); see Section 2.9. Remark 4. Actually the above construction of X can always be carried out, independently of whether (7) is satisfied or not. The previous reasoning yields that X has to be of class M∗ (Γ ∗ , S ∗ ), and by Theorem 1 it follows that in any case X is edge creeping, i.e. X ∈ E(Γ ∗ , S ∗ ). The proof above has the advantage that it does not use the asymptotic expansion of Section 2.8 and that it makes the edge creeping geometrically evident, but it yields a seemingly weaker result. In fact, we may conclude that (7) is always satisfied if the angle ψ between L and S (i.e. between L and the z-axis) is larger than π/2; otherwise we would have X ∈ T(Γ ∗ , S ∗ ) which is impossible according

96


Fig. 5. The edge-creeping minimal surface X in the boundary configuration Γ, S ∗

to Theorem 1. However, if 0 < ψ ≤ π/2, then Γ ∗ faces the convex side of S ∗ , and by Section 2.9 we obtain X ∈ T(Γ ∗ , S ∗ ); thus (7) is no longer satisfied.

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces Now we return to the notations and assumptions introduced in Section 2.1. The aim of the present section is to generalize the existence and uniqueness results to boundary configurations Γ, S whose supporting surface S = Σ0 × R has only a piecewise smooth directrix Σ0 ; precisely speaking, the subarc Σ of Σ0 between the points p1 and p2 is a polygon with the vertices q1 , . . . , ql 3 while Σ1 , Σ2 and the arcs p 1 q1 , q 1 q2 , . . . , q l−1 ql , q l p2 are C -smooth. The basic properties of surfaces in M(Γ, S) and M∗ (Γ, S) are listed in Section 2.1, Propositions 2–4. In the sequel we rely on the results of Sections 2.1–2.6. The basic strategy (n) will be to approximate Σ0 in a suitable manner by smooth directrices Σ0 (n) and then to check which properties of the minimizers Xn in C(Γ, S ) with (n) S (n) := Σ0 × R can be carried over to minimizers in C(Γ, S). Let us recall the notation f := π ◦ X, f˜ := f |I , and I ∗ = I ∗ (X) := I \ f˜−1 (Q) where Q := {q1 , q2 , . . . , ql }. Let (1)

I = I (X) := I \ {branch points of X on I}, I = I (X) := I ∗ \ {branch points of X on I ∗ }.

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces

97

Then Proposition 4 of Section 2.4 can be generalized by the same reasoning in the following way: Proposition 1. The component N 3 = N ·e3 of the Gauss map N of a surface X ∈ M∗ (Γ, S) satisfies Nv3 = ρN 3

(2)

on I

with ρ := κ(X)[N · τ (X)]2 (Xv · n(X)).

(3)

We now introduce two subclasses of surfaces X whose Gauss maps point into the upper hemisphere: (4)

M+ (Γ, S) := {X ∈ M(Γ, S) : N 3 ≥ 0 in B}, 3 M+ ∗ (Γ, S) := {X ∈ M∗ (Γ, S) : N ≥ 0 in B}.

3 Proposition 2. If X ∈ M+ (Γ, S) or M+ ∗ (Γ, S) respectively, then N (w) > 0 3 in B and N (u, 0) > 0 for u ∈ I or I respectively.

Proof. From ΔN 3 + |∇N |2 N 3 = 0 and N 3 ≥ 0 we obtain ΔN 3 ≤ 0 in B. If N 3 (w0 ) = 0 for some w0 ∈ B, the maximum principle implies N 3 (w) ≡ 0, i.e. X would be a planar surface, a contradiction. If N 3 (w0 ) = 0 for some w0 ∈ I or I , E. Hopf’s lemma yields Nv3 (w0 ) > 0, while (2) implies Nv3 (w0 ) = 0, a contradiction. In Section 2.6, Proposition 2, we have proved that N 3 (w) > 0 on B if N is the Gauss map of a freely stable surface X ∈ M(Γ, S) where S ∈ C 3 and Condition (B) or (M1 ) is fulfilled. Now we shall strengthen this estimate on subintervals of I which are mapped by X onto subarcs of Σ that are convex with respect to the domain G bounded by Γ and Σ. Proposition 3. Let Σ0 be of class C 3 , and suppose that Condition (B) or (M1 ) is satisfied. Moreover, let X be a minimizer of D (and therefore of A) in C(Γ, S) with the following property: There are numbers m > 0, w0 ∈ I, and r ∈ (0, 1 − |w0 |) such that (5)

κ(X(w)) ≥ 0

for w ∈ Ir (w0 )

and

N 3 (w) ≥ 2m

for w ∈ Cr (w0 ).

Then it follows that (6)

N 3 (w) ≥ m

for all w ∈ S r (w0 ).

Here we have used the following notations: Sr (w0 ) := {w ∈ C : |w − w0 | < r, Im w > 0}, Ir (w0 ) := {w ∈ C : |w − w0 | < r, Im w = 0}, Cr (w0 ) := {w ∈ C : |w − w0 | = r, Im w ≥ 0}.

98


Proof. We define a function η : B → R by 3 N (w) − m for w ∈ S r (w0 ) with N 3 (w) < m, η(w) := 0 otherwise, and we set −

B − := {w ∈ Sr (w0 ) : N 3 (w) < m},

B + := B \ B .

It follows that η ∈ C 0 (B) ∩ H21 (B), η ≤ 0, and supp η ⊂ Sr (w0 ) ∪ Ir (w0 ). Let us first assume that X has no branch points on I; then η(w) = N 3 (w) − m,

ηv (w) = Nv3 (w)

on ∂ B − := ∂B − ∩ Ir (w0 ),

and furthermore η(w) = N 3 (w) − m, ∇η(w) = ∇N 3 (w), η(w) = 0, ∇η(w) = 0 a.e. on B \ B − .

Δη(w) = ΔN 3 (w)

on B − ,

We obtain (|∇η|2 − |∇N |2 η 2 ) du dv

2

δ A(X, η) =

B

= B−

|∇η|2 − |∇N |2 η 2 du dv,

and an integration by parts yields η · (Δη + |∇N |2 η) du dv − δ 2 A(X, η) = − B−

∂ B−

ηηv du.

Since ΔN 3 + N 3 |∇N |2 = 0 on B, it follows that Δη + |∇N |2 η = −|∇N |2 m

a.e. on B,

and therefore δ 2 A(X, η) = B−

ηm|∇N |2 du dv −

∂ B−

ηNv3 du.

Recall that the free second variation of X is given by δ 2 A(X, η) = δ 2 A(X, η) + ρ(X)η 2 du. I


It follows that δ 2 A(X, η)

ηm|∇N | du dv + 2

= B−

99

∂ B−

η · [−Nv3 + ρ(X)(N 3 − m)] du,

and (2) implies (7)

δ 2 A(X, η)

ηm|∇N | du dv − 2

= B−

ηmρ(X) du. ∂ B−

Since κ ≥ 0 it follows that −ρ ≥ 0 on Ir (w0 ), and because of η(w) < 0 on B − we obtain (8)

ηm|∇N |2 ≤ 0 on B − ,

−ηmρ(X) ≤ 0 on ∂ B − .

Thus we have δ 2 A(X, η) ≤ 0. On the other hand, X is freely stable since it is a minimizer of A, and so we also obtain δ 2 A(X, η) ≥ 0 since η ∈ H 1,2 (B) ∩ C 0 (B). Hence we have δ 2 A(X, η) = 0. Because of (7) and (8) this equation implies that one of the following two cases has to occur: (i) B − is void; (ii) |∇N | = 0 on B − . We claim that (i) must hold. Otherwise B − would be nonvoid and (ii) were true. Since N (w) is real analytic on B, we would have ∇N (w) ≡ 0 on B and therefore N (w) ≡ const on B. Thus N 3 (w) ≥ 2m in B on account of assumption (5) whence B − were void, a contradiction. So we are in case (i), that is, N 3 (w) ≥ m for all w ∈ S r (w0 ). Now we can formulate the first main result of this section, an existence result. Theorem 1. Let Σ be piecewise smooth, and suppose that Condition (B) or (M1 ) (or (M2 ) respectively) are satisfied. Then we have: (i) There is a minimizer of D and A in C(Γ, S) which is of class M+ ∗ (Γ, S), and so it has the regularity properties stated in Section 2.1, Proposition 4. In addition X is H¨ older continuous on B. (ii) The mapping f = π ◦ X with X from (i) maps B onto G. Moreover, ˚ ∪ I ∗ (X) onto G \ ({p1 , p2 } ∪ Q) and a f is a diffeomorphism of B ∪ C −1 ˜ homeomorphism of B \ f (Q ) onto G \ Q , where Q is the subset of E-type vertices in Q.

100


(iii) There is an equivalent nonparametric representation Z(x1 , x2 ) = (x1 , x2 , z(x1 , x2 )) of X on G, where z is a real analytic solution of div(∇z/ 1 + |∇z|2 ) = 0 in G, which is of class C 0 (G \ Q ) ∩ C 2 (G \ [Q ∪ {p1 , p2 }]) and satisfies z=γ

on Γ ,

∂z =0 ∂ν

˚ \ Q. on Σ

(iv) Each convex vertex of Σ (with respect to G) is of T-type for X. (v) Suppose that Γ is not contained in a plane parallel to Π and that Σ is a polygon with k convex and n ≥ 1 nonconvex vertices q1 , . . . , ql between p1 and p2 , l = k + n. Assume also that the height function ξ 3 : [0, 1] → R of Γ has at most n − 1 interior extrema (i.e. maxima or minima in (0, 1)). Then at least one nonconvex vertex qj is of E-type for X, i.e. X creeps along the edge Ej = {qj } × R where S is nonconvex. More generally, if ξ 3 : [0, 1] → R has no more than n − r interior extrema, 1 ≤ r ≤ n, then at least r of the nonconvex vertices are of E-type for X. Proof. We approximate Σ by smooth curves Σ (n) which do not meet Σ1 , Σ2 , (n) and Γ except in P1 and P2 , and we set Σ0 := Σ1 ∪Σ (n) ∪Σ2 . We may assume (n) that the parameter representations σ (n) : R → R2 of Σ0 are of class C 3 with (n) (n) (n) (n) |σ˙ | = 1. We have p1 = σ (s1 ); suppose that p2 = σ2 (s2 ). Then the (n) lengths of Σ and Σ (n) are given by L(Σ) = s2 − s1 and L(Σ (n) ) = s2 − s1 respectively, and we arrange for L(Σ (n) ) → L(Σ), and

(n)

i.e. s2

→ s2 ,

as n → ∞

(n)

g(Σ0 , Σ0 ) → 0 as n → ∞, (n)

(n)

where g(Σ0 , Σ0 ) denotes the greatest distance between Σ0 and Σ0 , i.e. (n) (n) g(Σ0 , Σ0 ) := max{dist(p, Σ) : p ∈ Σ0 }. Furthermore we choose Σ (n) so (n) (n) that G ⊂ G where G and G are the inner domains of the Jordan curves Γ ∪ Σ (n) and Γ ∪ Σ, and that σ − σ (n) C 3 (Jε ) → 0 as n → ∞ l for any ε > 0, where Jε := R \ j=1 (tj − ε, tj + ε); in other words, we approximate S = Σ0 ×R away from its edges Ej = {qj }×R smoothly in C 3 by (n) smooth surfaces S (n) := Σ0 ×R. We can also assume that the approximating Σ (n) preserve local convexity. This means the following: Suppose that qj = σ(tj ) is a convex corner of G, i.e. the interior angle of ∂G at qj is less than π and κ(σ(s), 0) ≥ 0 for s ∈ [tj − δ, tj ) ∪ (tj , tj + δ]


101

for some small δ > 0; then the curvatures κ(n) of S (n) satisfy κ(n) (σ (n) (s), 0) ≥ 0 for s ∈ [tj − δ, tj + δ]. Finally we can assume that all configurations Γ, S (n) satisfy Condition (B) (or (M1 ), or (M2 ) respectively), and that the sets S (n) ∪ Γ fulfil a uniform chord-arc condition, i.e. there are numbers M ≥ 1 and δ > 0 such that any two points P , P of S (n) with |P − P | ≤ δ can be connected in S (n) ∪ Γ by a rectifiable arc Γ ∗ whose length L(Γ ∗ ) satisfies L(Γ ∗ ) ≤ M |P − P |. For every n ∈ N there exists a minimizer Xn of D in C(Γ, S (n) ). By Sec(n) ) tion 2.6 every surface Xn is a freely stable minimal surface of class M+ ∗ (Γ, S without branch points in B \ {±1}, and each mapping fn := π ◦ Xn (n)

yields a diffeomorphism of B \ {±1} onto G \ {p1 , p2 } and a homeomorphism of B onto G(n) , and Xn : B → R3 has an equivalent representation (n) Zn (x1 , x2 ) = (x1 , x2 , zn (x1 , x2 )), (x1 , x2 ) ∈ G , as a graph over G(n) of a solution zn = Xn3 ◦ fn−1 of the minimal surface equation on G(n) satisfying zn = γ

on Γ

and

∂zn =0 ∂ν n

˚(n) on Σ

where ν n denotes the exterior normal to ∂G(n) . Set m(γ) := maxΓ |γ| = max[0,1] |ξ 3 |, and let L(Γ ) and L(∂G(n) ) be the lengths of the curves Γ and ∂G(n) = Γ ∪ Σ (n) respectively. We claim that the area A(Zn ) = Wn dx1 dx2 with Wn := 1 + |∇zn |2 G(n)

is bounded by A(Zn ) ≤ m(γ)L(Γ ) +

(9)

1 2 L (∂G(n) ). 4π

To see this we multiply the minimal surface equation for zn by zn and obtain −1 0 = zn div(zn /Wn ) = div(W−1 n zn ∇zn ) − Wn + Wn .

Integrating this equation over G(n) we obtain 1 2 Wn dx1 dx2 ≤ W−1 dx dx + n G(n)

G(n)

∂G(n)

W−1 n zn

∂zn dH1 . ∂ν n

˚(n) , it follows Since Wn ≥ 1 and |∂zn /∂ν n | < Wn as well as ∂zn /∂ν n = 0 on Σ that A(Zn ) ≤ meas Gn + m(γ)L(Γ (n) ), and the isoperimetric inequality yields 4π meas G(n) ≤ L2 (∂G(n) ). Thus we have proved (9).

102


There is a number c0 such that m(γ)L(Γ ) +

1 2 L (∂G(n) ) ≤ c0 4π

for n ∈ N.

Since D(Xn ) = A(Xn ) = A(Zn ) it follows that D(Xn ) ≤ c0

(10)

for n ∈ N.

Moreover, as Xn maps C onto Γ , there is a number c1 with sup |Xn | ≤ c1

(11)

for n ∈ N.

C

On account of (10), (11), and Poincaré’s inequality, there is a number c such that Xn H21 (B) ≤ c

(12)

for n ∈ N.

We know that Xn (1) = P1 , Xn (−1) = P2 , and we can assume that Xn (i) = P3 where P3 is an arbitrarily chosen point on Γ between P1 and P2 . Every Xn maps C homeomorphically onto Γ . Thus we can extract a subsequence {Xnj } which converges weakly in H21 (B, R3 ) and uniformly on C to some X ∈ H21 (B, R3 ) ∩ C 0 (∂B, R3 ). By Rellich’s theorem we also obtain X − Xnj L2 (B) + X − Xnj L2 (∂B) → 0

as j → ∞.

Then there is a subsequence {nj } of {nj } such that Xnj (w) → X(w) L1 -a.e. on I. Since g(S (n) , S) → 0 as n → ∞ it follows that X ∈ C(Γ, S) whence e(Γ, S) ≤ D(X), where we have set e(Γ, S) := inf D = inf A, C(Γ,S)

C(Γ,S)

and similarly e(Γ, S (n) ) :=

inf

C(Γ,S (n) )

D=

inf

C(Γ,S (n) )

A.

By “adding collars” between S (n) and S to surfaces spanning Γ, S (n) or Γ, S it follows from g(S (n) , S) → 0 as n → ∞ that inf C(Γ,S (n) ) A → inf C(Γ,S) A, and so we have e(Γ, S (n) ) → e(Γ, S)

as n → ∞.

On the other hand, Xnj X in H 1,2 (B, Rn ) implies D(X) ≤ lim inf D(Xnj ), j→∞


103

and so we obtain e(Γ, S) ≤ D(X) ≤ lim D(Xnj ) = lim e(Γ, S (nj ) ) = e(Γ, S). j→∞

j→∞

Thus (13)

e(Γ, S) = D(X) = lim D(Xn ) = lim e(Γ, S (n) ) n→∞

n→∞

and X − Xn H 1,2 (B) → 0

(14)

as n → ∞

if we change notation and call the subsequence {Xn } instead of {Xnj }. In particular we have proved that X is a minimizer of D and A in C(Γ, S). On account of Harnack’s theorem and of well-known estimates for harmonic functions we have lim X − Xn C k (Ω) = 0 for Ω B

(15)

n→∞

and any k = 0, 1, 2, . . . . From (13) we infer X ∈ M(Γ, S), and (15) implies (16)

lim |N (w) − Nn (w)| = 0 for w ∈ B \ {branch points of X}

n→∞

where N and Nn denote the Gauss maps of X and Xn respectively. Since Nn3 (w) > 0 for any w ∈ B and N : B → S 2 is continuous we arrive at N 3 (w) ≥ 0 on B, i.e. X ∈ M+ (Γ, S). By assumption, Γ, S (n) satisfy a uniform chord-arc condition. Then there are numbers β ∗ ∈ (0, 1), r∗ ∈ (0, 1) and c∗ > 0 such that ∗

|Xn (w) − Xn (w )| ≤ c∗ r−β |w − w |β

∗

for n ∈ N and w, w ∈ B with |w−w | ≤ r. Together with (14) and the Arzelà– ∗ Ascoli theorem we infer that X ∈ C β (B, R3 ) and X − Xn C τ (B,R3 ) → 0 for any τ ∈ (0, β ∗ ). It follows that fn := π ◦ Xn converges uniformly on B to f := π ◦ X. Since Xn ∈ M∗ (Γ, S), fn |I maps I monotonically onto Σn , and Σn converges in the sense of Fréchet to Σ, we conclude that f |I maps I weakly monotonically onto Σ. Thus X ∈ M∗ (Γ, S), and together with N 3 (w) ≥ 0 on B we arrive at X ∈ M+ ∗ (Γ, S). Because of ΔN 3 = −N 3 |∇N |2 ≤ 0 on B we infer that N 3 (w) > 0 on B. Now we may proceed as in the proof of Proposition 2 in Section 2.6, obtaining f (B) = G. Since f |∂B maps ∂B weakly monotonically onto ∂G, Proposition 1 of Section 2.3 implies that f maps B diffeomorphically onto G, and Jf > 0 in B. Since X lies on one side of S, there are no branch points of X on I. Then Proposition 2 of Section 2.4 yields Jf > 0 on I ∗ and therefore |fu | > 0 on I ∗ . Furthermore, by Proposition 3 of

104


˚ contains no branch points of X and that Jf > 0 Section 2.4 we see that C ˚ Then it follows that f maps B \ f˜−1 (Q ) homeomorphically onto G\Q on C. ˚ ∪ I ∗ (X) diffeomorphically onto G \ ({p1 , p2 } ∪ Q). Thus we have and B ∪ C proved (i) as well as (ii), and (iii) follows by the reasoning of Section 2.5. Now we turn to the proof of (iv). Suppose that qj is a convex corner. By construction we have κ(n) (σ (n) (s), 0) ≥ 0 for s ∈ [tj − δ, tj + δ]. Suppose that qj were of E-type for X, i.e. f˜−1 (qj ) = Ij = [uj , uj ]. Set w0 := 12 (uj + uj ). Then there are numbers r, r1 , r2 with 0 < r1 < r < r2 such that Ij ⊂ (w0 − r1 , w0 + r1 ), f ([w0 − r2 , w0 + r2 ]) ⊂ σ((tj − δ, tj + δ)), [w0 − r2 , w0 − r1 ] ∪ [w0 + r1 , w0 + r2 ] ⊂ I ∗ . By Proposition 2 we have N 3 (w) > 0 for all w ∈ Cr (w0 ). Hence there is some m > 0 such that N 3 (w) ≥ 3m for all w ∈ Cr (w0 ). Because of (16) it follows that Nn3 (w) ≥ 2m

for w ∈ Cr (w0 ) if n 1.

Since Xn tends to X uniformly on B we have κ(n) (Xn (w)) ≥ 0

for w ∈ Ir (w0 ) if n 1.

By Proposition 3 we obtain Nn3 (w) ≥ m for all w ∈ S r (w0 ) if n 1. As X has no branch points in B, relation (15) implies Nn (w) → N (w) for all w ∈ B, and so we arrive at (17)

N 3 (w) ≥ m for all w ∈ S r (w0 ).

On the other hand, it follows from X ∈ C 0 (B, R3 ) and X(Ij ) ⊂ Ej that X is real analytic on B ∪ ˚ Ij . As X(B) lies on one side of S, the surface X has no Ij . From branch point on Ij , and therefore X and N are real analytic on B ∪ ˚ 3 X(u, 0) = (qj , X (u, 0)) for u ∈ Ij it follows that Xu (u, 0) = (0, 0, Xu3 (u, 0)) Ij , while for u ∈ ˚ Ij with Xu3 (u, 0) = 0; thus N 3 (w) = N (w) · e3 = 0 for w ∈ ˚ (17) implies N 3 (u, 0) ≥ m for u ∈ Ij , a contradiction. Thus qj is not of E-type, but of T-type, and (iv) is verified.


105

Statement (v) is proved by the same reasoning as Theorem 1 of Section 2.8; so we can be brief. Suppose that the height function of Γ has at most n − 1 extrema, and that all nonconvex vertices qj are of T-type for X (here “nonconvex” means: the angle between the two edges of Σ meeting at qj on the side of Γ is larger than π). Let uj be pre-images f˜−1 (qj ) of the 3 qj , j = 1, . . . , n. The function Xw (w) is continuous on B, holomorphic in B, 3 3 and satisfies Xw (w) = Xw (w) for w ∈ D where D = {w ∈ C : |w| < 1}. By a straight-forward extension of the asymptotic expansion in Section 2.7 (see (16) and (18) of Section 2.7) we obtain 3 (w) = cj (w − uj ) + · · · Xw

as w → uj

and therefore (18)

X 3 (w) = aj + Re{cj,2 (w − uj )2 + cj,3 (w − uj )3 + · · · }

for w ∈ D with |w − uj | 1, where aj ∈ R and cj,2 , cj,3 , . . . ∈ C. Denote by a and b the minimum and maximum respectively of the height function ξ 3 : [0, 1] → R; we have a < b since Γ is nonplanar. From X(C) = Γ and X(w) = X(w) we infer that a = min X 3 |∂D ,

b = max X 3 |∂D ,

and the maximum principle implies a < X 3 (w) < b for all w ∈ D, in particular a < X 3 (uj ) = aj < b for j = 1, . . . , n. Denote by γj the connected component of the level set {w ∈ D : X 3 (w) = aj } which contains the point w = uj . For j = k the sets γj and γk cannot intersect if aj = ak ; but they can also not intersect if aj = ak , because in this case we would obtain a nonempty open subset Ω of D such that X 3 (w) = aj on ∂Ω, taking X 3 (w) = X 3 (w) into account. This would imply X 3 (w) ≡ aj on Ω whence X 3 (w) ≡ aj on D, which is impossible since Γ does not lie in a plane parallel to Π. From (18) and X(w) = X(w) as well as from γj ∩ γk = ∅ for j = k it follows that each γj meets C in at least two points wj1 = exp(iθj1 ) and wj2 = exp(iθj2 ) with θj1 < θj2 such that 0 ≤ θn1 < θn2 < · · · < θ21 < θ22 < θ11 < θ12 ≤ π, if the uj are enumerated in such a way that −1 < u1 < u2 < · · · < un < 1. So we have (19)

X 3 (wj1 ) = X 3 (wj2 ) = aj .

Since X maps C homeomorphically onto Γ , there are uniquely determined points τj1 , τj2 ∈ [0, 1] which all are different (i.e. τjα = τkβ if and only if j = k and α = β) and satisfy

106

(20)


X(wj1 ) = ξ(τj1 ),

X(wj2 ) = ξ(τj2 )

for 1 ≤ j ≤ n.

From (19) and (20) it follows that ξ 3 (τj1 ) = ξ 3 (τj2 ). If there is no extremum of ξ 3 between τj1 and τj2 , the function ξ 3 (t) is either increasing or decreasing between τj1 and τj2 and therefore a constant. It follows that X 3 (eiθ ) ≡ aj

for θj1 ≤ θ ≤ θj2 ,

and as before we obtain a nonempty open set Ω ⊂ D with X 3 (w) ≡ aj on ∂Ω, which again is impossible. We conclude that each interval I(j) bounded by τj1 and τj2 must contain an inner extremum of the function ξ 3 (t), and so ξ 3 (t) possesses at least n extrema in (0, 1). This contradicts our assumption that there are at most n − 1 extrema of ξ 3 (t) in (0, 1). Therefore X is edge-creeping for at least one of the nonconvex vertices qj . In the same way it follows that at least r of the nonconvex vertices are of E-type for X if ξ 3 (t) possesses at most n − r extrema in (0, 1), 1 ≤ r ≤ n. Theorem 2. Let Σ be piecewise smooth, and suppose that Condition (R) is satisfied (which is, for instance, the case when Condition (A) holds true). Then we have: (i) There is a freely stable minimal surface X of class M+ ∗ (Γ, S) which is H¨ older continuous on B and has the regularity properties stated in Section 2.1, Proposition 4. (ii) This surface has the properties (ii)–(v) of Theorem 1. Proof. This result is an immediate consequence of Theorem 1; see the proof of Theorem 2 in Section 2.6. Remark 1. The surface X obtained in Theorem 1 is (in some restricted sense) a relative minimizer of A and D in C(Γ, S), and so it is freely stable. (Here the definition of the free stability has to be carried over in a suitable way to piecewise smooth Σ.) Note, however, that X need not be the absolute minimizer of A and D. This can be seen by a suitable modification of the example following Remark 1 in Section 2.6. Remark 2. If Σ is nonpolygonal, i.e. not piecewise straight, the angle β between two subarcs Σ and Σ of Σ meeting at a corner q of Σ can be less than π on the side of Σ containing Γ , without Σ being convex with respect to G close to q. This is for instance the case if Σ is “thorn-shaped” in a neighbourhood of q. One might conjecture that also such vertices are of T-type for minimizers X of D in C(Γ, S). It seems that one can adjust a regularity theorem for solutions to the capillarity problem, obtained by L. Simon [13], to solutions of the free boundary problem, thus obtaining (iv) for any corner with an angle less than π facing Γ . For piecewise smooth Σ we have less general uniqueness results than for smooth Σ. Let us state two “restricted” uniqueness results applying to surfaces


107

˚ in of the class M∗ (Γ, S). In the sequel we assume X(i) = P3 for some P3 ∈ Γ order to exclude conformal equivalence of surfaces in M(Γ, S). Theorem 3. Suppose that Σ0 is piecewise smooth and that the two straight lines L1 and L2 normal to Σ0 at p1 and p2 do not meet G except for p1 and p2 . Then there is at most one surface X in the class Mos ∗ (Γ, S) := {Y ∈ M∗ (Γ, S) : Y (B) lies on the same side of S as Γ }. This surface is a graph over G \ Q . Proof. If X ∈ Mos ∗ (Γ, S) then f (B) lies on the same side of Σ0 as Γ . Moreover, f := π ◦ X yields a weakly monotonic mapping of ∂B onto ∂G. Using the maximum principle and the assumptions on L1 and L2 we infer that f (B) ⊂ G. By Section 2.3, Proposition 1, it follows that f maps B diffeomorphically onto G. In particular we have Jf > 0 and N 3 > 0. Now we can proceed as in Section 2.5 to achieve the desired uniqueness result. Corollary 1. If we also assume Condition (R), then there is exactly one X ∈ M∗ (Γ, S) whose trace X(B) lies on one side of S, the freely stable surface X described in Theorem 2. From Corollary 1 we infer Corollary 2. If X ∈ M∗ (Γ, S) is a graph above G, then X meets all convex edges of S transversally. 3 Remark 3. If X ∈ M+ ∗ (Γ, S) we have N ≥ 0 in B, and

ΔN 3 = −N 3 |∇N |2 ≤ 0 ˚ As in the proof of Proposition 2 of implies N 3 > 0 on B (since N 3 > 0 on C). Section 2.6 it follows now that f (B) ⊂ G; thus X ∈ Mos ∗ (Γ, S), and we have os M+ ∗ (Γ, S) ⊂ M∗ (Γ, S).

Therefore Theorem 3 applies to M+ ∗ (Γ, S), whence we find: Theorem 4. There is at most one X of class M+ ∗ (Γ, S). Now we want to prove a uniqueness theorem for freely stable minimal surfaces of class M∗ (Γ, S); here we only consider the case where Σ is polygonal. Given X ∈ M∗ (Γ, S), let Q := Q \ Q and Q be the set of corners qj of Σ which are of type T or E respectively with respect to X. As usual we set f = π ◦ X and f˜ = f |I ; we know that N 3 (w) = 1 for w ∈ f˜−1 (Q ),

N 3 (w) = 0 for w ∈ f˜−1 (Q ),

and one can prove that D(N ) < ∞. (See Section 2.7, Theorem 4, where the case l = 1 is treated. The general case can be handled in the same way.) Thus the normal variation

108


X ε (w) := X(w) + εη(w)N (w),

|ε| 1,

is admissible, i.e. X ε ∈ C(Γ, S), provided that η ∈ C 1 (B) vanishes on an open subset J of ∂B containing C ∪ f˜−1 (Q ), and we do not need correct X + εηN as in Section 2.6. Thus we define: If Σ is a polygon, a surface X ∈ M∗ (Γ, S) is said to be freely stable if (21)

δ 2 A(X, η) ≥ 0 for all η ∈ C 1 (B) with η = 0 on C ∪ f˜−1 (Q ).

Minimizers X of A in C(Γ, S) satisfy δ 2 A(X, η) ≥ 0 for all η ∈ C 1 (B) with η = 0 on J, where J is an open subset of ∂B containing C ∪ f˜−1 (Q ), and a simple approximation argument leads to (21). Therefore Minimizers of A in C(Γ, S) are freely stable. Now we can formulate our last uniqueness result. Theorem 5. Let Σ be a polygon, and suppose that one of the Conditions (B), (M1 ), or (M2 ) is satisfied. Then there is exactly one X ∈ M∗ (Γ, S) which is freely stable, namely the minimizer determined by Theorem 1. The proof of this result is carried out in essentially the same way as the one of Theorem 1 in Section 2.6; see Hildebrandt and Sauvigny [6], pp. 97–98.

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge In this section we want to prove a Bernstein theorem for solutions ζ ∈ C 2 (Ω∞ ) of the minimal surface equation (1)

(1 + ζy2 )ζxx − 2ζx ζy ζxy + (1 + ζx2 )ζyy = 0

in a sectorial domain (2)

Ω∞ := {(x, y) ∈ R2 : x = ρ cos ϕ, y = ρ sin ϕ, ρ > 0, 0 < ϕ < β}

with (3)

β := π + απ,

satisfying the Neumann boundary condition (4)

∂ζ = 0 on ∂ Ω∞ := ∂Ω∞ \ {0} ∂ν

where ν denotes the exterior normal to ∂ Ω∞ . The following result is true: Theorem 1. Suppose that ζ ∈ C 0 (Ω ∞ ) ∩ C 1 (Ω ∞ \ {0}) satisfies (1) in Ω∞ as well as the boundary condition (4), and assume that 0 < |α| < 1. Then ζ is a constant function, i.e. ζ(x, y) ≡ ζ(0, 0) in Ω ∞ .

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

109

Remark 1. If α = 0, the domain Ω∞ is a halfplane. By applying the reflection principle of Vol. 1, Section 4.8, it is not difficult to see that ζ can be extended to a solution of (1) on R2 . Then the Bernstein theorem of Section 2.4 yields that ζ is affine, i.e. (5)

ζ(x, y) = ax + by + c

on R2

with a, b, c ∈ R. The same holds true if β = π/n for n ∈ N, as one sees by a similar reasoning. For n = 2, 3, . . . one actually obtains a = b = 0, i.e. ζ(x, y) = c on R2 , since in these cases the surface normal of graph ζ at (0, 0) turns out to be e3 = (0, 0, 1). For n = 1 this is not true. If β = (m/n)π with an arbitrary rational number in (0, 2π), the above reasoning does not work. We shall derive Theorem 1 from an a priori estimate on the Gauss map 1 · (−ζx , −ζy , 1), W := 1 + ζx2 + ζy2 (6) Y := W of a minimal graph (7)

Z(x, y) := (x, y, ζ(x, y))

where ζ is a solution of the minimal surface equation L(ζ) = 0 in a sectorial domain ΩR . Here L(ζ) is the minimal surface operator (8)

L(ζ) := (1 + ζy2 )ζxx − 2ζx ζy ζxy + (1 + ζx2 )ζyy .

To this end we define the domain ΩR in R2 by ΩR := {(x, y) ∈ R2 : x = ρ cos ϕ, y = ρ sin ϕ, 0 < ρ < R, 0 < ϕ < π + απ}. The boundary of ΩR consists of the origin 0 = (0, 0), the circular arc Sr := {(R cos ϕ, R sin ϕ) : 0 ≤ ϕ ≤ π + απ}

Fig. 1. The domain ΩR

110


∗ with the boundary ∂Ω ∗ = [0, Q ] ∪ S ∗ ∪ [0, P ] Fig. 2. The set ΩR R R R R

and the two straight segments ΓR0 and ΓRβ , β := π + απ, where ΓRϕ := {(ρ cos ϕ, ρ sin ϕ) : 0 < ρ < R} for ϕ ∈ R. Let ΓR := ΓR0 ∪ ΓRβ , and denote by ν the exterior normal to ∂ΩR on ΓR . Moreover we set Ω R := Ω R \ {0}. For any R > 0 we introduce the following condition:

Condition (∗R ). The function ζ is of class C 0 (Ω R ) ∩ C 2 (Ω R ) and satisfies L(ζ) = 0 in ΩR , ∂ζ/∂ν = 0 on ΓR , and ζ(0, 0) = 0. Theorem 2. Suppose that 0 < |α| < 1. Then there are numbers c > 0 and ϑ ∈ (0, 1) with the following property: For any R > 0 and any ζ satisfying Condition (∗R ), the Gauss map Y of the minimal graph Z corresponding to ζ satisfies sup |Y − e3 | ≤ c · (r/R)μ

(9)

for all r ∈ (0, ϑR]

Ωr

where μ ∈ [0, 1] is given by (10)

μ :=

1 min{|α|, 1 − |α|}. 1+α

Proof of Theorem 1. Suppose that ζ ∈ C 0 (Ω ∞ ) ∩ C 1 (Ω ∞ \ {0}) satisfies L(ζ) = 0 in Ω∞ and ∂ζ/∂ν = 0 on ∂Ω∞ \ {0}. The reflection principle yields that ζ is real analytic on ∂Ω∞ \ {0}. Assume also that ζ(0, 0) = 0. Then (∗R ) is satisfied for any R > 0, and so we obtain from Theorem 2 for R → ∞ that Y (x, y) ≡ e3 on Ω∞ , whence ∇ζ(x, y) ≡ 0 on Ω∞ and therefore ζ(x, y) ≡ 0. To obtain the general result we apply the above reasoning to ζ˜ := ζ − ζ(0, 0). Another implication of Theorem 2 is


111

Corollary 1. Any ζ satisfying Condition (∗R ) is of class C 1 (Ω R ). Now we turn to the proof of Theorem 2, which will be based on several Lemmata. Lemma 1. If ζ fulfils Condition (∗R ) for some R > 0, the function Φ := |Z|2 does not assume a relative maximum on ΩR ∪ ΓR . Proof. If Φ would attain a relative maximum at (ξ, η) ∈ ΩR ∪ ΓR then also the function Ψ : Ω R → R defined by Ψ (x, y) := Z(x, y) · Z(ξ, η),

(x, y) ∈ Ω R ,

would have a relative maximum at (ξ, η), because Φ(x, y) ≤ Φ(ξ, η) for (x − ξ)2 + (y − η)2 1 implies Ψ (x, y) ≤

Φ(x, y)

Φ(ξ, η) ≤ Φ(ξ, η) = Ψ (ξ, η)

on account of Schwarz’s inequality. The function Ψ (x, y) is real analytic in ΩR . Hence local constancy of Ψ would imply Ψ (x, y) ≡ const on ΩR , and since Ψ ∈ C 0 (Ω R ), we would have Ψ (x, y) ≡ Ψ (0, 0) = 0 on Ω R ; therefore 0 = Ψ (ξ, η) = Φ(ξ, η) = ξ 2 + η 2 + ζ 2 (ξ, η) ≥ ξ 2 + η 2 > 0, as (ξ, η) ∈ ΩR ∪ ΓR . Thus Ψ is nowhere locally constant on ΩR ∪ ΓR . Let us introduce the linear operator L := a(x, y)Dx2 + 2b(x, y)Dx Dy + c(x, y)Dy2 with a := 1 + ζy2 , b := −ζx ζy , c := 1 + ζx2 , which is uniformly elliptic on each

compact subset of Ω R . Since LΨ = L(ζ) · ζ(ξ, η) and

in ΩR

∂ζ ∂Ψ = ν · (ξ, η) + ζ(ξ, η) ∂ν ∂ν

on ΓR ,

we have LΨ = 0 in ΩR as well as

∂Ψ (ξ, η) = 0. ∂ν By the maximum principle and E. Hopf’s boundary point lemma we then obtain a contradiction.

112


∗ Lemma 2. Let ΩR be the connected component of the set {(x, y) ∈ ΩR : ∗ , and suppose that ζ fulfils Condition (∗R ). Then |Z(x, y)| < R} with 0 ∈ ∂ΩR ∗ ∗ ∗ = ΓR∗ ∪ SR where ΓR∗ is the union of two ΩR is simply connected, and ∂ΩR 0 ∗ closed segments [0, PR ] and [0, QR ] on ΓR and ΓRβ respectively, and SR is a β 0 ∗ closed set meeting ΓR and ΓR at PR and QR respectively, and SR \{PR , QR } ⊂ ΩR ∪ SR . ∗ with the interior G. Then Proof. Let γ be a closed Jordan curve in ΩR 2 Φ(x, y) < R on γ. By Lemma 1 we obtain Φ(x, y) < R2 on G, and so ∗ ∗ . Hence ΩR is simply connected. G ⊂ ΩR Since |Z(x, y)| < R for (x, y) ∈ Ω r with 0 < r 1, there is a point Q on ∗ ΓRβ such that the closed interval [0, Q] belongs to ∂ΩR . Thus the intersection ∗ ∗ of the closure SR of the set ∂ΩR \ ∂Ω∞ with ΓRβ \ {0} is nonempty. Let QR ∗ be the point in this intersection with the largest distance from 0. As ΩR is open and connected, there exists a simple continuous curve γ : [0, 1) → R2 ∗ with γ(t) ∈ ΩR for 0 < t < 1, γ(0) = Q, and γ(tj ) → QR for some sequence of points tj ∈ (0, 1) with tj → 1. Then it follows that the bounded open subset ∗ (here [Q, QR ] is the G of ΩR with ∂G = [Q, QR ] ∪ γ([0, 1]) belongs to ΩR β closed interval on ΓR between Q and QR ). Otherwise supG |Z| ≥ R, and so Φ|G would have a maximizer ζ ∈ G. On account of Lemma 1, the point ζ can neither lie in G nor on the open interval (Q, QR ) between Q and QR ; moreover, Φ(γ(t)) < R2 for 0 ≤ t < 1. Therefore only QR can be a maximizer of Φ|G . Because of Φ(QR ) = R2 it follows that maxG Φ = R2 and therefore |Z(x, y)|2 = Φ(x, y) < R2 for all (x, y) ∈ G, a contradiction. Hence all of G ∗ ∗ belongs to ΩR , and so we obtain ∂ΩR ∩ ΓRβ = [0, QR ]. 0 In the same way we argue for ΓR .

R > 0, the area A(R) := Lemma 3. If ζ satisfies Condition (∗R ) for some ∗ W(x, y) dx dy of its graph Z on the set Ω defined in Lemma 2 is esti∗ R ΩR mated by 3 A(R) ≤ (1 + α)πR2 . 2 Proof. We have

1 ∇ζ div W

=0

in ΩR .

Multiplying this equation by the truncated function ⎧ −R ≤ ζ(x, y) ≤ R, ⎨ ζ(x, y) η(x, y) := R if ζ(x, y) > R, ⎩ −R ζ(x, y) < −R, which is of class Lip(Ω R ), and integrating the resulting equation

1 1 η∇ζ = ∇η · ∇ζ a.e. in ΩR div W W


113

over the domain ΩR \ Ω ε for 0 < ε < R, we obtain (η/W)(ζx dy − ζy dx) = (1/W)∇η · ∇ζ dx dy, (11) ∂(ΩR \Ω ε )

ΩR \Ω ε

using an integration by parts. Since ∇η(x, y) = 0 on {|ζ(x, y)| > R} and ∇η(x, y) = ∇ζ(x, y) a.e. on {|ζ(x, y)| ≤ R}, the right-hand side of (11) is equal to (1/W)|∇η|2 dx dy, ΩR \Ω ε ∗ ∗ implies |∇η| = |∇ζ| a.e. on ΩR . and ζ 2 ≤ |Z|2 ≤ R2 on ΩR Thus 2 (1/W)|∇ζ| dx dy ≤ (1/W)∇η · ∇ζ dx dy. (12) ∗ \Ω ΩR ε

ΩR \Ω ε

Furthermore the integrand of the left-hand side of (11) vanishes on ΓR because of ∂ζ/∂ν = 0 on ΓR . So we infer from (11) and (12) the estimate 1 1 2 ηζr r dϕ − ηζr r dϕ (1/W)|∇ζ| dx dy ≤ ∗ W W ΩR \Ω ε SR Sε where r, ϕ are polar coordinates about the origin. Because of |ζr |/W ≤ 1 and |η| ≤ R it follows that (1/W)|∇ζ|2 dx dy ≤ (1 + α)πR(R + ε). ∗ \Ω ΩR ε

Passing to the limit ε → +0 we obtain (1/W)|∇ζ|2 dx dy ≤ (1 + α)πR2 . ∗ ΩR

Furthermore we have 1 1 |∇ζ|2 = W − W W and

∗ ΩR

1 dx dy ≤ W

dx dy = ΩR

Thus we finally arrive at ∗ ΩR

W dx dy ≤

1 (1 + α)πR2 . 2

3 (1 + α)πR2 . 2

114


Now we want to represent the graph Z of a solution ζ to (∗R ) by conformal parameters (u, v) varying in B := {(u, v) ∈ R2 : u2 + v 2 < 1, v > 0} with ∂B = I ∪ C, I = (−1, 1), C = {w ∈ C : |w| = 1, v > 0}. As usual we use here the complex notation w = u + iv, identifying R2 with C. Set I − := (−1, 0),

I + := (0, 1).

For any ϕ ∈ [0, 2π] we introduce the halfplanes Hϕ := {(x, y, z) : x = r cos ϕ, y = r sin ϕ, r > 0, z ∈ R}

(13)

and the planes Πϕ := {(x, y, z) : x = r cos ϕ, y = r sin ϕ, r, z ∈ R}.

(14)

Recall that e1 , e2 , e3 denote the unit vectors (1, 0, 0), (0, 1, 0), (0, 0, 1), and let ∗ ∗ , SR , ΓR∗ , PR , QR be defined as in Lemma 2. ΩR ∗ \ S ∗ which is Lemma 4. There exists a homeomorphism f of B ∪ I onto ΩR R 0 2 2 − + 2 of class C (B ∪ I, R ) ∩ C (B ∪ I ∪ I , R ) and has the following properties:

(i) f |I is continuous, f (0) = 0, f (1) = PR , f (−1) = QR and (15)

lim |Z(f (wn ))| = R

n→∞

wn → w0 ∈ C

for any sequence of points wn ∈ B with

as n → ∞.

(ii) The Jacobian Jf of f satisfies Jf (u, v) > 0

(16)

for all w = u + iv ∈ B ∪ I − ∪ I + .

(iii) The reparametrized graph of Z, X := Z ◦ f , which we write as (17)

X(u, v) = (X 1 (u, v), X 2 (u, v), X 3 (u, v)) = Z(f (u, v)),

is of class C 0 (B ∪ I, R3 ) ∩ C 2 (B ∪ I − ∪ I + , R3 ) ∩ H 1,2 (B, R3 ) and satisfies the boundary conditions (18)

X(I + ) ⊂ H0 ,

Xv (u, 0) ⊥ H0

X(I − ) ⊂ Hπ+απ ,

if u ∈ I + ,

Xv (u, 0) ⊥ Hπ+απ

if u ∈ I −

as well as (19)

ΔX = 0,

|Xu |2 = |Xv |2 ,

Moreover, X|I is continuous.

Xu · X v = 0

in B.


115

(iv) There is a holomorphic function h1 : D → C in the unit disk D := {w ∈ C : |w| < 1} such that Xw (w) = wα h1 (w)j + o(|w|α ) as w → 0, √ where j = (1/ 2)(e1 − ie2 ) and h1 (0) > 0. (v) The Dirichlet integral of X is estimated by 1 3 (21) |∇X|2 du dv ≤ (1 + α)πR2 . 2 B 2 (20)

∗ with the propProof. The existence of a diffeomorphism f from B onto ΩR erties (i), (iii) follows, for instance, from J. Jost [17], Chapter 3, in particular Theorem 3.2.1 and Corollary 3.2.1. Moreover, Proposition 2 of Section 2.4 yields Jf (w) > 0 on I − ∪ I + , and Section 2.7 implies (20). Since f is a ∗ ∪ ΓR , we obtain h1 (0) = 0, and the boundhomeomorphism of B ∪ I onto ΩR ary conditions (18) imply that h1 (w) is real valued on I. From this we infer h1 (0) > 0. Finally (21) follows from Lemma 3 since A(R) = 12 B |∇X|2 du dv.

Because of (ii), the minimal surface X has no branch points on B ∪I − ∪I + , and the Gauss map N : B ∪ I − ∪ I + → S 2 of X, given by (22)

N := |Xu ∧ Xv |−1 (Xu ∧ Xv ),

is connected with the Gauss map Y : Ω R → S 2 of Z by (23)

N (w) = Y (f (w))

for any w ∈ B ∪ I − ∪ I + .

On account of (15) we can extend N to a continuous mapping B∪I → S 2 ⊂ R3 by setting N (0) := e3 , see Section 2.7. Then we infer that also Y can be extended continuously to Ω R by setting Y (0, 0) = e3 . This implies ζ ∈ C 1 (Ω R ) and ∇ζ(0, 0) = 0. By Theorem 3 of Section 2.7 we obtain also Lemma 5. For any ρ ∈ (0, 1) the Gauss map N of the conformal reparametrization X of Z from Lemma 4 satisfies 1 |∇N |2 du dv < ∞. (24) 2 Bρ Now we agree upon the following Notation (N). In the sequel, c1 , c2 , . . . , and r1 , r2 , . . . denote positive numbers which are independent of R > 0 and of the solution ζ of (∗R ) that we are considering. Lemma 6. There is a constant c1 > 0 such that (25)

|N (w) − N (0)| ≤ c1 |w|min{|α|,1−|α|}

for any w ∈ B ∪ I.

116


2 Proof. Note that N maps B ∪ I into the open upper hemisphere S+ of S 2 , 3 2 i.e. N (w) > 0 on B ∪ I. Let σ be the stereographic projection of S from the south pole −e3 . Then g := σ ◦ N defines an antiholomorphic map of B into the unit disk D. (Note that σ is orientation preserving whereas the stereographic projection from the north pole is orientation reversing. Moreover, N is orientation reversing because of K ≤ 0.) From (18) we obtain the boundary conditions g(I + ) ⊂ Π0 and g(I − ) ⊂ Ππ+απ , and (24) yields g ∈ H21 (Bρ , C) for any ρ ∈ (0, 1). Then the reasoning used in Section 2.7 yields the asymptotic expansion g(w) = cwγ + o(|w|γ ) for w → 0, w ∈ B,

with a constant c ∈ C and an exponent γ ∈ {|α|, 1 − |α|}. Now we consider a function ψ(w), w ∈ B ∪ I − ∪ I + , which is defined by ψ(w) := (w)−γ g(w)

for w ∈ B ∪ I − ∪ I + .

The function ψ is bounded and antiholomorphic in B as well as continuous on B ∪ I − ∪ I + . Furthermore the boundary condition (18) implies that ψ(w) is real-valued on I − ∪ I + . Then by Schwarz’s reflection principle, we can extend ψ to an antiholomorphic function Ψ on D \ {0} which satisfies Ψ (w) = c + o(1)

as w → 0.

Hence the singularity w = 0 of Ψ (w) is removable, and Ψ (w) is antiholomorphic on D. Thus |Ψ (w)| is continuous and subharmonic in D. Moreover it follows from the construction of Ψ that for every > 0 there is some δ > 0 such that |Ψ (w)| < 1 + if 1 − δ < |w| < 1. Then the maximum principle implies |Ψ (w)| < 1 + on D for any > 0, whence |Ψ (w)| ≤ 1 on D and therefore |g(w)| ≤ |w|γ

for all w ∈ B.

Since N (0) = e3 we also have g(0) = σ(N (0)) = σ(e3 ) = 0, and consequently |g(w) − g(0)| ≤ |w|γ

for all w ∈ B.

Since σ is a bi-Lipschitz map of the closed upper hemisphere onto D, there is a constant c1 > 0 such that |P − P | ≤ c1 |σ(P ) − σ(P )| 2 , and therefore for any two points P , P on S+

|N (w) − N (0)| ≤ c1 |w|γ

for all w ∈ B ∪ I.


117

Now we come to the main estimate. We want to find a lower bound for f of the form |f (w)| ≥ cR|w|1+α

for w ∈ Br := {w ∈ B : |w| < r}

with constants c > 0 and r > 0 independent of R. In conjunction with Lemma 6 this estimate will lead to Theorem 2. As before, ζ is assumed to be a solution of (∗R ) for some R > 0, Z(x, y) = (x, y, ζ(x, y)), and X = Z ◦ f is the conformal representation of Z|ΩR∗ determined in Lemma 4. We begin by estimating E := |Xu |2 from below in a single point w∗ ∈ B which is in a suitable position. Lemma 7. There exist constants c2 > 0 and r1 , r2 ∈ (0, 1) with r1 < r2 such that the inequality E(w∗ ) ≥ c2 R2

(26)

is satisfied for at least one point w∗ ∈ B with (27)

r1 ≤ |w∗ | ≤ r2

and

Re w∗ = 0.

Proof. Without loss of generality we can assume that R = 1 because we may ˜ always pass from X(w) to the rescaled map X(w) := (1/R)X(w). From (21) we infer that (28) |∇X|2 du dv ≤ 3(1 + α)π, B

and so we can apply the Courant–Lebesgue lemma to the semicircles Cρ := {w ∈ C : |w| = ρ, Im w ≥ 0}. We obtain that, for any δ ∈ (0, 1), there is a number δ ∗ ∈ (δ, the length of the curve X|Cδ∗ can be estimated by (29) |dX| ≤ 2 3π 2 (1 + α)/ log(1/δ).

√

δ) such that

Cδ∗

Let X(w) = (X 1 (w), X 2 (w), X 3 (w)). The component of X in z-direction, X 3 (w), is of class C 2 (B ∪ I − ∪ I + ) and satisfies X 3 (0) = 0 as well as Xv3 (w) = 0 on I − ∪ I + . By Schwarz’s reflection principle we extend X 3 (w) to a harmonic function on the disk D = {w : |w| < 1}, since the singularity w = 0 is removable. In particular we have X 3 ∈ C 2 (B ∪ I) and Xv3 (w) = 0

on I.

118


On account of Hopf’s lemma and the maximum principle it follows that the maximum M and the minimum m of X 3 on B δ∗ = {w : |w| ≤ δ ∗ , Re w ≥ 0} are both assumed on Cδ∗ , and that m < X 3 (w) < M for all w ∈ I. Therefore X 3 (0) = 0 implies m < 0 < M . In conjunction with (29) we infer that |dX 3 | ≤ 1/4 max{−m, M } ≤ M − m ≤ Cδ∗

if 0 < δ ≤ δ1 and δ1 is a sufficiently small positive number. It follows that sup |X 3 (w)| ≤ 1/4. Bδ

For the planar 1 − 1 map f we have f (0) = 0,

f (I + ) = (0, P1 ) ⊂ Γ10 ,

f (I − ) = (0, Q1 ) ⊂ Γ1β ,

since R = 1, and (29) yields ∗

∗

|df | ≤ 2 3π 2 (1 + α)/ log(1/δ).

|f (δ ) − f (−δ )| ≤ Cδ∗

Then there is a number δ2 ∈ (0, 1) such that sup |f (w)| ≤ 1/4 Bδ

if 0 < δ ≤ δ2 . Setting r1 := min{δ1 , δ2 } it follows that r1 ∈ (0, 1) and |X(w)| ≤ 1/2 for all w ∈ B r1 , in particular |X(w1 )| ≤ 1/2 for w1 = ir1 .

(30)

Next we recall that |Z(x, y)| → 1 as (x, y) ∈ Ω1∗ tends to a boundary point on S1∗ . Thus by (15) and R = 1 we have |X(eiθ )| = 1 for almost all θ ∈ (0, π), see M. Tsuji [2], p. 135. Applying the Courant–Lebesgue lemma to the circular arcs γδ := {w ∈ B : |w − i| = δ}, 2 we obtain √ as before that for any δ > 0 with δ < (1 − r1 ) there exists a δ ∈ (δ, δ) such that |dX| ≤ 2 3π 2 (1 + α)/ log(1/δ). γδ

Then we can choose δ > 0 so small that for the corresponding δ ∈ (δ, and w2 := (1 − δ )i we have |w2 | < r2 := 1 − δ, 0 < r1 < r2 < 1, and

√

δ)


119

|X(w2 )| ≥ 3/4.

(31) This implies

1/4 ≤ |X(w2 ) − X(w1 )| ≤

r2

|Xv (0, v)| dv.

r1

Furthermore, there is some v ∗ ∈ [r1 , r2 ] such that r2 |Xv (0, v)| dv = (r2 − r1 )|Xv (0, v ∗ )|. r1

Then for w∗ := iv ∗ and c2 := 1/[4(r2 − r1 )]2 we obtain Re w∗ = 0, Im w∗ > 0, r1 ≤ |w∗ | ≤ r2 , and E(w∗ ) ≥ c2 > 0. Now we are going to estimate the gradient of f from below by means of Harnack’s inequality and Lemma 7. Lemma 8. There exists a constant c3 > 0 such that (32)

|∇f (w)|2 ≥ c3 R2 |w|2α

for all w ∈ Br2 .

Proof. Again we can assume that R = 1. Now we associate with f (w) = (X 1 (w), X 2 (w)) the mapping F : B ∪ I → C defined by F(w) := X 1 (w) + iX 2 (w). Since f is harmonic in B, the complex derivative Fw is holomorphic in B; therefore also (33)

g(w) := 2w · Fw (w)

is holomorphic in B ∪ I − ∪ I + (cf. Lemma 4). The Jacobian Jf =

∂(X 1 , X 2 ) ∂(u, v)

of f can be written as (34)

Jf = |Fw |2 − |Fw |2 ,

and by Lemma 4 we know that (35)

Jf > 0 on B ∪ I − ∪ I + .

The conformality relations for X are equivalent to Xw · Xw = 0, that is, 3 3 · Xw =0 fw · fw + X w

whence |∇X 3 |2 ≤ |∇f |2

120


and therefore E = |Xu |2 =

1 |∇X|2 ≤ |∇f |2 . 2

Furthermore (34) and (35) imply |Fw |2 < |Fw |2 , and therefore |∇f |2 = 2|Fw |2 + 2|Fw |2 < 4|Fw |2 .

(36) Thus we arrive at

E < 4|Fw |2

(37)

on B.

In addition we infer from (34) and (35) that Fw (w) = 0 on B ∪ I − ∪ I + , and therefore also g(w) = 0 on B ∪ I − ∪ I + .

(38)

Next we introduce the function h : B ∪ I − ∪ I + → C by h(w) := w−1−α g(w) = 2w−α Fw (w),

(39)

which is holomorphic on B ∪ I − ∪ I + . The asymptotic expansion (20) yields that g(w) = γw1+α + o(w1+α ) as w → 0, w ∈ B ∪ I − ∪ I + , with some constant γ > 0. Thus we can extend h(w) to a continuous function on B ∪ I by setting h(0) := γ, and we obtain from (38) and (39) that h(w) = 0 on B ∪ I.

(40)

Now we transform g(w) to polar coordinates ρ, ϕ about the origin via w = ρeiϕ , and we get (41)

g(ρeiϕ ) = ρ

∂ ∂ [F(ρ cos ϕ, ρ sin ϕ)] − i [F(ρ cos ϕ, ρ sin ϕ)]. ∂ρ ∂ϕ

On account of the boundary conditions (18) we infer from (41) that h(w) ∈ R for w ∈ I − ∪I + . Thus, by Schwarz’s reflection principle, h(w) can be extended to a continuous function on D which is holomorphic on D \ {0} where D = {w ∈ C : |w| < 1}. The singularity of h(w) at w = 0 is removable, and therefore h(w) is holomorphic on D, and (42)

h(w) = 0 for all w ∈ D


121

on account of (38)–(40). Moreover, by Lemma 7 and estimate (37) there is a point w∗ ∈ B satisfying r1 ≤ |w∗ | ≤ r2 ,

Re w∗ = 0,

and |Fw (w∗ )|2 ≥ c2 /4. For w ∈ B ∪ I − ∪ I + we have |h(w)| = 2|w|−α |Fw (w)|, and therefore √ |h(w∗ )| ≥ min{r1−α , r2−α } c2 .

(43) Now we set r :=

1−r2 2

and

D := {w ∈ C : |w| < r2 }, D := {w ∈ C : |w| < r2 + r}, Dr (w0 ) := {w ∈ C : |w − w0 | < r}. Suppose that w0 ∈ C satisfies |w0 | = r2 + r. Then Dr (w0 ) ⊂ D \ D , and we have |h(w)| ≤ 2 max{r2−α , 1}|Fw (w)| for all w ∈ B \ D . The mean value theorem for harmonic functions yields 1 h(w) du dv h(w0 ) = 2 πr Dr (w0 ) whence 1 |h(w)|2 du dv πr2 Dr (w0 ) 1 ≤ |h(w)|2 du dv πr2 D\D 2 = |h(w)|2 du dv πr2 B\D 8 max{r2−2α , 1} |Fw |2 du dv. ≤ πr2 B

|h(w0 )|2 ≤

Moreover, by (36), |Fw (w)|2 ≤

1 1 |∇f |2 ≤ |∇X|2 , 2 2

and so the estimate (21) yields 3 |Fw (w)|2 du dv ≤ (1 + α)π. 2 B

122


This leads to (44)

|h(w0 )|2 ≤ 12(1 + α)r−2 max{r2−2α , 1}

provided that w0 ∈ ∂D .

By Harnack’s inequality there is a number k > 1 such that sup H ≤ k inf H

(45)

D

D

holds for all nonnegative harmonic functions H : D → R. We apply this estimate to H(w) := log M − log |h(w)|, w ∈ D , where the constant M is defined as (46) M := 12(1 + α)r−1 max{r2−α , 1}. Note that H(w) is well-defined and harmonic for all w ∈ D since h(w) = 0 on D, and the maximum principle implies |h(w)| ≤ M on D , and therefore H(w) ≥ 0 for all w ∈ D . We obtain sup log D

M M ≤ k inf log D |h| |h|

whence log

M inf D |h|

≤ log

Mk supD |h|k

and consequently |h(w∗ )|k ≤ sup |h|k ≤ M k−1 |h(w)| D

for all w ∈ D .

In conjunction with (43) we arrive at (47)

|Fw (w)| =

√ k 1 α 1 |w| |h(w)| ≥ M 1−k min{r1−α , r2−α } c2 |w|α 2 2

for all w ∈ B ∩ D = Br2 , and (36) yields (48)

|∇f (w)|2 ≥ 2|Fw (w)|2

on B.

Now we infer from (46)–(48) that there is a constant c3 such that |∇f (w)|2 ≥ c3 |w|2α

for all w ∈ B with |w| ≤ r2 .

The general concept of the preceding proof is taken from work of E. Heinz, cf. in particular Heinz [5], Lemma 5. A dilation estimate is given in the following Lemma 9. There exist constants c5 > 0 and r4 ∈ (0, 1) such that |f (w)| ≥ c5 R|w|1+α

for all w ∈ B r4 .


123

Proof. Once again we may assume that R = 1. By (32) we have |∇f (w)|2 ≥ c3 |w|2α

for all w ∈ B r2 ,

and the conformality relations imply |Xu |2 = |Xv |2 = therefore |Xu (w)|2 = |Xv (w)|2 ≥

1 1 |∇X|2 ≥ |∇f |2 ; 2 2 c3 2α |w| 2

for all w ∈ Br2 .

In addition we have Xu (w) · Xv (w) = 0 for all w ∈ B. Furthermore, by Lemma 6 there is a radius r3 ∈ (0, 1) such that |N (w) − N (0)| ≤ 1

for all w ∈ B r3 ,

and we also have N (0) = e3 and Xu (w) · N (w) = Xv (w) · N (w) = 0

for all w ∈ B.

Then it is not difficult to show that there is a constant c4 > 0 such that the 2 × 2-Jacobi matrix ∇f of the plane map f satisfies (49)

|∇f (w)ξ| ≥ c4 |w|α

for all w ∈ Br4 and all ξ ∈ S 1

where r4 := min{r2 , r3 } and S 1 := {ξ ∈ R2 : |ξ| = 1}. Let us introduce the domains B ∗ := Br4 , Ω ∗ := {z = w1+α : w ∈ B ∗ }, and Ω ∗∗ := {ζ = f (w) : w ∈ B ∗ }. We note that f yields a homeomorphism of B ∗ onto Ω ∗∗ , and the mapping φ(w) := w1+α = ˆ (Re w1+α , Im w1+α ) furnishes ∗ ∗ a homeomorphism of B onto Ω . Then g := f ◦ φ−1 is a homeomorphism of Ω ∗ onto Ω ∗∗ with g(0) = 0, and the restriction of g to Ω ∗ \ {0} is a diffeomorphism. From f (w) = g(φ(w)),

w ∈ B∗,

we obtain for w = 0 and z = φ(w) that ∇f (w) = ∇g(z)∇φ(w) and therefore ∇f (w)ξ = ∇g(z)∇φ(w)ξ

for all ξ ∈ S 1 .

Since the mapping φ : w → z = w1+α is conformal for w = 0, the polar decomposition of the nonsingular 2 × 2-matrix ∇φ(w) is given as ∇φ(w) = AU

124


with U ∈ SO(2),

ρ = (1 + α)|w|α

A = diag(ρ, ρ),

where ρ = |φ (w)| is the dilation factor of the conformal map φ(w). On account of (49) we arrive at for all z ∈ Ω ∗ \ {0} and all η ∈ S 1

(1 + α)|∇g(z)η| ≥ c4

since the mapping ξ → η = U ξ is a 1-1 map of S 1 onto itself. Consequently the inverse h = g −1 satisfies |∇h(ζ)ξ| ≤

(50)

1+α c4

for all ζ ∈ Ω ∗∗ \ {0} and all ξ ∈ S 1 .

Now we observe that Ω ∗ = {z = reiθ : 0 < r < r41+α , 0 < θ < (1 + α)π}. Let r ∈ (0, r41+α ] and suppose that z∗ ∈ Ω ∗ satisfies |z∗ | = r and |g(z∗ )| = min{|g(z)| : z ∈ Ω ∗ and |z| = r}. We set ζ∗ := g(z∗ ) and consider the path t → ζ(t) with 0≤t≤1

ζ(t) := tζ∗ ,

which connects 0 with ζ∗ in Ω ∗∗ . Then the curve γ(t) := h(tζ∗ ),

0 ≤ t ≤ 1,

connects 0 with z∗ = h(ζ∗ ) within Ω ∗ , and for any ∈ (0, 1) we have

1

|z∗ − h( ζ∗ )| ≤

|γ(t)| ˙ dt =

1

|∇h(tζ∗ )ζ∗ | dt.

By (50) it follows that |z∗ − h( ζ∗ )| ≤ (1 − )(1 + α)c−1 4 |ζ∗ |. Set c5 := (1 + α)−1 c4 and let → +0. Then h( ζ∗ ) → 0, and we obtain |z∗ | ≤ c−1 5 |ζ∗ |. Then we get c5 r ≤ |g(z∗ )| ≤ |g(z)|

for all z ∈ Ω ∗ with |z| = r.

Thus we obtain c5 |z| ≤ |g(z)|

for all z ∈ Ω ∗ ,

and finally |f (w)| = |g(φ(w))| ≥ c5 |φ(w)| = c5 |w|1+α

for all w ∈ B ∗ .


125

Proof of Theorem 2. Suppose that 0 < |α| < 1, and define μ ∈ [0, 1] by μ :=

1 min{|α|, 1 − |α|}. 1+α

Then we choose c1 as in Lemma 6 and c5 , r4 as in Lemma 9, and we set c := c1 c−μ 5 .

ϑ := min{1, c5 r41+α },

Next we fix some R > 0. For r ∈ (0, ϑR] we set 1+α . ρ := [c−1 5 · (r/R)] 1

Then we obtain 0 < ρ < r4 , and so the arc Cρ = {w ∈ B : |w| = ρ} is contained in the semidisk B r4 = {w ∈ B : |w| ≤ r4 }. Suppose now that ζ satisfies Condition (∗R ). Let X be the conformal ∗ determined by Lemma 4, reparametrization of Z(x, y) = (x, y, ζ(x, y)) on ΩR and denote by Y and N the Gauss maps of Z and X respectively. Then X =Z ◦f

and

N = Y ◦ f,

∗ is the diffeomorphism from Lemma 4. By Lemma 9 we where f : B → ΩR have |f (w)| ≥ c5 R|w|1+α for all w ∈ B r4 .

Since 0 < ρ < r4 it follows for w ∈ Cρ that |f (w)| > c5 Rρ1+α = r. This implies Ω r ⊂ f (B ρ )

(51)

∗ \S ∗ with f (0) = 0; in particular since f is a homeomorphism of B ∪I onto ΩR R we have ϑ < 1. From (51) we infer

sup{|Y (x, y) − e3 | : (x, y) ∈ Ω r } ≤ sup{|Y (x, y) − e3 | : (x, y) ∈ f (B ρ )} = sup{|Y (f (w)) − e3 | : w ∈ B ρ } = sup{|N (w) − e3 | : w ∈ B ρ }. On account of Lemma 6 and N (0) = e3 we obtain for w ∈ B ρ that μ μ |N (w) − e3 | ≤ c1 ρmin{|α|,1−|α|} = c1 c−μ 5 (r/R) = c · (r/R) .

It follows that sup{|Y (x, y) − e3 | : (x, y) ∈ Ω r } ≤ c · (r/R)μ if 0 < r ≤ ϑR, which is the assertion of Theorem 2.

126


2.11 Scholia 1. References to the Literature. The results of this chapter are taken from joint work of S. Hildebrandt and F. Sauvigny that appeared between 1991 and 1999. Several basic ideas can already be found in Sauvigny’s earlier work on surfaces of prescribed mean curvature, in particular the idea to estimate the component N 3 := N · e3 of the normal vector from below, using the stability inequality and an estimate of |Xu ∧ Xv | from below. We refer to Sauvigny [7–11] and also to Vol. 1, Chapters 5 and 7. Sections 2.2–2.4 are essentially a condensed version of Hildebrandt and Sauvigny [1] and [2]; some parts are taken from [3] and [6]. The uniqueness results in Sections 2.5, 2.6, and 2.9 were proved in [1,3], and [6]. The asymptotic expansions of Section 2.7 were first developed in [4], whereas the example of an edge-creeping surface and the criterion ensuring edge-creeping can be found in [5]. The results of Sections 2.6 and 2.9 on the existence of embedded solutions are published in [3] and [6]. The Bernstein-type theorem presented in Section 2.10 was proved in Hildebrandt and Sauvigny [7]. For a convex corner (i.e. −1 < α < 0) this result was somewhat earlier derived by C.-C. Lee [1] using a very different argument that we will now sketch. Let us suppose that ζ ∈ C 1 (Ω \ {0}) ∩ C 2 (Ω) is a solution of the minimal surface equation (1) div T ζ = 0 in Ω, T ζ := ∇ζ/W, W := 1 + |∇ζ|2 where Ω = Ω∞ is a sectorial domain with the vertex 0 and an opening angle less than π which satisfies the Neumann condition ∂ζ/∂ν = 0 on ∂Ω \ {0} where ν is the exterior normal of ∂Ω. Then Lee notes first that, on account of a regularity result due to L. Simon [13], the solution ζ is of class C 1 (Ω). Then the mapping ψ : (x, y) → (ξ, η) with ξ = x + F (x, y), and

∇F = (1 + ζx2 , ζx ζy )W−1 ,

η = y + G(x, y) ∇G = (ζx ζy , 1 + ζy2 )W−1

defines a C 1 -diffeomorphism of Ω onto Ω = ψ(Ω) which introduces conformal coordinates ξ, η on the minimal surface represented by ζ. It turns out that also Ω is a sectorial domain of an opening angle less than π; see also Sections 2.2–2.4. Because of (1) there is a function u ∈ C 1 (Ω) such that uy = −ζy /W. Obviously |∇u| < 1, whence u(x, y) < x2 + y 2 . Define v(ξ, η) on Ω by −1 v := u ◦ ψ . Then it follows that v(ξ, η) < ξ 2 + η 2 since the map ψ can be seen to be expanding, i.e. ux = ζx /W,

2.11 Scholia

127

(ξ1 − ξ2 )2 + (η1 − η2 )2 > (x1 − x2 )2 + (y1 − y2 )2 for (ξ1 , η1 ) = ψ(x1 , y1 ), (ξ2 , η2 ) = ψ(x2 , y2 ), (x1 , y1 ), (x2 , y2 ) ∈ Ω. It turns out that v ∈ C 1 (Ω ) ∩ C 2 (Ω ) is harmonic in Ω and constant on ∂Ω . Since v(ξ, η) = O( ξ 2 + η 2 ), the Phragmén–Lindel¨ of theorem for harmonic functions on a sector domain implies v(ξ, η) ≡ const whence u(x, y) ≡ const, and so ζ(x, y) ≡ const; therefore ζ(x, y) ≡ 0 if ζ(0, 0) = 0. Lee’s theorem is a variation of an earlier result due to J.C.C. Nitsche [11], who in 1965 stated that a solution ζ ∈ C 0 (Ω) ∩ C 2 (Ω) of (1) in a sectorial domain Ω vanishes identically if it has zero boundary values on ∂Ω and if the opening angle of Ω is less than π. This result suggests that a solution of the minimal surface equation in an unbounded domain Ω depends only on the shape of Ω and the boundary values ζ|∂Ω , provided Ω is properly contained in a halfplane. Theorems of this kind were proved by J.-F. Hwang [1,2]. We also note that Simon’s gradient bound follows for a sectorial domain from Theorem 1, which can be stated in the following equivalent form: There are constants c > 0 and ϑ ∈ (0, 1) such that for R > 0 the gradient of any solution ζ of (∗R ) can be estimated by sup |∇ζ| ≤ c(r/R)μ

for all r ∈ (0, ϑR)

Ωr

with μ := (1 + α)−1 min{|α|, 1 − |α|} provided that 0 < |α| < 1. The phenomenon of edge-creeping was apparently discovered by Y.-W. Chen [3] in 1956 when he solved the exterior free boundary value problem for minimal surfaces whose free boundary value lies on a cylinder surface S = Σ × R with a closed planar, convex polygon Σ as directrix. It seems that Chen’s interesting work went unnoticed for some time. An earlier paper by Chen [2] treated the exterior free boundary problem for smooth cylinders S as supporting surfaces; in this case no edge creeping appears. The phenomenon of edge-creeping was rediscovered by Hildebrandt and Nitsche [3] in 1982; the results of this paper are described in Chapter 1. In 1992 K.A. Brakke [1] discussed the phenomenon by means of the Weierstrass representation, admitting also nonperpendicular intersection of the free boundary. The results of the present chapter were generalized by G. Turowski [1–3] to minimal surfaces of annulus type spanning a boundary configuration Γ, S consisting of a cylinder surface S = Σ × R and a closed Jordan curve Γ around S where Σ is a simple closed polygon in a plane Π and Γ has a convex projection Γ into Π. Her approach is quite different from that of Hildebrandt–Sauvigny as it uses ideas by Y.-W. Chen [3] as a starting point. Nevertheless the methods of Sections 2.1–2.9 are fundamental for Turowski’s work that we shall sketch in the sequel (see No. 2). T. Nehring [1] proved the existence of embedded minimal surfaces of annulus-type spanning a configuration Γ, S where Γ is a closed Jordan curve surrounding a convex supporting surface S.

128


Sections 2.5, 2.6, and 2.9 contain existence results for solutions ζ of the minimal surface equation (1) in G satisfying the “mixed” boundary value problem ˚ ζ = γ on Γ , ∂ζ/∂ν = 0 on Σ, where ∂G = Γ ∪ Σ. By entirely different methods, E. Giusti [6] treated this problem (as well as the analogous one for surfaces of prescribed mean curvature) on convex domains G; the convexity of G is crucial in his approach. Disk-type embedded minimal surfaces with a free boundary on a convex polyhedron were constructed by J. Jost [13]. In particular, he proved that for his solutions the edges of the supporting convex polyhedron are of transversal type T. This corresponds to statement (iv) of Theorem 1 in Section 2.9. 2. Embedded Minimal Surfaces of Annulus-Type with a Partially Free Boundary. Now we briefly describe the work of G. Turowski [1–3] which can be viewed as an extension of the results presented in Sections 2.1–2.9 to doubly connected minimal surfaces. She studied the following situation: Let S = Σ × R be a vertical cylinder above a simple closed polygon Σ in the x1 , x2 -plane Π which is surrounded by a closed Jordan curve Γ that is a generalized graph of γ above a closed convex curve Γ containing Σ in its interior. Courant has established the existence of annulus-type minimal surfaces spanning the configuration Γ, S under natural and fairly weak assumptions which are satisfied in the present situation; cf. Courant [15], p. 210, Theorem 3.6. However it is not investigated whether this work can be combined with the method of the present Chapter to obtain embedded solutions which lie as graphs above the ring-like domain G in Π which is bounded by Γ and Σ. Instead Turowski’s approach is based on ideas due to Y.-W. Chen [3], and so Courant’s method cannot be used as it stands. We sketch the necessary modifications. By Rr,1 we denote the annulus {w ∈ C : r < |w| < 1} bounded by the two circles Cr and C1 of radius r and 1 respectively which are positively oriented with respect to Rr,1 . Let π be the orthogonal projection of R3 onto the plane Π. We assume that Γ and Σ are positively oriented with respect to G, and that Γ induces the orientation of Γ . Then we set C∗ (Γ, S) := X ∈ H 1,2 (Rr,1 , R3 ) ∩ C 0 (Rr,1 , R3 ) for some r ∈ (0, 1) : X : C1 → Γ and π ◦ X : Cr → Σ are weakly monotonic and respecting the orientations of Γ and Σ . Note that r is not fixed; it is allowed to vary among admissible surfaces in C∗ (Γ, S). The problem of minimizing the Dirichlet integral 1 |∇X|2 du dv D(X) := 2 Rr,1

2.11 Scholia

129

in the class C∗ (Γ, S) can be treated like the Douglas problem for two boundary curves; see e.g. Nitsche [28], pp. 556–566. Detailed proofs of the following results are given in Turowski [1], Kap. 2. The standard condition of cohesion is replaced by the following condition: Definition 1. A sequence {Xn } in C∗ (Γ, S) with parameter domains Rrn ,1 is said to fulfil the strong condition of cohesion if there is an α > 0 such that diam fn (γ) ≥ α for all n ∈ N, where fn := π ◦ Xn and γ is an arbitrary closed Jordan curve in Rrn ,1 containing the origin in its interior. Then the following is proved: Theorem 1. If there exists a sequence {Xn } in C∗ (Γ, S) with D(Xn ) → d∗ (Γ, S) :=

inf

C∗ (Γ,S)

D

that satisfies the strong condition of cohesion then D possesses a minimizer in C∗ (Γ, S). Any such minimizer X : Rr,1 → R3 is harmonic and satisfies the conformality relations |Xu |2 = |Xv |2 ,

Xu · Xv = 0.

The boundary values X|C1 map C1 homeomorphically onto Γ . Finally the component X 3 of X = (X 1 , X 2 , X 3 ) can be continued across Cr as a harmonic function, and the radial (= normal) derivative Xρ3 vanishes on Cr . When does there exist a minimizing sequence for D satisfying the strong condition of cohesion? This will be ensured by a suitable Douglas condition which is stated in the next result. Proposition 1. Suppose that d∗ (Γ, S) satisfies the Douglas condition (2)

d∗ (Γ, S) < a(Γ ) + a(Σ)

where a(Γ ) and a(Σ) are the areas enclosed by Γ and Σ respectively. Then D possesses a minimizing sequence satisfying the strong condition of cohesion. The proofs of Theorem 1 and Proposition 1 are given in Turowski [1], pp. 11–25. Theorem 2. The Douglas condition (2) follows from each of the two inequalities (3)

1 2 L (Γ ) < 2a(Σ) + a(Γ ) 2π

and (4)

1 R(Γ )L(Γ ) < 2a(Σ) + a(Γ ) 2

where R(Γ ) is the radius of the smallest ball containing Γ .

130


Proof. Let X ∈ C(Γ ) be area minimizing in C(Γ ). Then f := π ◦ X maps B := {w : |w| < 1} diffeomorphically onto the domain enclosed by Γ (cf. ˜ onto ˜ := f −1 (G); there is a conformal map τ of R Vol. 1, Section 4.9). Set R −1 ∗ some annulus Rr,1 , and so Z := X ◦ τ ∈ C (Γ, S). Let d(Γ ) := inf D = inf A. C(Γ )

C(Γ )

It follows that d(Γ ) = D(X) ≥ D(Z) + a(Σ) ≥ d∗ (Γ, S) + a(Σ). The isoperimetric inequalities (cf. Vol. 1, Section 4.14) yield 1 2 1 (5) d(Γ ) ≤ min L (Γ ), R(Γ )L(Γ ) , 4π 2 and so (2) is a consequence of (3) or of (4).

We note that the Douglas condition (2) also follows from the assumption d(Γ ) < 2a(Σ) + a(Γ ),

(6)

which is more advantageous than (3) or (4) if one can find a better bound for d(Γ ) than (5). Another condition implying (2) can be found in the following way: Let m1 (Γ ) and m2 (Γ ) be the smallest and the largest value of x3 on Γ , and denote by Π1 and Π2 the planes through (0, 0, m1 (Γ )) and (0, 0, m2 (Γ )) respectively that are parallel to Π. Moreover, Γ1 and Γ2 be the curves in Π1 and Π2 respectively parallel to Π, and G1 and G2 be the annuli in Π1 and Π2 parallel and congruent to G. We form two surfaces Y1 and Y2 in C(Γ, S) where Yj consists of Gj and the “collar” Kj between Γj and Γ . Then d∗ (Γ, S) < A(Yj ) = area(G) + area(Kj ) = a(Γ ) − a(Σ) + area(Kj ). Thus (2) is satisfied if (7)

area(Kj ) ≤ 2a(Σ),

and in particular if (8) m2 (Γ ) − m1 (Γ ) · length(Γ ) ≤ 2a(Σ). By the results of Chapter 2 of Vol. 2 each minimal surface X : Rr,1 → R3 of class C∗ (Γ, S) belongs to C m,σ (Rr,1 ∪ C1 , R3 ) if Γ ∈ C m,σ , σ ∈ (0, 1). For the regularity at Cr one cannot simply quote this result as, by definition of C∗ (Γ, S), a certain monotonicity is required along Cr . However, one knows already that X 3 is real analytic on (Rr,1 ∪ Cr ) and satisfies Xρ3 = 0 on Cr .

2.11 Scholia

131

Using the conformality relations and Schwarz’s reflection principle, one obtains that X ∈ C 1,1/2 (Rr,1 ∪ c, R3 ) for every open arc c of Cr that is mapped by f := π ◦ X into a closed linear piece of Σ, and for each w0 ∈ c there are integers ν, μ ≥ 0 and numbers α, β ∈ C \ {0} such that ν/2 1 (w) = α w − w0 + O |w − w0 |(ν+1)/2 Xw as w → w0 . 2 Xw (w) = β(w − w0 )μ/2 + O |w − w0 |(μ+1)/2 Definition 2. Let X ∈ C∗ (Γ, S) be a minimal surface such that X 3 is real analytic on Rr,1 ∪ Cr and satisfies Xρ3 = 0 on Cr , and let p be a vertex of Σ. Then the pre-image f −1 (p) of p under the mapping f := π ◦ X is either a closed arc s or consists of a single point w. In the first case, s is called a stationary arc; in the second w is said to be a vertex point. A regular arc in Cr is a maximal open subarc of Cr that is homeomorphically mapped by f into an open side (= linear piece) of Σ. Consider an arbitrary straight line L in Π given by the equation px1 + qx = c with p2 + q 2 = 1, and let f = π ◦ X be the harmonic mapping associated with the minimal surface X. Then C1 ∩ f −1 (Γ ∩ L) is either empty or equal to b1 or b1 ∪ b2 with b1 , b2 being closed arcs in C1 that can be degenerated to single points, and Cr ∩ f −1 (Σ ∩ L) is empty or = s = s1 ∪ · · · ∪ sl , where the sj are closed arcs, possibly degenerated to single points. Let us call each of the analytic curves in (f |Rr,1 )−1 (L) a level curve of L in Rr,1 . Then one proves (cf. Turowski [2], pp. 244–248): 2

Proposition 2. (i) Each level curve of L in Rr,1 is either an in Rr,1 homotopically nontrivial closed Jordan curve, or it connects two uniquely determined points on Cr ∪ C1 . (ii) Let w0 ∈ Cr be the endpoint of a stationary arc, and suppose that f (B (w0 ) ∩ Cr ) ⊂ L for some with 0 < 1. Then at least one level curve of the straight line L through w0 perpendicular to L starts in w0 . (iii) There are only finitely many stationary arcs for X along Cr , and each vertex point is the common endpoint of two regular arcs. Now we consider the normal N of X ∈ C∗ (Γ, S) with the properties stated above. Using arguments similar to those in Sections 2.1–2.9 one obtains that N can be extended continuously to every point of Cr . On stationary arcs we have N 3 = 0 whereas N is tangent to S on regular arcs. Furthermore the Dirichlet integral of N restricted to Rr,1−δ is finite for any δ ∈ (0, 1 − r). On C1 we have the following situation: Suppose that Γ is a C 1,μ generalized graph, μ ∈ (0, 1). Then X ∈ C 1,μ (Rr,1 ∪ C1 , R3 ) and X has no branch points on C1 . Moreover, N is continuous on Rr,1 , and N 3 ≥ 0 on C1 . If Γ is nonvertical on the open arc s in Γ then N 3 > 0 on f −1 (s) ∩ C1 , in particular, N 3 |C1 > 0 if Γ is a C 1,μ -graph over Γ . Proposition 3. If Γ is a C 1,μ -graph or a C 2,μ -generalized graph then the negative part (N 3 )− of N 3 is of class H21 (Rr,1 ) ∩ C 0 (Rr,1 ).

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Definition 3. X is called monotonely freely stable if |∇λ|2 + 2λ2 EK du dv ≥ 0 Rr,1

holds for every λ ∈ Cc∞ (B \ S), where the “exceptional set” S consists of the vertex points, the stationary arcs, the branch points, and the points of regular arcs with fθ = 0; E := |Xu |2 , and K is the Gauss curvature of X. It follows that every minimizer of D (and so of A) in C∗ (Γ, S) is monotonely freely stable. By using the ideas of Sections 2.6 and 2.9 one obtains Theorem 3. (i) Let Γ be a C 1,μ -graph or a C 2,μ -generalized graph, μ ∈ (0, 1), and suppose that X : Rr,1 → R3 with X ∈ C∗ (Γ, S) is monotonely freely stable. Then N 3 > 0 on Rr,1 , and f := π ◦ X is a diffeomorphism of Rr,1 onto the domain G with ∂G = Σ ∪ Γ . The pre-image of each open side of Σ under f = π ◦X is a regular arc. Moreover X has no branch points on C1 , on regular arcs, and on stationary arcs which are mapped by f onto intruding vertices of the polygon Σ. Finally the Jacobian Jf is positive on regular arcs in Cr and on f −1 (s) ⊂ C1 where s denotes open arcs in Γ over which Γ lies as a graph. (ii) The surface X can be represented as a graph of a scalar function z given by z := X 3 ◦ (f |Rr,1 )−1 which satisfies the minimal surface equation in G and the boundary conditions z = γ on Γ or on Γ \ {q1 , . . . , ql } if Γ has vertical parts over the points qj , and

∂z = 0 on Σ \ {vertices of Σ}. ∂ν Moreover, z is real analytic on (G ∪ Σ) \ {vertices of Σ}, and z is of class C 1 (G \ {q1 , . . . , ql , vertices of Σ}). (iii) Every other minimal surface Y satisfying the same assumptions as X is conformally equivalent to X. The proof of Propositions 2, 3 and Theorem 3 is sketched in Turowski [2], pp. 248–255, and in detail given in [1], pp. 252–303. A cornerstone of the proof is a generalization of Kneser’s reasoning to doubly connected domains that is stated in Bshouty and Hengartner [1], yet essentially without a proof. This is provided in Turowski [1], pp. 92–95. The corresponding Kneser-type result reads as follows: Proposition 4. Let Γ and Σ be two closed Jordan curves in the x1 , x2 -plane R2 such that Σ lies in the interior of Γ . Denote by G1 the interior domain of the contour Σ, by G2 the exterior of Γ , and by G the annulus-type domain between Σ and Γ . Assume f ∈ C 0 (Rr,1 , R2 ) ∩ C 2 (Rr,1 , R2 ) is harmonic in

2.11 Scholia

133

Rr,1 and lifts the positive orientation from ∂Rr,1 to ∂G. Moreover, suppose that f |Rr,1 is an open mapping satisfying Jf ≥ 0 on Rr,1 . Finally let f |Cr and f |C1 be weakly monotonic mappings of Cr and C1 onto Σ and Γ respectively. Then f |Rr,1 is a diffeomorphism from Rr,1 onto G, and Jf > 0 on Rr,1 . In her papers [1] (pp. 104–131) and [3], G. Turowski derived results about the behaviour of minimizers X ∈ C∗ (Γ, S) along the edges of S which are analogous to properties proved in Sections 2.7–2.9: Theorem 4. (i) There is no edge-creeping along edges of S sitting over intruding vertices (i.e. along “convex edges”). (ii) Let Γ be composed of 2n arcs of monotonicity for x3 , and suppose that Σ has 2n + l protruding (i.e. “concave”) vertices. Then there occurs edgecreeping at least along l protruding vertices. Of Turowski’s other results we quote Theorem 5. (i) If Γ, S is symmetric with respect to a plane or a straight line, so is X. (ii) If Γ, S is invariant under a rotation about the x3 -axis, so is X. (iii) Suppose that E1 , . . . , Ek are symmetry planes for Γ, S perpendicular to the x1 , x2 -plane. Then either none of the protruding vertices of Σ has an edge with edge-creeping, or we have edge-creeping at least over 2k vertices. Thus edge-creeping is excluded if Σ has less than 2k protruding edges.

Chapter 3 Bernstein Theorems and Related Results

In Chapter 2 of Vol. 1 we have already presented a proof of the following striking result of Bernstein [4]. Theorem 1. Suppose u ∈ C 2 (R2 ) is an entire solution of the minimal surface equation. Then u is affine linear, i.e. u(x, y) = ax + by + c with constants a, b, c ∈ R. Bernstein has indeed proved some stronger and more geometric version, namely that a bounded C 2 -function u defined on the whole of R2 , whose graph has nonpositive Gauss curvature K ≤ 0, necessarily has vanishing curvature K ≡ 0 on R2 . From this he was able to derive a Liouville type theorem for a general class of second order elliptic equations which then implies his famous result using a particular substitution. Furthermore we have derived a local sharpening of Bernstein’s Theorem 1 due to E. Heinz, cp. Theorem 2 in Section 2.4 of Vol. 1: Theorem 2. Let u ∈ C 2 (BR (x0 )) be a solution of the minimal surface equation. Then the Gauss curvature K(x0 ) of graph u at (x0 , u(x0 )) fulfills |K(x0 )| ≤

16 . R2

It is the aim of this chapter to extend these results to higher dimensions including also certain types of singular minimal surfaces. However, we note at the outset that the description of Bernstein type results here is far from being complete. Therefore we have added in the Scholia to this chapter a number of beautiful further results in this direction, which however are not proved in this monograph. Some of the proofs rely on methods from geometric measure theory, and we refer to the monograph by Giusti [4] for a lucid presentation of many of the relevant techniques. In Section 3.1 we discuss a very elegant proof of Theorem 1 which is due to Nitsche [2] and combines ideas of H. Lewy, E. Heinz and K. J¨ orgens, namely U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften 341, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11706-0 3,

135

136

3 Bernstein Theorems and Related Results

to invoke a deep connection between the minimal surface equation on the one hand and the Monge–Ampère equation on the other. We then present Moser’s form of Bernstein’s theorem which is a consequence of a Harnack inequality for linear differential equations. Applying the fundamental interior gradient estimate due to Bombieri, de Giorgi, and Miranda [1] one obtains another interesting form of Bernstein’s theorem valid in all dimensions for solution which are bounded on one side by a cone. In Section 3.2 we discuss first and second variation formulae for the area and a related singular integral which will then be introduced. In particular we discuss the notion of stability for n-dimensional surfaces in Rn+k , giving special emphasis to the case of hypersurfaces. Basic geometric identities are introduced in Section 3.3, in particular we give proofs of the important identity of J. Simons [1] on the Laplacian of the second fundamental form, Codazzi’s equation and Jacobi’s field equation which was already considered in Section 2.4 in the case of two-dimensional minimal surfaces (see equation (18)). In Section 3.4 we present some examples of minimal cones which are stable and prove Simons’s celebrated result on the nonexistence of nontrivial stable cones in Rn+1 for n ≤ 6. We also introduce the symmetric and singular minimal surface equations and extend Simons’s result to the case of “α-minimal” cones. Then we discuss (Theorem 3) the important integral curvature estimate for minimal submanifolds in Rn+1 due to Schoen, Simon, and Yau [1]. These estimates immediately imply Bernstein’s theorem up through dimension 5, see Theorem 4. An extension of this technique due to U. Dierkes [12] is developed to prove curvature estimates for α-stable hypersurfaces (Theorem 5). As a consequence one obtains Bernstein theorems for the singular or symmetric minimal surface equation in low dimensions, see Theorem 6 and Corollary 3. Monotonicity and mean value formulae are derived in Section 3.5. They constitute the main tool for proving generalizations of the classical Sobolev inequality for functions defined on submanifolds of Rn+k , cp. Theorem 3. We follow here the method of Michael and Simon [1] and in particular the lecture notes by L. Simon [8]. In Theorem 4 a version of the Michael–Simon inequality for α-minimal surfaces is derived. In Section 3.6 pointwise curvature estimates for minimal and α-minimal hypersurfaces are proved, cp. Theorems 1 and 2. In particular we derive a higher dimensional analogue of Heinz’s inequality for minimal and α-minimal surfaces. These estimates are established by a Moser-type iteration argument “on the surface”. To make this scheme work the Michael–Simon inequalities are used in an essential way. Finally we prove in Theorem 3 a pointwise estimate due to Ecker and Huisken [1] for the product |A|v, where A denotes the second fundamental form, v = 1 + |Du|2 , and M = graph u is a minimal graph. This estimate implies a Bernstein theorem in arbitrary dimension provided the solution u fulfills the growth assumption |Du| = o(|x| + |u(x)|) as |x| → ∞.

3.1 Entire and Exterior Minimal Graphs of Controlled Growth

137

3.1 Entire and Exterior Minimal Graphs of Controlled Growth 3.1.1 J¨ orgens’s Theorem The following result of K. Jörgens [1] implies the preceding Theorem 1. Theorem 1. Suppose u ∈ C 2 (R2 ) solves the Monge–Ampère equation (1)

det D2 u = uxx uyy − u2xy = 1

in R2 .

Then u(x, y) is a quadratic polynomial. Proof. Equation (1) implies that u or −u is a convex function. Without loss of generality, suppose that u is convex and consider the mapping φ : R2 → R2 , (x, y) → φ(x, y) = (ξ(x, y), η(x, y)), defined by ξ(x, y) := x + ux (x, y), η(x, y) := y + uy (x, y), or (ξ, η) = (x, y) + Du(x, y). We claim that φ is a C 1 -diffeomorphism of R2 . Indeed, by definition of φ and because of (1) we have ∂(ξ, η) 1 + uxx uxy = det = 2 + uxx + uyy ≥ 2 det uxy 1 + uyy ∂(x, y) since u is convex. Therefore φ is locally invertible near every point of R2 . Furthermore, since 2 D u is positive definite we obtain x1 − x2 ≥ 0. (2) [Du(x1 , y1 ) − Du(x2 , y2 )] y1 − y2 This yields the estimate 2 ξ(x1 , y1 ) ξ(x2 , y2 ) − η(x1 , y1 ) η(x2 , y2 ) 2 x 1 − x2 + Du(x1 , y1 ) − Du(x2 , y2 ) = y1 − y2 x 1 − x2 2 x1 − x2 [Du(x1 , y1 ) − Du(x2 , y2 )] = +2 y1 − y2 y1 − y2 2 x1 x2 − + |Du(x1 , y1 ) − Du(x2 , y2 )|2 ≥ y1 y2

138


by (2). Therefore φ is injective and, for topological reasons, φ : R2 → R2 is a C 1 -diffeomorphism from R2 onto itself. Let ψ : R2 → R2 , (ξ, η) → ψ(ξ, η) = (x(ξ, η), y(ξ, η)) denote the C 1 -inverse of φ. We compute the Jacobian matrix of ψ, −1 ∂(ξ, η) 1 ∂(x, y) 1 + uyy −uxy = = 1 + uxx ∂(ξ, η) ∂(x, y) 2 + Δu −uxy and introduce the function h : R2 → R2 given by x − ux (x, y) , (ξ, η) → h(ξ, η) := −y + uy (x, y) where we put (x, y) = ψ(ξ, η). By the chain rule we find ∂h ∂h ∂(x, y) = ∂(ξ, η) ∂(x, y) ∂(ξ, η) 1 1 − uxx 1 + uyy −uxy = uxy −1 + uyy −uxy 2 + Δu 1 uyy − uxx −2uxy , = 2uxy uyy − uxx 2 + Δu

−uxy 1 + uxx

where the argument is (x, y) = ψ(ξ, η). Thus the function h(ξ, η) = (h1 (ξ, η), h2 (ξ, η)) satisfies the Cauchy–Riemann equations ∂h2 uyy − uxx ∂h1 = = , ∂ξ ∂η 2 + Δu ∂h1 ∂h2 −2uxy =− = ∂η ∂ξ 2 + Δu and defines an entire holomorphic function on C = ˆ R2 . On the other hand we have for ζ := ξ + iη h (ζ) =

∂h1 ∂h2 ∂h2 ∂h1 dh = +i = −i dζ ∂ξ ∂ξ ∂η ∂η

and 2 2 2 ∂h1 ∂h + |h (ζ)| = ∂ξ ∂ξ 1 2 ∂h1 ∂h2 ∂(h1 , h2 ) ∂h ∂h − = det = ∂ξ ∂η ∂η ∂ξ ∂(ξ, η) 1 Δu − 2 = < 1. ((Δu)2 − 4) = (2 + Δu)2 Δu + 2

2


139

Therefore the entire holomorphic function h : C → C is bounded, and by Liouville’s theorem h (ζ) = const. Since h (ζ) =

uxy uyy − uxx + 2i 2 + Δu 2 + Δu

and |h (ζ)|2 = const =

Δu − 2 < 1, Δu + 2

we obtain Δu = const and then uxx = const, uxy = const and uyy = const, whence u(x, y) = ax2 + bxy + cy 2 + dx + ey + f for suitable constants a, b, c, d, e, f ∈ R.

Proof of Bernstein’s theorem. (See Theorem 1 of the introduction to this chapter.) The minimal surface equation div

Dz 1 + |Dz|2

=0

is equivalent to (compare Vol. 1, Chapter 2) T := (1 + q 2 )r − 2pqs + (1 + p2 )t = 0. Equation (15) in Section 2.2, Vol. 1, or a straight forward computation, show that this implies (3)

1 + q2 W

= x

pq W

and y

1 + p2 W

= y

pq W

,

x

where we have put W = 1 + p2 + q 2 , p = zx , q = zy , r = zxx , s = zxy , t = zyy . Equations (3) yield the existence of a twice differentiable function u = u(x, y) ∈ C 2 (R2 ) with (4)

uxx =

1 + p2 , W

uxy =

pq , W

uyy =

1 + q2 W

and det D2 u = uxx uyy − u2xy = 1. By Theorem 1 we conclude that D2 u is a constant matrix, whence by (4) also p and q must be constant, and so z is an affine linear function.

140


3.1.2 Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser’s Bernstein Theorem Now we consider weak solutions u ∈ H21 (Ω), Ω ⊂ Rn open, of a linear elliptic equation (5)

Lu =

n

Di (aij (x)Dj u) = 0,

i,j=1

in Ω, i.e. we have the integral identity

n

aij Di uDj ϕ dx = 0

Ω i,j=1

for all test functions ϕ ∈ Cc∞ (Ω). Here aij = aij (x) ∈ L∞ (Ω) are measurable and bounded coefficients satisfying the ellipticity bounds λ|ξ|2 ≤

n

aij (x)ξi ξj ≤ Λ|ξ|2

i,j=1

for all ξ ∈ Rn and a.e. x ∈ Ω, where 0 < λ ≤ Λ denote fixed real numbers. We recall the Moser–Harnack inequality for positive weak solutions of equation (5), cp. Moser [2] or Gilbarg and Trudinger [1] for a detailed proof. Theorem 2 (Moser–Harnack inequality). Suppose u ∈ H21 (Ω) is a positive solution of equation (5). Then for any compact set Ω ⊂ Ω we have the estimate u ≤ c min u, max Ω

Ω

where c depends only on Ω, Ω and λ. Remarks. (i) max and min stand for the essential maximum (supremum) and minimum (infimum) respectively. Also “u positive” means that the set {x ∈ Ω : u(x) ≤ 0} has zero n-dimensional Lebesgue measure. (ii) An immediate consequence of Theorem 2 is the local H¨ older continuity of weak solutions of (5), see e.g. Gilbarg and Trudinger [1]. (iii) If Ω = B2R (0) ⊂ Rn is a ball of radius 2R about zero and Ω = BR (0) then, by homogeneity, we have c independent of R. For solutions u of (5) we put M (r) := max u(x), |x|=r

m(r) := min u(x) |x|=r

1 (Rn ) ∩ which are continuous functions of r, whenever defined. If u ∈ H2,loc C 0,α (Rn ) is an entire (i.e. globally defined) solution of (5) it follows easily from the maximum principle (which is also a consequence of Theorem 2) that


141

(i) M (r) is increasing, and (ii) m(r) is decreasing for all r ≥ 0. More generally we have for exterior solutions (defined on Rn \ K for some 1 (Rn \ K) ∩ C 0,α (Rn \ K) of equation (5) the compact set K ⊂ Rn ) u ∈ H2,loc following Theorem 3. Suppose that for a nonconstant exterior solution u of (5) and some r0 ≥ 0 we have for all r > r0 that M (r) is increasing and m(r) is decreasing. Then the oscillation ω(r) := M (r) − m(r) tends to infinity at least like a power of r. 1 (Rn )∩ Corollary 1. The oscillation of a nonconstant entire solution u ∈ H2,loc 0,α n C (R ) of (5) grows at least like a power of r.

Proof of Theorem 3. Invoking Theorem 2 we put Ω = {x ∈ Rn : |x| = r} and Ω = {x ∈ Rn : r/2 < |x| < 2r}, where we assume r > 2r0 . By assumption and the maximum principle u assumes its maximum and minimum at the outer boundary ∂B2r (0) of Ω; whence M (2r) − u and u − m(2r) are positive in Ω. Theorem 2 yields the inequalities M (2r) − m(r) ≤ c[M (2r) − M (r)] and M (r) − m(2r) ≤ c[m(r) − m(2r)] for some constant c depending only on n and λ and not on r. Adding these inequalities we obtain for ω(r) = M (r) − m(r) the estimate ω(2r) + ω(r) ≤ c[ω(2r) − ω(r)], or

c+1 ω(r) for all r ≥ 2r0 . c−1 Iterating this relation one finds for ω(2r) ≥

Θ :=

c+1 > 1 and c−1

all ν ∈ N

that ω(2ν r) ≥ Θν ω(r), which (by a standard device) finally leads to ω(r) ω(R) ≥ Θ−1 α · Rα r for all 2r0 ≤ r < R < ∞ and α := log2 Θ > 0.

1 (Rn ) is an entire solution then we find from Proof of Corollary 1. If u ∈ H2,loc the maximum principle that M (r) is increasing and m(r) decreasing for all r ≥ 0. The assertion then follows from Theorem 3.

142


With a similar technique one proves Theorem 4. Suppose that u is a bounded nonconstant solution of (5) in the domain {x ∈ Rn : |x| > r0 } for some r0 > 0. Then the oscillation ω(r) = M (r) − m(r) tends to zero as r → ∞. In particular, lim|x|→∞ u(x) = u∞ exists and is finite. Combining Theorems 3 and 4 we obtain Corollary 2. If u denotes an exterior solution of (5) in the domain {|x| > r0 } for some r0 > 0 then the oscillation of u on {|x| = ro } tends either to infinity or to zero. Proof of Theorem 4. Again we put M (r) = max u, |x|=r

m(r) = min u. |x|=r

From the maximum principle applied to a suitable “annulus” it is clear that M cannot have two relative minima in the interval (r0 , ∞). It is possible that one relative minimum of M occurs at r = r1 ≥ r0 . In this case we consider r > r1 where M must be increasing. Otherwise M is decreasing on (r0 , ∞). Similarly m(r) can have only one maximum. If this occurs at r = R1 then m is decreasing in (R1 , ∞). Otherwise m(·) must be increasing for all r > r0 . Altogether four cases of monotonicity behaviour at infinity of the functions m and M are possible. Suppose first that M is increasing for r > r1 and m is decreasing for r > R1 . By Theorem 3 and the boundedness of u this leads to a contradiction. Hence only the following three cases are possible: (i) M decreasing and m increasing for r > r0 ; (ii) M decreasing and m decreasing for r > R1 ; (iii) M increasing and m increasing for r > r1 . Let (i) hold and consider u in the annulus Ω = {x ∈ Rn : r < |x| < 4r} and on the sphere

Ω = {x ∈ Rn : |x| = 2r},

where r > r0 . Since u takes on its maximum and minimum on the inner sphere {|x| = r} the same argument as in the proof of Theorem 3 yields the relation ω(2r) ≤ with Θ :=

c−1 c+1

c−1 ω(r) = Θω(r) c+1

< 1 and ω(r) = M (r) − m(r).


143

Iterating this inequality we infer ω(2ν r) ≤ Θν ω(r) for all ν ∈ N and r > r0 . We now take arbitrary real numbers r < R and select ν ∈ N such that 2−ν−1 R ≤ r < 2−ν R. Since ω(r) is decreasing we have ω(R) ≤ α ω(2ν r) ≤ Θν ω(r). Now put α := log2 Θ−1 > 0, then Θν = 2−να ≤ ( 2r R ) and we conclude (7)

ω(R) ≤ R−α (2α rα ω(r))

holding for all r, R ∈ (r0 , ∞) with r ≤ R. Clearly (7) implies the assertion of Theorem 4. In case (ii) we can assume that limr→∞ m(r) = A exists and is finite (because u is bounded). Without loss of generality let A = 0, so that u > 0 in the exterior domain {|x| > R1 }. Applying Theorem 2 to the solution u in the annulus n r Ω = x ∈ R : < |x| < 2r 2 and the sphere

Ω = {x ∈ Rn : |x| = r}

we obtain M (r) ≤ cm(r) → 0 as r → ∞. Hence ω(r) = M (r) − m(r) ≤ (c − 1)m(r) → 0 as r → ∞ which is the desired result. Finally, the conclusion in case (iii) follows as in (ii) by considering −u (instead of u). The existence of the limit lim|x|→∞ u(x) follows from ω(r) → 0 as r → ∞ and the monotonicity properties of the functions M and m respectively. Theorem 4 is proved. The proof of Corollary 2 follows from the proof of Theorem 4 and the conclusions in Theorems 3 and 4 respectively. From Theorem 4 we conclude the following result of Moser [2]: Theorem 5. Let u ∈ C 2 (Rn \ B r0 (0)) be an exterior solution of the minimal surface equation n Di u (8) Di = 0. 1 + |Du|2 i=1 Assume that the gradient |Du| is bounded on the set {|x| > r0 }. Then the limit lim Du(x) = A ∈ Rn |x|→∞

exists.

144


This implies Moser’s Bernstein Theorem: Corollary 3. Suppose u ∈ C 2 (Rn ) is an entire solution of the minimal surface equation (8) in Rn and that the gradient |Du| is bounded. Then u is affine linear, i.e. u(x) = ax + b for constants a ∈ Rn , b ∈ R. Proof of Theorem 5. For k = 1, . . . , n put w := Dk u = (8) with respect to xk we obtain n

∂u . ∂xk

Differentiating

Dj [aij (x)Di w] = 0,

i,j=1

pi pj ∂ 2 W δij − = aij (x) = i j ∂p ∂p p=Du(x) W W 3 p=Du(x) and W (p) = 1 + |p|2 . The largest and the smallest eigenvalues of the matrix (aij ) are W −1 (Du) and W −3 (Du) respectively. Under the assumption of boundedness of |Du| the coefficient matrix aij (x) is uniformly elliptic with ellipticity constants where

λ := (1 + |Du|20,Rn )−3/2 > 0 and

Λ := (1 + |Du|20,Rn )−1/2 .

Theorem 5 now follows from Theorem 4.

Corollary 3 follows from Corollary 1 applied to Dk u, k = 1, . . . , n. Later in this chapter we prove a more general version of Corollary 3, allowing a moderate growth of the gradient, cp. Theorem 3 in Section 3.6. Furthermore, Theorem 5 holds without the assumption of boundedness if the dimension n ≤ 7, cp. Theorem 4 of Section 3.4 and the Scholia to this chapter. 3.1.3 The Interior Gradient Estimate and Consequences We recall the following fundamental estimate which was first proved by Finn [3] (n = 2) and Bombieri, de Giorgi, and Miranda [1] (n ≥ 3). For a proof we refer to the monographs by Giusti [4] or Gilbarg and Trudinger [1]. Theorem 6. Let u ∈ C 2 (BR (x0 )) be a solution of the minimal surface equation Du div = 0 in BR (x0 ) ⊂ Rn . 1 + |Du|2 Then there is an absolute constant C = C(n) such that the estimate supBR (x0 ) u − u(x0 ) (9) |Du(x0 )| ≤ C exp C R holds true.

3.2 First and Second Variation Formulae

145

Estimate (9) is sharp as far as the exponential growth is concerned. It is interesting to note that the proof of (9) makes an intensive use of an “analysis upon the graph u” rather than “down in Rn ”. There are several striking consequences of the above a priori bound, and we refer for instance to Gilbarg and Trudinger [1] for a detailed exposition. We mention here the following Bernstein type result which easily follows from (9) and Corollary 3. Theorem 7. Let u ∈ C 2 (Rn ) be an entire solution of the minimal surface equation and suppose that there is a constant K such that (10)

u(x) ≤ K + K|x|

holds for every x ∈ Rn . Then u is affine linear, i.e. u(x) = Ax+b with suitable constants A ∈ Rn , b ∈ R. Proof. From (9) and (10) we infer K + K|x0 | + KR + |u(x0 )| |Du(x0 )| ≤ C exp C R K + K|x0 | + |u(x0 )| . ≤ C exp CK + C R Letting R → ∞ and recalling that x0 ∈ Rn is arbitrary, we conclude that |Du|0,Rn ≤ C exp(CK) which—by Corollary 3—implies the conclusion.

Remark. Of course, instead of (10) the one sided bound (10 )

u(x) ≥ −K − K|x|

could have been assumed in Theorem 7.

3.2 First and Second Variation Formulae In this section we consider smooth (C 1 or C 2 ) n-dimensional, connected, oriented submanifolds M in Rn+k without boundary, and particularly hypersurfaces in Rn+1 , i.e. the case of codimension equal to one. The main topics are the first and second variation formulas for the area integral and a related singular integral which will be introduced later. Only basic knowledge of measure theory is required, since the objects which are considered here are smooth manifolds; however we have used the notion of Hausdorff measure to indicate that the formulas also hold for more general objects such as nrectifiable sets and varifolds. We refer the interested reader to the monograph of Simon [8] for a more comprehensive information on geometric measure theory. Our derivation of the first and second variation formulae for the area integral (Propositions 1 and 2) are also taken from these notes.

146


Notation. In Sections 3.2–3.4 we work with the mean curvature H and mean

curvature vector H which are given by H := nH

and

H := nH

respectively, where H and H denote the mean curvature and the mean curvature vector which were introduced in Chapter 4, Vol. 2. In particular we have for any n-dimensional C 2 -submanifold M ⊂ Rn+k the formulae

H=−

k (divM Nj )Nj j=1

and

ΔM x = H, compare Section 4.3 in Vol. 2. 3.2.1 First and Second Variation of the Area Integral In the sequel we shall make use of the area formula, for a proof of which we refer to the monograph of Federer [1], cp. Corollary 3.2.20, or Hardt [2]. Suppose that M ⊂ Rn+k is an embedded n-dimensional C 1 -submanifold and let f : M → RN , n ≤ N , be a given differentiable one-to-one mapping. The area formula states that the n-dimensional Hausdorff measure (= ndimensional volume) Hn (f (S)) for any Hn -measurable set S ⊂ M is given by

(1) Hn (f (S) = Jf (x) dHn (x), S

where Jf denotes the Jacobian of f defined by (2)

Jf (x) := {det[df ∗ (x) ◦ df (x)]}1/2 .

Here df (x) : Tx M → RN denotes the differential of f at x, cp. Def. 6 in Section 4.3, Vol. 2, and df ∗ (x) : RN → Tx M stands for the adjoint linear map. For example, let u : Ω → R, Ω ⊂ Rn open, be some C 1 -function. Formula (1) applied to the mapping f : Ω → Rn+1 ,

x → f (x) := (x, u(x))

gives the well-known relation for the nonparametric area Hn (f (Ω)) = Hn (graph u|Ω ) = area(graph u|Ω )

Jf (x) dHn (x) = {det( fxi , fxj )}1/2 dHn (x) = Ω Ω

1/2 = {det(Id + (uxi uxj ))} dx = 1 + |Du|2 dx Ω

Ω


147

since dHn (x) = dLn (x) = dx on Rn . More generally, and most important for the application we have in mind, let g denote a nonnegative Hn -measurable function on M ; then we have the area formula

g(f −1 (y)) dHn (y) = g(x)Jf (x) dHn (x). (3) f (A)

A

Now let U ⊂ Rn+k be an open set with U ∩M = ∅ and such that Hn (K ∩M ) < ∞ for each compact set K ⊂ U . Consider M ∈ C 2 and a one-parameter family Φ , −1 < < 1, of C 2 -diffeomorphisms from U into U with the following properties: (i) Φ(, x) := Φ (x) ∈ C 2 ((−1, 1) × U, U ); (ii) Φ0 (x) = x for every x ∈ U ; (iii) for some compact set K ⊂ U there holds Φ |U \K = Id|U \K

for all ∈ (−1, 1).

∂Φ (, x) X(x) := ∂ =0

Put

∂2Φ (, x) Z(x) := 2 ∂ =0

and

to denote the initial velocity and acceleration vector fields of the variation Φ respectively. Then X and Z have compact support K ⊂ U , and by Taylor’s formula we have Φ (x) = x + X(x) +

(4)

2 Z(x) + o(2 ). 2

The first and second variation of the area A = Hn (M ∩ K) are given by the following expressions d δA(M, X) := Hn (Φ (M ∩ K)) d =0 and

d2 n δ A(M, X, Z) = 2 H (Φ (M ∩ K)) . d =0 2

Observe that, by the area formula (1) we have

n (5) H (Φ (M ∩ K)) = JΨ (x) dHn (x), M ∩K

where we have put Ψ := Φ |M ∩U .

148


To derive an expression for the first and second variation, we therefore need to compute the derivatives d2 d (JΨ ) and (JΨ ) . 2 d d =0 =0 To this end we select an orthonormal basis t1 , . . . , tn of the tangent space Tx M and let e1 , . . . , en+k stand for the canonical basis of Rn+k . By (4) we have for every x ∈ M , t ∈ Tx M and ∈ (−1, 1) the relation dΨ (x)(t) = Dt Ψ (x) = DΦ (x)t = t + Dt X(x) +

2 Dt Z(x) + o(2 ), 2

assuming also Dt Ψ ∈ C 2 . Hence with respect to the selected basis the linear map dΨ (x) : Tx M → Rn+k has the matrix ai := ti + Dti X (x) +

2 Dt Z (x) + o(2 ) 2 i

), X = (X 1 , . . . , X n+k ) and i = 1, . . . , n; = 1, . . . , where ti = (t1i , . . . , tn+k i ∗ n + k. The product dΨ ◦ dΨ (x) possesses the matrix (bij )i,j=1,...,n , where n+k n+k ai aj = δij + (ti Dtj X + tj Dti X ) bij := =1

=1

n+k 1 2 (t Dt Z + tj Dti Z + 2Dti X Dtj X ) + 2 i j =1

2

+ o( ). Using the formula det(Id + A + 2 B) = 1 + trace A 1 1 2 2 2 + trace B + (trace A) − trace(A ) + O(3 ) 2 2 we infer (cp. Section 4.3, Vol. 2, in particular Definitions 6 and 7). (JΨ )2 = det bij = 1 + 2 divM X + 2 divM Z +

n

|Dti X|2 + 2(divM X)2

i=1 n

1 − (ti Dtj X + tj Dti X)2 2 i,j=1 + o(2 ).


Denoting the normal part of Dti X by (Dti X)⊥ = Dti X − we find (JΨ )2 = 1 + 2 divM X + 2 divM Z + 2(divM X)2 + −

n

n

n

j=1 tj (Dti X

149

· tj )

|(Dti X)⊥ |2

i=1

(ti Dtj X)(tj Dti X)

i,j=1

+ o(2 ). Moreover, since

√ 1+x=1+

x 2

−

x2 8

1 + A + 2 B = 1 +

+ O(x3 ) it follows for small ||

A 2 + (B − A2 /4) + O(3 ), 2 2

whence 2 (divM Z + (divM X)2 2 n n + |(Dti X)⊥ |2 − (ti Dtj X)(tj Dti X)) + o(2 ).

JΨ = 1 + divM X +

i=1

i,j=1

Now we obtain from (5) the formulas for the first and second variation of the area:

d = divM X dHn (x), (6) δA(M, X) = Hn (Φ (M ∩ K)) d M =0 and (7)

d2 n δ A(M, X, Z) = 2 H (Φ (M ∩ K)) d =0

n 2 divM Z + (divM X) + |(Dti X)⊥ |2 = 2

M n

−

i=1

(ti Dtj X)(tj Dti X) dHn (x).

i,j=1

This motivates the following Definition 1. Let U ⊂ Rn+k be open. An n-dimensional C 1 -submanifold M ⊂ Rn+k is stationary (with respect to the area integral) in U if Hn (M ∩ K) < ∞ for each compact set K ⊂ U and

(8) divM X dHn (x) = 0 for all X ∈ Cc1 (U, Rn+k ). M

150


Proposition 1. Let M ⊂ Rn+k be a C 2 -submanifold and U ⊂ Rn+k be an open set such that M ∩ U = ∅, ∂M ∩ U = ∅ and Hn (M ∩ K) < ∞ for each compact subset K ⊂ U . Then M is stationary in U if and only if the

mean curvature vector H = 0 in M ∩ U , i.e. if and only if M is a minimal submanifold. Proof. Suppose M is stationary in U and select a local orthonormal field N1 , . . . , Nk ∈ Tx M ⊥ together with arbitrary functions ξ1 , . . . , ξk ∈ Cc1 (U, R). We compute the expression divM X for the normal field X=

k

ξi N i

i=1

and obtain divM

k i=1

ξi N i

=

k k k (∇M ξi ) · Ni + ξi divM Ni = ξi divM Ni . i=1

i=1

i=1

Relation (8) and the fundamental lemma in the calculus of variations yield

that the mean curvature vector H satisfies

H=−

k (divM Ni )Ni ≡ 0, i=1

i.e. M is minimal (cp. Section 4.3, Vol. 2).

On the other hand, suppose that M has the mean curvature vector H ≡ 0 in U , and let X ∈ Cc1 (U, Rn+k ) be an arbitrary vectorfield. We decompose X = X T + X ⊥ into tangential and normal part along M , i.e. X T ∈ Tx M while k (X · Nj )Nj ∈ (Tx M )⊥ . X⊥ = j=1

One finds divM X ⊥ =

k (X · Nj ) divM Nj = −(X · H) ≡ 0 j=1

on M ∩U . Furthermore the Divergence Theorem of Gauss (Section 4.3, Vol. 2) states that

n n T divM X dH = divM X dH + divM X ⊥ dHn = 0 M

M

M

where we have used that X has compact support in U . Therefore M is stationary in U . Next we derive a simple expression for the second variation δ 2 A(M, X, Z) for a stationary manifold M and normal perturbations X.


151

Proposition 2. Let U ⊂ Rn+k be an open set and suppose M is an ndimensional stationary C 2 -submanifold such that ∂M ∩ U = ∅, M ∩ U = ∅ and Hn (M ∩ K) < ∞ for each compact subset K ⊂ U . Assume that X ∈ Cc1 (U, Rn+k ) with X(x) ∈ Tx M ⊥ for every x ∈ M ∩ U . Then the second variation is given by

n n |(Dti X)⊥ |2 − (X · A(ti , tj ))2 dHn (9) δ 2 A(M, X) = M

i=1

i,j=1

where t1 , . . . , tn denotes an orthonormal basis of Tx M and A = Ax : Tx M × Tx M → Tx M ⊥ stands for the second fundamental form of M . In particular, if k = 1 and X = ξN for some function ξ ∈ Cc1 (U, R), formula (9) reduces to

|∇M ξ|2 − ξ 2 |A|2 dHn , (10) δ 2 A(M, X) = M

where |A|2 :=

n

|A(ti , tj )|2 =

i,j=1

n

| dN (ti ), tj |2

i,j=1

denotes the squared length of the second fundamental form. Remark. In the case n = 2, formula (10) agrees with equation (17) in Section 2.8, Vol. 1. Proof. Since M is stationary and Z has compact support K ⊂ U it follows that M divM Z dHn = 0. Moreover, because of X = X ⊥ one finds divM X = k divM X ⊥ = j=1 (X · Nj ) divM Nj = −X · H ≡ 0 on M ∩ U . Relation (9) now follows from (7) by observing that k (X · N )N t i D tj X = t i D tj = ti

=1 k

(X · N )Dtj N

=1

= X·

k (ti Dtj N )N = −XAx (ti , tj ) =1

and from the definition of the second fundamental form Ax (t, τ ) = −

k dN (t), τ N . =1

Finally, if k = 1 and X = ξN then (Dti X)⊥ = (Dti (ξN ))⊥ = (Dti ξ)N , since Dti N ∈ Tx M for each x ∈ M ; hence (10) follows from (9).

152


Equation (10) motivates the following Definition 2. Let U ⊂ Rn+1 be open and suppose M ⊂ Rn+1 is a C 2 -minimal submanifold such that ∂M ∩ U = ∅, M ∩ U = ∅ and Hn (M ∩ K) < ∞ for each compact subset K ⊂ U . Then M is called stable in U if

{|∇M ξ|2 − ξ 2 |A|2 } dHn ≥ 0 (11) M

for each function ξ ∈ Cc1 (U, R). 3.2.2 First and Second Variation Formulae for Singular Minimal Surfaces Let M be an n-dimensional submanifold of class C 2 which is contained in the open half space Rn × R+ ⊂ Rn+1 , where R+ = {t > 0} and suppose U ⊂ Rn × R+ is open with U ∩ M = ∅, (M \ M ) ∩ U = ∅ and Hn (M ∩ K) < ∞ for each compact set K ⊂ U . In this section we tacitly assume that M and U satisfy this assumption. For α ∈ R we consider the functional

Eα (M ) = |xn+1 |α dHn (x), M

where x = (x1 , . . . , xn+1 ), and compute the first and second variation δEα (M, X) and δ 2 Eα (M, X, Z) respectively, following the procedure outlined in the last subsection. Using the same terminology as before we put d δEα (M, X) = Eα (Φ (M ∩ K)) , d =0 and δ 2 Eα (M, X, Z) =

d2 , E (Φ (M ∩ K)) α 2 d =0

to denote the first and second variation of Eα respectively. Proposition 3. Let M ⊂ Rn × R+ be a C 2 -submanifold, Φ : U → U ⊂ Rn × R+ and put ∂Φ (x) X(x) = (X 1 (x), . . . , X n+1 (x)) = ∈ Cc1 (U, Rn+1 ) ∂ =0 and 1

Z(x) = (Z (x), . . . , Z

n+1

∂ 2 Φ (x)) = (x) ∈ Cc1 (U, Rn+1 ). 2 ∂ =0

Then the first and second variation of Eα are given by

|xn+1 |α {divM X + αX n+1 /xn+1 } dHn (12) δEα (M, X) = M


153

and

(13) δ Eα (M, X, Z) =

|x

2

|

n+1 α

α(α − 1)(xn+1 )−2 (X n+1 )2

M

+ α(xn+1 )−1 Z n+1 + 2α(xn+1 )−1 X n+1 divM X n |(Dti X)⊥ |2 + divM Z + (divM X)2 + i=1

−

n

(ti Dtj X)(tj Dti X) dHn .

i,j=1

Remark. Formally, if α = 0, (12) and (13) reduce to relations (6) and (7) respectively. Proof. The area formula (3) yields

|xn+1 |α dHn (x) Eα (Φ (M ∩ K)) = Φ (M ∩K)

= |Ψn+1 (x)|α JΨ dHn (x), M ∩K

where K ⊂ U is compact, Ψ = Φ |M ∩U , and JΨ denotes the Jacobian of Ψ (x) = (Ψ1 (x), . . . , Ψn+1 (x)). Following the discussion in Section 3.2.1 we recall the relation 2 divM Z + (divM X)2 JΨ = 1 + divM X + 2 n n + |(Dti X)⊥ |2 − (ti Dtj X)(tj Dti X) + o(2 ). i=1

i,j=1

Similarly one obtains |Ψn+1 (x)|α = |xn+1 |α 1 + α(xn+1 )−1 X n+1 +

2 [α(α − 1)(xn+1 )−2 (X n+1 )2 + α(xn+1 )−1 Z n+1 ] + o(2 ). 2

Relations (12) and (13) now follow immediately by computing the coefficients of and 2 /2 in the product |Ψn+1 (x)|α JΨ . Remark. The proof also shows that the variational formulas continue to hold for arbitrary C 2 -submanifolds M ⊂ Rn+1 (rather than M ⊂ Rn × R+ ) and U ⊂ Rn+1 open, provided we assume additionally α > 1 or α > 2 respectively.

154


Definition 3. A C 1 -submanifold M ⊂ Rn × R+ is called α-stationary in U ⊂ Rn × R+ , U open, if Eα (M ∩ K) < ∞ for each compact set K ⊂ U and

|xn+1 |α {divM X + α(xn+1 )−1 X n+1 } dHn = 0 (14) M

holds for all vector fields X = (X 1 , . . . , X n+1 ) ∈ Cc1 (U, Rn+1 ). Remark. In case α > 1 this definition extends to arbitrary C 1 -submanifolds M ⊂ Rn+1 and open sets U ⊂ Rn+1 (instead of Rn × R+ ). Proposition 4. Let M ⊂ Rn × R+ be a C 2 -submanifold and U ⊂ Rn × R+ be an open set such that M ∩ U = ∅, ∂M ∩ U = ∅ and Hn (M ∩ K) < ∞ for each compact set K ⊂ U . Then M is α-stationary in U if and only if the mean curvature H of M with respect to the unit normal ν = (ν 1 , . . . , ν n+1 ) is given by H(x) = α(xn+1 )−1 ν n+1 .

(15)

Proof. Take some arbitrary function ξ ∈ Cc1 (U, R) and put X = ξ · ν. Then divM X = ξ · divM ν = −ξH and (15) follows from (14) and the fundamental lemma in the calculus of variations. Now let M have mean curvature given by (15) and let X ∈ Cc1 (U, Rn+1 ) be arbitrary. Then we put X = X ⊥ + X T with X ⊥ = (X · ν)ν and X T ∈ Tx M ; whence

divM X ⊥ = (X · ν) divM ν = −H · X = −H · X ⊥ = −α(xn+1 )−1 ν n+1 (X · ν), and |xn+1 |α divM X T = divM [|xn+1 |α X T ] − ∇M |xn+1 |α X T = divM [|xn+1 |α X T ] − α|xn+1 |α−1 (∇M xn+1 · X T ) = divM [|xn+1 |α X T ] − α|xn+1 |α−1 X n+1 + α|xn+1 |α−1 ν n+1 (X · ν) where we have used the identity ∇M xn+1 · X T = [en+1 − (en+1 · ν)ν] · X = X n+1 − ν n+1 (X · ν). The Divergence Theorem of Gauss (Section 4.3, Vol. 2) implies

|xn+1 |α {divM X + α|xn+1 |−1 X n+1 } dHn M

{divM [|xn+1 |α X T ] − α|xn+1 |α−1 X n+1

= M

+ α|xn+1 |α−1 ν n+1 (X · ν) − α|xn+1 |α−1 ν n+1 (X · ν) + α|xn+1 |α−1 X n+1 } dHn = 0 since X has compact support in U .


155

Remark. (i) If α > 1 Proposition 4 extends to C 2 -submanifolds M ⊂ Rn+1 and open sets U ⊂ Rn+1 as follows: M is α-stationary in U ⊂ Rn+1 , U open, if and only if either the mean curvature H of M ∩ U is given by H(x) = α(xn+1 )−1 ν n+1 or M ∩ U ⊂ {xn+1 = 0}. Indeed, if one of the above conditions hold, then (14) follows as in the proof of Proposition 4. On the other hand, if we assume (14) then the fundamental lemma in the calculus of variations implies that |xn+1 |α {H − α(xn+1 )−1 ν n+1 } = 0 on M ∩ U. Thus for all x ∈ M ∩ U with xn+1 = 0 we have (15). Furthermore since H ∈ C 0 (M ) it is clear that ν n+1 (x) → 0 as xn+1 → 0, xn+1 = 0, i.e. the C 2 submanifold M intersects the plane xn+1 = 0 perpendicularly (if it intersects at all). This implies H(x) = α(xn+1 )−1 ν n+1 for all x ∈ M ∩ U . Clearly the other alternative is xn+1 ≡ 0 on M ∩ U . (ii) There are “α-stationary” Lipschitz surfaces which are not of class C 2 and for which the above alternative does not hold. For example, let M ⊂ Rn+1 consist of two vertical parallel halfplanes E1 and E2 and the horizontal strip {xn+1 = 0} between E1 and E2 . Then M is piecewise of class C 2 and satisfies the variational equation |xn+1 |α {H − α(xn+1 )−1 ν n+1 } = 0,

α > 1,

for H -almost all x ∈ M . Further examples of α-stationary Lipschitz surfaces are given by the cones α n+1 {(x1 )2 + · · · + (xn )2 }1/2 . = x n−1 n

Proposition 5. Let U ⊂ Rn ×R+ be open and suppose M ⊂ Rn ×R+ is an ndimensional α-stationary C 2 -submanifold with ∂M ∩ U = ∅ and M ∩ U = ∅. Let X = ξ · ν where ξ ∈ Cc1 (U, R) and ν = (ν 1 , . . . , ν n+1 ) denotes a unit normal to M . Then the second variation δ 2 Eα (M, X) is given by

2 (16) δ Eα (M, X) = |xn+1 |α {|∇M ξ|2 − α−1 H2 ξ 2 − |A|2 ξ 2 } dHn . M

Proof. Observe that

divM X = −X H = −α(xn+1 )−1 ν n+1 ξ = −Hξ and α(α − 1)(xn+1 )−2 (X n+1 )2 + 2α(xn+1 )−1 X n+1 divM X + (divM X)2 = α(α − 1)(xn+1 )−2 (ν n+1 )2 ξ 2 − 2α2 (xn+1 )−2 (ν n+1 )2 ξ 2 + α2 (xn+1 )−2 (ν n+1 )2 ξ 2 = −α(xn+1 )−2 (ν n+1 )2 ξ 2 = −α−1 H2 ξ 2 ,

156


whence (16) follows from (13) and (12), similarly as in the proof of Proposition 2. Remark. The same proof shows that Proposition 5 also holds for α-stationary C 2 -submanifolds M ⊂ Rn+1 and U ⊂ Rn+1 (instead of Rn × R+ ) provided we assume α > 2. This motivates the following Definition 4. Let α > 0. A C 2 -submanifold M ⊂ Rn+1 is α-stable in U ⊂ Rn+1 if either (a) M ∩ U ⊂ Rn × {0} ⊂ Rn+1 or (b) The following two conditions hold: (i) The mean curvature H of M ∩ U fullfils H(x) = α(xn+1 )−1 ν n+1

|x

(ii)

| {α

n+1 α

−1

H + |A| }ξ dH ≤ 2

2

M

2

|xn+1 |α |∇M ξ|2 dHn

n

M

for every function ξ ∈ Cc1 (U, R), where U ⊂ Rn+1 is open, U ∩∂M = ∅, U ∩ M = ∅ and Hn (M ∩ K) < ∞ for each compact subset K ⊂ U .

3.3 Some Geometric Identities In this section we collect some basic geometric relations which are crucial for the subsequent development, such as e.g. J. Simons’s [1] identity and “Jacobi’s field equation”. In addition we recall some preparatory differential geometric material for smooth, embedded hypersurfaces in Rn+1 , e.g. the concept of covariant derivatives of tensors, the Riemann curvature tensor and the Codazzi equations. We assume the reader’s acquaintance with basic Riemannian geometry, although our exposition is mostly selfcontained. For more detailed information we refer to the standard literature on differential geometry, e.g. to the monographs by Gromoll, Klingenberg, and Meyer [1], Jost [18], and K¨ uhnel [2]. Let M ⊂ Rn+1 denote a smooth, embedded hypersurface and T be a (0, s)tensor-field on M , that is, a relation which assigns to each x ∈ M a multilinear mapping Tx : Tx M × Tx M × · · · × Tx M → R. s-fold product of tangent spaces Tx M

In local coordinates we may write T = Tx =

n i1 ,i2 ,...,is =1

Ti1 i2 ···is dxi1 ⊗ · · · ⊗ dxis

3.3 Some Geometric Identities

157

with functions Ti1 ···is = Tx ( ∂x∂i1 , . . . , ∂x∂is ), where dx1 , . . . , dxn ∈ (Tx M )∗ is ∂ ∂ i ∂ i dual to the basis ∂x 1 , . . . , ∂xn of the tangent space Tx M , i.e. dx ( ∂xγ ) = δγ . Then (dxi1 ⊗ · · · ⊗ dxis )i1 ,...,is =1,...,n constitute a basis of the space of all (0, s)-tensors, in particular we have ∂ ∂ (dxi1 ⊗ · · · ⊗ dxis ) = δγi11 · · · δγiss . , . . . , ∂xγ1 ∂xγs Similarly, a (1, s)-tensorfield is a relation which assigns to each x ∈ M a multilinear mapping Tx : Tx M × · · · × Tx M → Tx M which, in local coordinates, is expressed by n

Tx =

Tik1 ···is

k,i1 ,...,is =1

∂ ⊗ dxi1 ⊗ · · · ⊗ dxis . ∂xk

The tensor T is called differentiable, if all functions Ti1 ···is or Tik1 ···is are differentiable. For example, a Riemannian metric g : Tx M × Tx M → R is a (0, 2)-tensorfield on M , that is g=

n

gij dxi ⊗ dxj .

i,j=1

Similarly, the second fundamental form (cp. Chapter 1, Vol. 1 or Chapter 4, Vol. 2) A = II : Tx M × Tx M → R, (t, τ ) → IIx (t, τ ) = − dNx (t), τ is the (0, 2)-tensor given by A = II =

n

hij dxi ⊗ dxj ,

i,j=1

where

A

∂ ∂ , j i ∂x ∂x

= II

∂ ∂ , j i ∂x ∂x

= hij = −

∂2Φ ,N ∂xi ∂xj

and Φ denotes a local parametrization of M near the point x, cp. Section 4.3, Vol. 2. Note that here we do not distinguish between the slightly different notions of the second fundamental forms A and II, cp. Section 4.2, Vol. 2, and prefer the symbol A in the sequel since this is also more commonly used in the relevant literature. Furthermore we have chosen the symbols hij to denote the coefficients of A, instead of bij . The Weingarten map

158


S = −dNx : Tx M → Tx M may be written as a (1, 1)-tensor as follows S=

g ij hjk

i,j,k

which means that S

∂ ∂x

=

∂ ⊗ dxk , ∂xi

g ij hj

i,j

∂ , ∂xi

compare the discussion in Chapter 1, Vol. 1. In particular we obtain for the mean curvature H the formula g ij hij , H = trace S = i,j

which coincides with equation (3) in Section 4.3, Vol. 2 (keeping in mind that we have set H = n1 H). The Riemann curvature tensor R is the (1, 3)-tensor whose components are given by (1)

m := Rijk

that is

R

m m ∂Γjk ∂Γik m − + Γim Γjk − Γj Γik , i ∂x ∂xj

∂ ∂ , j i ∂x ∂x

n ∂ ∂ m = Rijk , k ∂x ∂xm m=1

see also Section 1.3, Vol. 1. Note that here and in the sequel we freely use Einstein’s summation convention, i.e. repeated Latin indices are to be summed from 1 to n, if not determined otherwise. Then we obtain the (0, 4)-tensor (again denoted as Riemann curvature tensor) m Rijk := gm Rijk that is

∂ ∂ ∂ ∂ R = Rijk . , , ∂xi ∂xj ∂xk ∂x

The Gauss equation states that (2)

Rijk = hi hjk − hik hj ;

for a proof we refer to Section 1.3, Vol. 1, in particular equation (25). It is equivalent to (3)

m Rijk = g m Rijk = g m (hi hjk − hik hj ).


159

3.3.1 Covariant Derivatives of Tensor Fields Let T denote a (0, s)- or (1, s)-tensor, and X be a fixed vector field on M . If v1 , . . . , vs are tangential vector fields along M we define the “covariant derivative” of T in the direction of X by (4)

(∇X T )(v1 , . . . , vs ) := ∇X (T (v1 , . . . , vs )) − T (∇X v1 , v2 , . . . , vs ) − T (v1 , ∇X v2 , . . . , vs ) − · · · − T (v1 , . . . , vs−1 , ∇X vs ),

where ∇ is the covariant differentiation on M (Section 1.5, Vol. 1). In local coordinates this can be expressed as follows: j ∂ k ∂v k i j + x v Γik , ∇X v = x ∂xk ∂xj where v = vi

∂ , ∂xi

X = xj

∂ ∂xj

or, equivalently, ∇X v = xk ∇k v j

∂ ∂xj

with ∇k v j =

(5)

∂v j j + v i Γik , ∂xk

that is, ∇k v j denotes the j-th component of the derivative of the vector field v = (v 1 , . . . , v n ) with respect to the k-th variable. ∇X T defined by (4) is again a (0, s)- or (1, s)-tensor respectively, and by ∇T we denote the (0, s + 1)- or (1, s + 1)-tensor given by ∇T (X, v1 , . . . , vs ) := (∇X T )(v1 , . . . , vs ), which in local coordinates for a (0, s + 1)-tensor may be written as ∇T = ∇k Ti1 ···is dxk ⊗ dxi1 ⊗ · · · ⊗ dxis with components (6)

∇k Ti1 ···is = ∇T =

∂ ∂ ∂ , i1 , . . . , is k ∂x ∂x ∂x

∂Ti1 ···is − Γki T − · · · − Γki T , 1 i2 ···is s i1 ···is−1 ∂xk

by (4) and since

∇

∂ ∂xk

∂ ∂xi

= Γki

∂ . ∂x

160


Examples. 1. Let T be the (0, 1)-tensor ∂f dxi =: ∇i f dxi , ∂xi where f : M → R denotes some differentiable function, which via a suitable parametrization of M may be considered as a function defined on an open subset of Rn . The covariant derivative ∇T is the (0, 2)-Hesse tensor of f which in local coordinates is given by T =

D2 f = ∇k ∇i f dxk ⊗ dxi with coefficients

∂2f ∂f − Γik . ∂xk ∂xi ∂x Taking the trace of D2 f we obtain the Laplace–Beltrami-operator (cp. Section 1.5, Vol. 1) 2 ∂ f ∂f . − Γ (7) ΔM f = g ki ∇k ∇i f = g ki ik ∂xk ∂xi ∂x ∇k ∇i f =

2. Let A = hij dxi ⊗ dxj denote the second fundamental tensor; then we want to compute the coefficients ∇k hij of the (0, 3)-tensor ∇A. Using (6) we find ∂hij − Γki hj − Γkj hi . ∇k hij = ∂xk On the other hand, we get ∇i hjk =

∂hjk − Γij hk − Γik hj , ∂xi

hj = Γik hj we obtain the Codazzi equations and since Γki

∇k hij = ∇i hjk ,

(8)

which are of central importance. Note that (8) is equivalent to (21) in Section 1.3 of Vol. 1. 3. Similarly we obtain the covariant derivative of the metric 2-tensor g, ∇k gij =

∂gij − Γki gj − Γkj gi = 0, ∂xk

by equation (6) in Section 1.3 of Vol. 1. Thus we arrive at Ricci’s Lemma: ∇g ≡ 0. 4. Definition (1) for the Riemann curvature tensor and (4) for the covariant derivative imply for any (0, 2)-tensor Tij dxi ⊗ dxj the identity (9)

m m ∇k ∇ Tij = ∇ ∇k Tij − Rki Tmj − Rkj Tim ;

in particular the second covariant derivatives do not commute and the curvature tensor “measures” the degree of non-commutativity.


161

3.3.2 Simons’s Identity and Jacobi’s Field Equation Of crucial importance is the following identity due to J. Simons [1]. Theorem 1 (Simons Identity). Suppose M ⊂ Rn+1 is a smooth embedded hypersurface. Then we have the identity (10)

ΔM hij = ∇i ∇j H + Hhik g k hj − |A|2 hij

where hij are the coefficients of the second fundamental form A = hij dxi ⊗dxj , ΔM is the Laplacian on M (i.e. the Laplace–Beltrami operator), and |A|2 stands for the squared length of A, i.e. |A|2 = g ij g k hik hj = hij hji

with hij := g ik hkj

and H = g k hk = hii is the mean curvature of M . Proof. The Codazzi equation (8) ∇ hij = ∇i hj implies ∇k ∇ hij = ∇k ∇i hj and by (9) we infer m m ∇k ∇i hj = ∇i ∇k hj − Rkij hm − Rki hjm .

The Gauss equation (2) yields m hm = g ms (hks hij − hkj his )hm , Rkij

and m hjm = g ms (hks hi − hk his )hjm , Rki

whence we obtain for the Laplace–Beltrami operator ΔM hij = g k ∇k ∇ hij = g k ∇i ∇k hj − g k g ms (hks hij − hkj his )hm − g k g ms (hks hi − hk his )hjm = g k ∇i ∇j hk − hij |A|2 + Hhis g ms hjm = ∇i ∇j H − hij |A|2 + Hhi g m hjm , where we have used Ricci’s Lemma ∇i g k = 0 and the relation g k g ms hkj his hm − g k g ms hks hi hjm = 0.

162


Proposition 1. Under the hypotheses of Theorem 1 we have the identity (11)

1 ΔM |A|2 = |∇M A|2 + hij ∇i ∇j H + Hhij hj hi − |A|4 , 2

where we have put hij = g ik hkj , hij := g is g jt hst and ∇M A = ∇i hjk dxi ⊗ dxj ⊗ dxk to denote the symmetric (0, 3)-tensor representing the covariant derivative of A with norm |∇M A|2 = g ir g js g kt ∇i hjk ∇r hst . Proof. We contract Simons’s identity ΔM hij = ∇i ∇j H + Hhik g k hj − |A|2 hij with hij := g is g jt hst and get (12)

hij ΔM hij = hij ∇i ∇j H + Hg is g jt hst hik g k hj − hij hij |A|2 = hij ∇i ∇j H + Hhit ht hi − |A|4 .

From |A|2 = hij hji we find j j m ki m j ki m j ΔM |A|2 = g ki ∇k ∇i [hm j hm ] = 2hm ΔM hj + g ∇i hj ∇k hm + g ∇k hj ∇i hm .

By virtue of Ricci’s lemma we have mk ΔM hm ΔM hkj j =g

and mr ∇i hrj , ∇i hm j =g

whence ΔM |A|2 = 2hij ΔM hij + g ki g mr g js ∇i hrj ∇k hsm + g ki g ms g jr ∇k hsj ∇i hrm = 2hij ΔM hij + 2|∇M A|2 , since ∇M A is the symmetric (0, 3)-tensor ∇i hjk dxi ⊗ dxj ⊗ dxk . Finally (11) follows from (12) and the above expression for ΔM |A|2 .

The unit normal field N of a surface M ⊂ Rn+1 is an important geometric quantity, and it is therefore desirable to study its Laplacian ΔM N . To this end we introduce Riemann normal coordinates at a given point x ∈ M , with orthonormal tangent vectors t1 , . . . , tn and such that Γijk = Γijk = 0 for all i, j, k at this particular point. We put Di = Dti to denote the directional

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

163

derivatives and observe that Di Dj = ∇i ∇j at this point. Furthermore by the Gauss representation and the Weingarten formulae (Section 1.3, Vol. 1, equations (1) and (2)) we may write at the point x Di tj = hij N

and

Di N = −hij tj .

Moreover we have in Riemann normal coordinates at x ΔM N = Di Di N, and the Codazzi equations yield Dk hij = Dj hik

for all indices i, j, k = 1, . . . , n.

Keeping this in mind, we compute at the point x: ΔM N = Di Di N = −Di [hij tj ] = −Di [hij ]tj − hij Di tj = −Dj [hii ]tj −

h2ij N

i,j

= −∇M H − |A| N. 2

Since x ∈ M is a fixed, but arbitrary point, we have proved Proposition 2. Let N be a unit normal field of a smooth embedded hypersurface M ⊂ Rn+1 . Then we have the identity (13)

ΔM N + |A|2 N + ∇M H = 0

on M.

Relation (13) is often referred to as Jacobi’s field equation; it is of crucial importance in many applications.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates. Further Bernstein Theorems In this section we consider stationary (H ≡ 0 or H = αν n+1 /xn+1 ) cones in Rn+1 with a point singularity at zero. In particular we describe J. Simons’s result [1] on the nonexistence of nontrivial stable minimal (H ≡ 0) cones in Rn+1 provided n ≤ 6. The nonexistence of such cones on the one hand is of importance for the regularity of sets with minimal perimeter or of area minimizing currents, see the monographs of Giusti [4], Federer [1], Massari and Miranda [1] and L. Simon [8] for a detailed exposition of these regularity results. On the other hand it is a preliminary step for proving a corresponding Bernstein type theorem. Firstly we verify the results for minimal cones (H = 0) following basically the argument given in Schoen, Simon, and Yau [1], or L. Simon [8] cp. also the

164


original paper by J. Simons [1]. Then we extend these methods to α-stationary (H = αν n+1 /xn+1 ) cones following Dierkes [10]. Finally we give proofs of the integral curvature estimates for minimal or αminimal hypersurfaces in Rn+1 due to Schoen, Simon, and Yau [1] and Dierkes [12] respectively. 3.4.1 Stability of Minimal Cones An n-dimensional cone C ⊂ Rn+1 with a singularity at zero is a nonempty set with the following properties: (i) if x ∈ C and t ≥ 0 then also tx ∈ C, (ii) M := C \ {0} is a smooth (at least C 2 ) hypersurface of Rn+1 . Such a cone C is called minimal if M = C \ {0} is stationary for the area integral or, equivalently, if the mean curvature H of M vanishes. Similarly, C is called α-minimal if either M = C \ {0} is α-stationary (that is, if the mean curvature H of M satisfies H = αν n+1 /xn+1 ), or if M ⊂ {xn+1 = 0}, cp. Section 3.2, Propositions 3 and 4. Furthermore, a minimal or α-minimal cone C ⊂ Rn+1 is called stable or α-stable respectively if M = C \ {0} is stable or α-stable, i.e. if

n 2 2 |A| ξ dH ≤ |∇M ξ|2 dHn (1) M

or

(2) M

M

|xn+1 |α (α−1 H2 + |A|2 )ξ 2 dHn ≤

|xn+1 |α |∇M ξ|2 dHn M

respectively holds for all ξ ∈ Cc1 (M, R). Clearly, there cannot exist nontrivial minimal cones C ⊂ Rn+1 which are described by a Lipschitz function u : Rn → R, i.e. C = {(x, u(x)) ∈ Rn × R}, since by elliptic regularity theory every Lipschitz solution of the minimal surface equation is already analytic, and hence C is a plane. The situation is different for α-minimal cones, since the Lipschitz function α [(x1 )2 + · · · + (xn )2 ]1/2 (x) := cα n n−1 for x ∈ Rn , α > 0, n ≥ 2 fulfills the mean curvature type equation α Du = (3) div 2 1 + |Du| u 1 + |Du|2


165

in Rn \ {0}. Equation (3) is equivalent to the fact that the mean curvature H of the graph of cα n, α n Cα n := {(x, cn (x)) ∈ R × R}, satisfies H = αν n+1 /xn+1

for x ∈ Rn+1 \ {0},

α i.e. Cα n is an α-minimal cone. It is also easily seen that the function cn is a n weak solution of (3) on all of R . On the other hand, there is a variety of minimal cones C in Rn in the sense given above if we drop the restriction that C be described as a graph of some function. For example one can readily check that the mean curvature of the cones over products of spheres

S n1 (r1 ) × S n2 (r2 ) × · · · × S nk (rk ) ⊂ S n1 +···+nk +k−1 ⊂ Rn1 +···+nk +k , √ ni ri := k

with the radii

i=1

ni

is equal to zero (for the computations see Chapter 4 of Vol. 2 and also G. Lawlor [1,2]). Restricting to the codimension-one case, we consider cones over products of two spheres, e.g. the cones over 1 1 1 1 1 1 3 4 2 2 √ √ √ √ ×S ⊂S ⊂R , S ×S ⊂ S 5 ⊂ R6 , S 2 2 2 2

or S3

1 √ 2

× S3

1 √ 2

⊂ S 7 ⊂ R8

etc.

Obviously these cones are given by the sets {(x1 , x2 , x3 , x4 ) ∈ R4 : (x1 )2 + (x2 )2 = (x3 )2 + (x4 )2 }, {(x1 , . . . , x6 ) ∈ R6 : (x1 )2 + (x2 )2 + (x3 )2 = (x4 )2 + (x5 )2 + (x6 )2 },

or

{(x , . . . , x ) ∈ R : (x ) + · · · + (x ) = (x ) + · · · + (x ) } 1

8

8

1 2

4 2

5 2

8 2

respectively. More generally the minimal cone C over the product S n1 (r1 ) × S n2 (r2 ) with n1 n2 r1 = and r2 = n1 + n2 n1 + n2 has the form C = {(x1 , . . . , xn1 +n2 +2 ) ∈ Rn1 +n2 +2 : n2 [(x1 )2 + · · · + (xn1 +1 )2 ] = n1 [(xn1 +2 )2 + · · · + (xn1 +n2 +2 )2 ]}.

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For the length of the second fundamental form A of C one obtains the expression |A(x)|2 =

(4)

dim C − 1 n1 + n2 = 2 |x| |x|2

for every x ∈ C \ {0}. Proposition 1 in Section 3.2 states that M = C \ {0} is stationary, i.e. we have

divM X dHn = 0 M

for every vector field X ∈ Cc1 (M, Rn+1 ), n := n1 + n2 + 1 = dim C. Put X(x) := |x|x 2 · ξ 2 where ξ ∈ Cc1 (M, R) is arbitrary. An easy computation, similar to the calculation in Chapter 4 of Vol. 2 shows that 1 ξ2 (∇M ξ · x) 2 . divM X = divM x + 2ξ + ξ x · ∇M |x|2 |x|2 |x|2 By Proposition 2, Section 4.3, Vol. 2 we have divM x =

n+1

n+1

j=1

j=1

ej , ∇M xj =

ej , P ej = trace P = n

where P : Rn+1 → Rn+1 denotes orthogonal projection onto the tangent space Tx M and e1 , . . . , en+1 stands for the canonical basis of Rn+1 . Furthermore ∇M

1 |x|2

−2 −2 = ∇M |x| = |x|3 |x|3

x |x|

T

−2 = |x|3

x⊥ x − |x| |x|

,

whence (5) x · ∇M

1 |x|2

x x⊥ −2 x − = |x|2 |x| |x| |x| ⊥ 2 ! −2 x −2 −2 1 − |(D|x|)⊥ |2 = 1− = = |x|2 |x| |x|2 |x|2

since M = C \ {0} is a cone. Collecting terms and using the stationarity of M we find

(∇M ξ · x) ξ2 n dH = −2 ξ dHn . (n − 2) 2 |x|2 M |x| M By Schwarz’s inequality we conclude n−2 2

M

ξ2 dHn ≤ |x|2

M

ξ2 dHn |x|2

12

|∇M ξ|2 dHn M

12


167

whence (6)

(n − 2)2 4

M

ξ2 dHn ≤ |x|2

|∇M ξ|2 dHn , M

which holds true for any stationary cone M = C \ {0} (and in fact for any minimal hypersurface in Rn+1 , cp. (5)). The stability condition (1) together with relation (6) imply stability of the cones, provided we assume |A|2 ≤

(n − 2)2 1 . 4 |x|2

By (4) we have |A|2 = n−1 |x|2 so we obtain stability of the cones C if the dimension n of C is at least 7. Proposition 1. The cones {(x1 , . . . , xn1 +n2 +2 ) ∈ Rn1 +n2 +2 : n2 [(x1 )2 + · · · + (xn1 +1 )2 ] = n1 [(xn1 +2 )2 + · · · + (xn1 +n2 +2 )2 ]} are stable for the area integral, if n1 + n2 ≥ 6. In particular the 7-dimensional cone over 1 1 × S3 √ , S3 √ 2 2 i.e. the so called “Simons cone” {x ∈ R8 : (x1 )2 + · · · + (x4 )2 = (x5 )2 + · · · + (x8 )2 } is stable. Also the cone over the product 1 5 × S5 S1 6 6 is stable. Remark. Bombieri, de Giorgi, and Giusti [1] showed that the Simons cone minimizes area in R8 in a very general sense. However, the 7-dimensional cone over 1 5 1 5 ×S in R8 , S 6 6 i.e. the set

x ∈ R8 : 5[(x1 )2 + (x2 )2 ] =

8

(xi )2

i=3

is stable, but not area-minimizing in R , as was proved by Simões [1]. A complete classification of cones over products of (more than two) spheres was given by Lawlor [1]. 8

168


Now we check the possibility of carrying over the above arguments to the cones α n Cα n := {(x, cn (x)) ∈ R × R}, where

cα n (x) =

α [(x1 )2 + · · · + (xn )2 ]1/2 , n−1

α > 0, n ≥ 2.

An elementary calculation shows that the second fundamental form A of Cα n has length |A(x)|2 =

(7)

αp −2 α r = , α+p |x|2

where n + x ∈ Cα n \ {0} ⊂ R × R ,

r2 := (x1 )2 + · · · + (xn )2

and

p = (n − 1).

Since the Cα n are α-stationary we have by Proposition 4, Section 3.2

(8) |xn+1 |α divMnα X + α(xn+1 )−1 X n+1 dHn = 0 α Mn

1 n+1 where Mnα := Cα ) ∈ Cc1 (Mnα , Rn+1 ) is arbitrary. n \{0} and X = (X , . . . , X As before we take the vectorfield X(x) := |x|x 2 ξ 2 , ξ ∈ Cc1 (Mnα , R) and infer from (8)

ξ2 ξ n+1 α |x | (n − 2 + α) 2 + 2(x · ∇Mnα ξ) 2 dHn = 0. |x| |x| α Mn

We apply Schwarz’s inequality and obtain 2

2 n−2+α n n+1 α ξ |x | dH ≤ |xn+1 |α |∇Mnα ξ|2 dHn . 2 |x|2 α α Mn Mn By virtue of (2) this implies stability of the cones Cα n if 2 n−2+α 1 ≥ |A|2 + H2 /α. 2 |x|2 From (7) we conclude |A|2 +

α H2 = +α α |x|2

ν n+1 xn+1

2 =

α+p |x|2

p p 2 2 n+1 2 since, along Cα ) = α+p . Hence the nonparametric n , r = α+p |x| and (ν √ α 2 cones Cn are α-stable provided that (α+p−1) ≥ 4(α+p) or (α+p) ≥ 3+ 8, where p := (n − 1).


169

Proposition 2. The nonparametric cones

Cα n =

n + α (x, cα n (x)) ∈ R × R : cn (x) =

are α-stable if α + n ≥ 4 +

√

α [(x1 )2 + · · · + (xn )2 ]1/2 n−1

8. In particular, the two-dimensional cones

√ 2 2 1/2 Cα ) ∈ R2 × R + } 2 = {(x, y, α(x + y ) are stable with respect to the integral Eα provided that α ≥ 2 +

√

8.

Remark 1. The minimizing properties of these cones in suitable classes of BV-functions were investigated by Dierkes [5,8]. Remark 2. Suppose u ∈ C 2 is positive and satisfies the Euler-equation (3) of the non-parametric integral

Eα (u) = uα 1 + |Du|2 dx Ω

i.e.

div

Du

1 + |Du|2

α = u 1 + |Du|2

on Ω ⊂ Rn ,

and assume that α = m for some integer m ∈ N. Consider the rotational symmetric graph Mrot of u, where Mrot = {(x, ω · u(x)) ∈ Rn × Rm+1 : x ∈ Ω and ω ∈ S m (1) ⊂ Rm+1 }. Introducing polar coordinates ω = (ω 1 , . . . , ω m+1 ) on S m one can show that the area A of the symmetric graph Mrot is given by

um 1 + |Du|2 dx A(Mrot ) = const Ω

(for details of the computation see e.g. Dierkes [14] or Barbosa and DoCarmo [5]). Therefore each positive solution u of the equation (3) above with α = m ∈ N has a symmetric graph with zero mean curvature H in the Euclidean space Rn+m+1 , while every nonnegative solution yields a minimal submanifold containing a singular set, namely the set {x ∈ Rn : u(x) = 0} (which of course is empty if u > 0). In particular the cones Cα n with α = m ∈ N correspond under rotation to the minimal cones over the product S

p

p α+p

×S

α

α α+p

⊂ S α+n (1) ⊂ Rα+n+1 ,

170


i.e. {α[(x1 )2 + · · · + (xn )2 ] = p[(xn+1 )2 + · · · + (xn+α+1 )2 ], where p = (n − 1), α = m ∈ N. We give a further argument which proves that the rotational graph Mrot of any solution u of (3) with α = m ∈ N has mean curvature H = 0. To this end let ϕ : Rn × Rm × R → R be given by ϕ(x1 , . . . , xn+m+1 ) = u2 (x1 , . . . , xn ) − (xn+1 )2 − · · · − (xn+m+1 )2 so that Mrot ⊂ Rn+m+1 is described by ϕ = 0. The mean curvature H of Mrot is then given by (cp. Proposition 4, Section 4.3 of Vol. 2, keeping in mind that H is (n + m)-times the mean curvature H) H=−

n+m+1 i=1

Di

Di ϕ |Dϕ|

.

Now Dϕ = 2(uux1 , . . . , uuxn , −xn+1 , . . . , −xn+m+1 ) and |Dϕ| = 2(u2 |Du|2 + (xn+1 )2 + · · · + (xn+m+1 )2 )1/2 , whence |Dϕ| = 2u(1 + |Du|2 )1/2 on Mrot . Therefore we find n i=1

Di

Di ϕ |Dϕ|

=

n

Di

i=1

⎛ = ⎝

n i=1

u xi (|Du|2 +

(xn+1 )2 +···+(xn+m+1 )2 1/2 ) u2

Di Di u(1 + |Du|2 ) −

n

⎞ uxi uxj uxi xj + u−1 |Du|2 ⎠

i,j=1

× (1 + |Du|2 )−3/2 Du |Du|2 = div + u(1 + |Du|2 )3/2 1 + |Du|2

on {ϕ = 0}.


171

Similarly we get on {ϕ = 0} n+m+1 i=n+1

Di

Di ϕ |Dϕ|

n+m+1

xi 2 2 n+1 2 (u |Du| + (x ) + · · · + (xn+m+1 )2 )1/2 i=n+1 n+m+1 (m + 1)u2 ( 1 + |Du|2 )2 − i=n+1 (xi )2 =− u3 (1 + |Du|2 )3/2 =−

Di

=−

(m + 1)u2 (1 + |Du|2 ) − u2 u3 (1 + |Du|2 )3/2

=−

m |Du|2 − , 2 1/2 u(1 + |Du| ) u(1 + |Du|2 )3/2

whence H = − div

Du 1+

|Du|2

+

m u(1 + |Du|2 )1/2

and the rotational graph is a minimal submanifold in Rn+m+1 if u satisfies the equation Du = αu−1 (1 + |Du|2 )−1/2 div 1 + |Du|2 with α = m. We shall henceforth refer to this equation as the symmetric minimal surface equation. Of course there is no such interpretation if α is not an integer although Eα and Eα = Ω uα 1 + |Du|2 dx are well defined. Remark 3. More generally we consider the nonparametric integral

|u|α 1 + |Du|2 dx, Eα (u) = Ω

where Ω ⊂ R is open and u : Ω → R is not necessarily positive. If α > 1 and u ∈ C 1 (Ω) we find easily the following expression for the first variation

DuDϕ α −1 α 2 δEα (u, ϕ) = α|u| u dx. 1 + |Du| ϕ + |u| 1 + |Du|2 Ω n

For ϕ ∈ Cc1 (Ω) and u ∈ C 2 (Ω) an integration by parts yields

α Du |u| α|u|α u−1 1 + |Du|2 ϕ − div ϕ dx. δEα (u, ϕ) = 1 + |Du|2 Ω

172


If u ∈ C 2 (Ω) is a stationary point of Eα (·), α > 1, then the fundamental lemma yields the singular elliptic equation |u|α Du = α|u|α u−1 1 + |Du|2 in Ω, div 1 + |Du|2 which (still assuming u ∈ C 2 , α > 1), is equivalent to Du α −1 2 −1/2 div − αu (1 + |Du| ) = 0. |u| 1 + |Du|2 We call this equation the singular minimal surface equation. Assume that u ∈ C 2 (Ω) is a solution of the singular minimal surface equation. We claim that either u ≡ 0, or u = 0 in Ω and u satisfies the symmetric minimal surface equation Du = αu−1 (1 + |Du|2 )−1/2 in Ω. div 1 + |Du|2 Otherwise there is some point x0 ∈ ∂Ω1 ∩ Ω where Ω1 denotes the open set {x ∈ Ω : u(x) = 0}. Then u(x0 ) = 0 and there is some sequence (xi )i∈N ∈ Ω1 with xi → x0 as i → ∞. Since on Ω1 we have Du div = αu−1 (1 + |Du|2 )−1/2 1 + |Du|2 and u ∈ C 2 (Ω), this implies |Du(xi )| → ∞ as i → ∞, an obvious contradiction, and hence the above alternative follows. However, despite this remark, we observe that there are nonsmooth (i.e. non C 2 ) minimizer u of the integral Eα which are weak solutions of the singular minimal surface equation, i.e. for which δEα (u, ϕ) = 0 holds for all ϕ ∈ Cc1 (Ω) and such that 0 < Ln ({u = 0}) < Ln (Ω). These minimizers of H¨ older class C 0,1/2 were found by Dierkes [5,8]. We recall that the weak Lipschitz solutions cα n of the symmetric minimal surface equation are clearly weak solutions of the singular minimal surface equation. 3.4.2 Nonexistence of Stable Cones Here we address the question whether the dimensional restrictions n ≥ 7 or √ α+n ≥ 4+ 8 for minimal or α-minimal cones respectively are sharp, in other words, are there nontrivial √ n-dimensional stable or α-stable minimal cones for n ≤ 6 or α + n < 4 + 8 respectively? We show in the following that this is not the case.


173

Theorem 1. Let C ⊂ Rn+1 be an n-dimensional stable embedded minimal cone with vertex at 0. If n ≤ 6 then C must be a hyperplane. Theorem 2. Let M = C \ {0} ⊂ Rn × R+ be an α-stable n-dimensional cone √ with vertex at 0, and suppose that α + n < 4 + 8. Then M is a hyperplane P . Furthermore, P must be perpendicular or equal to the coordinate plane {xn+1 = 0}. Proof of Theorem 1. To simplify notation it is convenient to work with an orthonormal frame given by an orthonormal basis t1 , . . . , tn of Tx M , M = C \ {0} and the unit normal ν of M at a certain point x of the cone C. Also we write hij,k for the covariant derivative ∇k hij of the two-tensor hij etc. In this notation the Codazzi equations are simply hij,k = hik,j

for all i, j, k = 1, . . . , n,

and the Simons’s identity (11) in Proposition 1 of Section 3.3.2 may be written as 1 ΔM |A|2 = h2ij,k + hij H,ij + Hhij hj hi − |A|4 , (9) 2 i,j i,j,k

where |A|2 =

i,j,

h2ij

and

|∇M A|2 =

i,j

h2ij,k .

i,j,k

Since the mean curvature of the cone vanishes we get ⎛ ⎞2 n 1 (10) ΔM |A|2 = h2ij,k − ⎝ h2ij ⎠ = |∇M A|2 − |A|4 . 2 i,j=1 i,j,k

To proceed further we need the following Lemma 1. Let M = C \ {0} be a cone. Then we have the estimate (11)

|∇M A|2 ≥ |∇M |A||2 + 2|A|2 /|x|2 .

Remark. For an arbitrary surface M one has the weaker estimate |∇M A|2 ≥ |∇M |A||2 , which follows easily from |∇M |A||2 = |A|−2

n k=1

⎛ ⎝

n

⎞2 hij hij,k ⎠

i,j=1

and Schwarz’s inequality. Hence inequality (11) relies on the fact that M is a cone.

174


Proof of Lemma 1. Let x ∈ M be a point with |A(x)| = 0, and introduce Riemann normal coordinates near x, so that at the point x the vectors t1 , . . . , tn are orthonormal, Γijk = 0, and hij = λi δij is diagonal. We can assume that x and λn = hnn = 0 (since M is a cone!). Furthermore, tn = |x| hij (λx) = λ−1 hij (x)

for all λ > 0

and hence hij,n (x) = −|x|−1 hij (x).

(12)

Then we compute at this point |∇M |A|| = |A| 2

−2

n n k=1

2 λi hii,k

≤

i=1

since |A|2 =

(hii,k )2 i,k

n

λ2i .

i=1

Therefore |∇M A|2 − |∇M |A||2 ≥

h2ij,k −

i,j,k

≥ 2

n−1 i=1

h2ii,k =

h2ij,k

i,j,k i=j

i,k

h2in,i = 2

n−1 i=1

h2ii,n =

n−1 2 2 2 h = |A|2 , |x|2 i=1 ii |x|2

where we have used the Codazzi equations and (12). Finally, (11) holds a.e. on the set {x ∈ M : A(x) = 0} since |A| is Lipschitz continuous and ∇M |A| = 0 for almost all x ∈ {A(x) = 0}. Concluding with the proof of Theorem 1 we infer from (10) and (11) the inequality 1 ΔM |A|2 ≥ |∇M |A||2 + 2|A|2 /|x|2 − |A|4 . 2

(13)

So far we have not used the stability inequality (1). First we replace ξ by ξ|A|, where ξ ∈ Cc1 (M, R) and obtain from (1)

(14) ξ 2 |A|4 dHn M

≤ |∇M (ξ|A|)|2 dHn M

= {|∇M ξ|2 |A|2 + 2(∇M ξ · ∇M |A|)ξ|A| + ξ 2 |∇M |A||2 } dHn . M


175

An integration by parts leads to

ξ|A|(∇M ξ · ∇M |A|) dHn 2 M

=

1 2

1 =− 2

∇M ξ 2 · ∇M |A|2 dHn M

ξ 2 [|A|4 − |∇M |A||2 − 2|x|−2 |A|2 ] dHn

ξ ΔM |A| dH ≤ 2

n

2

M

M

where we have used inequality (13). Now (14) and the last estimate yield

n 2 −2 2 ξ |x| |A| dH ≤ |∇M ξ|2 |A|2 dHn (15) 2 M

M

for all ξ ∈ Cc1 (M, R). We claim that (15) continues to hold for all functions ξ ∈ Cc1 (C, R) provided only that

ξ 2 |x|−2 |A|2 dHn < ∞. (16) M

To see this, replace ξ by ξ · γ where γ is a suitable cut-off function with 1 for |x| ∈ (, −1 ), γ = 0 for |x| < 2 or |x| > 2−1 and 0 ≤ γ ≤ 1, |∇γ (x)| ≤ 3|x|−1 for all x ∈ Rn+1 . Then ξγ is admissible in (15) and the claim follows by letting → 0+ . By the “coarea formula” (see Proposition 1 at the end of this proof) we can write

∞ (17) ϕ(x) dHn = rn−1 ϕ(r · ω) dHn−1 (ω) dr M

0

Σ

for all nonnegative functions ϕ ∈ C 0 (M, R), where Σ := M ∩ S n (1),

S n (1) ⊂ Rn+1

denoting the unit n-sphere.

Furthermore, since M is a cone, 2 x , |A(x)|2 = |x|−2 A |x| and using (17) it is easily verified that 1−n/2−2

ξ(x) := |x|1+ |x|1

with |x|1 := max(1, |x|)

176


is admissible in (15), because (16) is fulfilled. Observing that (1 + )2 |x|2 in M ∩ B1 (0), 2 |∇M ξ| ≤ n 2 2−n−2 (2 − 2 − ) |x| in M ∩ (Rn+1 \ B 1 (0)) and using inequality (15), one finds

n 2 2 |x| |A| dH + 2 |x|2−n−2 |A|2 dHn 2 M ∩B1 (0) M ∩(Rn+1 \B1 (0))

2 ≤ (1 + ) |x|2 |A|2 dHn M ∩B1 (0) 2

n + 2− − |x|2−n−2 |A|2 dHn . 2 M ∩{|x|≥1} For the dimensions n = 2, 3, 4, 5, 6 it is possible to choose > 0 such that (1 + )2 < 2 and (2 − n2 − )2 < 2 and hence |A|2 ≡ 0 follows, which implies that M is a hyperplane. Theorem 1 is proved. In the proof of Theorem 1 we used the following important coarea formula, which we quote without proof. Proposition 3 (Coarea Formula). Let M ⊂ Rn+k be an n-dimensional C 1 -submanifold and f : M → RN , N < n, be a C 1 map. Define the “Jacobian” J ∗ f (x) by J ∗ f (x) = {det(df (x) ◦ df ∗ (x))}1/2 , where df (x) : Tx M → RN denotes the differential of f at x and df ∗ (x) stands for the adjoint map. Suppose that g is a non-negative Hn -measurable function on M . Then the following formula holds true:

n n−N ∗ g(x)J f (x) dH (x) = g(z) dH (z) dLN (y). RN

M

f −1 (y)

For a proof we refer to Federer [1], Hardt [2] or Giaquinta, Modica, and Sou˘cek [1]. We apply the coarea formula as follows: Let f : M → R, M ⊂ Rn+1 , be given by f (x) = |x|, then T x x = ∇M f (x) = |x| |x| since M is a cone. Then J ∗ f (x) = 1, f −1 (r) = {x ∈ M : |x| = r}, and by Proposition 3:

∞ n n−1 ϕ(x) dH (x) = ϕ(z) dH (z) dr M ∩S n (r)

0

M

∞

= 0

rn−1 M ∩S n (1)

ϕ(rω) dHn−1 (ω) dr,


177

since dHn−1 (z) = rn−1 dHn−1 (ω),

z = rω,

which yields formula (17). Proof of Theorem 2. The following estimate is a consequence of Simons’s identity (9) and Lemma 1. Lemma 2. Let M = C \ {0} be a cone. Then we have the estimate 1 ΔM |A|2 ≥ |∇M |A||2 + 2|x|−2 |A|2 + hij H,ij + Hhij hj hi − |A|4 , 2 where H,ij denote the second covariant derivatives of the mean curvature H (and, of course, the summation convention is used here). In the following we assume that M = C \ {0} is an α-stable embedded cone in Rn × R+ , so that the stability inequality (2) is fulfilled. We replace ξ by |A|ξ in (2) and obtain

(18) |xn+1 |α {α−1 H2 |A|2 + |A|4 }ξ 2 dHn M

≤ |xn+1 |α {|A|2 |∇M ξ|2 + |∇M |A||2 ξ 2 M

+ 2ξ|A|(∇M ξ · ∇M |A|)} dHn . An integration by parts yields

(19) |xn+1 |α |A|ξ(∇M ξ∇M |A|) dHn 2 M

1 = |xn+1 |α ∇M ξ 2 ∇M |A|2 dHn 2 M

1 =− |xn+1 |α ξ 2 ΔM |A|2 dHn 2 M

1 − ξ 2 ∇M |xn+1 |α ∇M |A|2 dHn . 2 M Lemma 2, relations (18) and (19) imply the inequality

(20) |xn+1 |α {α−1 H2 |A|2 + 2|x|−2 |A|2 + hij H,ij + Hhij hj hi }ξ 2 dHn M

1 + (∇M |xn+1 |α · ∇M |A|2 )ξ 2 dHn 2 M

≤ |xn+1 |α |A|2 |∇M ξ|2 dHn , M

compare relation (15) for the case α = 0. To compute the additional terms in a convenient way, we select an orthogonal frame t1 , . . . , tn ∈ Tx M , so

178


x that tn = |x| and in addition that tn+1 = tn+1 = · · · = tn+1 1 2 n−1 = 0. Then −1 hin = hni = 0 for all i = 1, . . . , n, and since hij (λx) = λ hij (x), λ > 0, we find hij,n = −|x|−1 hij for all i, j = 1, . . . , n. Therefore we get ⎛ ⎞ 1 1 ⎝ ∇M |xn+1 |α ∇M |A|2 = |xn+1 |α (21) h2ij ⎠ ,k · 2 2 i,j

= α|x

|

n+1 α−1

,k

(tn+1 hij hij,k ) k

i,j,k

⎛

= α|xn+1 |α−1 ⎝

⎞ hij hij,n ⎠ tn+1 n

i,j

= −α|x

| |x|

n+1 α

−2

|A|2 .

Next we choose again a geodesic frame at the point x so that at this particular point we have ∇i ∇j H = Di Dj H where Di = Dti denotes directional derivative with respect to ti . Using this notation we find n+1 ν −1 α H,ij = Di Dj xn+1 = Di −(xn+1 )−2 Dj xn+1 ν n+1 + (xn+1 )−1 Dj ν n+1 = 2(xn+1 )−3 Di xn+1 Dj xn+1 ν n+1 − (xn+1 )−2 (Di Dj xn+1 )ν n+1 − (xn+1 )−2 Dj xn+1 Di ν n+1 − (xn+1 )−2 Di (xn+1 )Dj ν n+1 + (xn+1 )−1 Di Dj ν n+1 = 2(xn+1 )−3 tn+1 tn+1 ν n+1 − (xn+1 )−2 Di (tn+1 )ν n+1 i j j − (xn+1 )−2 tn+1 Di ν n+1 − (xn+1 )−2 tn+1 Dj ν n+1 j i + (xn+1 )−1 Di Dj ν n+1 . By virtue of Di ν = −hi t and Di tj = hij ν (which hold at this particular point x) we obtain tn+1 ν n+1 − (xn+1 )−2 hij (ν n+1 )2 α−1 H,ij = 2(xn+1 )−3 tn+1 i j + (xn+1 )−2 tn+1 hi tn+1 + (xn+1 )−2 tn+1 hj tn+1 j i − (xn+1 )−1 Di [hj tn+1 ]. Using the Codazzi equations (equations (8) in Section 3.3) we obtain for the last term ] = hj,i tn+1 + hj Di tn+1 = hij, tn+1 + hj hi ν n+1 , Di [hj tn+1


179

whence α−1 hij H,ij = 2(xn+1 )−3 hij tn+1 tn+1 ν n+1 i j − (xn+1 )−2 |A|2 (ν n+1 )2 + (xn+1 )−2 hij hi tn+1 tn+1 j + (xn+1 )−2 hij hj tn+1 tn+1 − (xn+1 )−1 hij hij, tn+1 i − (xn+1 )−1 hij hj hi ν n+1 = −|A|2 H2 α−2 + |x|−2 |A|2 − (xn+1 )−1 hij hj hi ν n+1 , and finally hij H,ij = α|x|−2 |A|2 − α−1 H2 |A|2 − Hhij hj hi .

(22)

Combining relations (21) and (22) we obtain in view of inequality (20) at the point x ∈ M |xn+1 |α {α−1 H2 |A|2 + 2|x|−2 |A|2 + hij H,ij + Hhij hj hi } 1 + (∇M |xn+1 |α · ∇M |A|2 ) = 2|xn+1 |α |x|−2 |A|2 , 2 an expression which is independent of the chosen coordinate system and is hence valid for all x ∈ M . Thus we infer from inequality (20) the estimate

(23)

|xn+1 |α |x|−2 |A|2 ξ 2 dHn ≤

2 M

|xn+1 |α |A|2 |∇M ξ|2 dHn M

which holds for all ξ ∈ Cc1 (M, R). Now we can argue similarly as in the proof of Theorem 1. Namely it is readily seen that (23) continues to hold for Lipschitz functions ξ with compact support in C (rather than M = C \ {0}) if only the condition

|xn+1 |α |x|−2 |A|2 ξ 2 dHn < −∞ M

holds true. Hence we may test (23) with 1+α−n/2−2

ξ(x) = |x|1+−α |x|1

,

where |x|1 := max(1, |x|) and > α/2. (Indeed, ξ is admissible since by the x 2 )| it easily follows that coarea formula and |A(x)|2 = |x|−2 |A( |x|

|x|α−2 |A|2 ξ 2 dHn < ∞.) M

180


From the estimate

|∇M ξ|2 ≤

(1 + − α)2 |x|2−2α (2 −

we infer using (23)

2 M ∩B1 (0)

− ) |x|

n 2

2

2−n−2

for x ∈ M ∩ B1 (0), for x ∈ M ∩ {|x| > 1}

|xn+1 |α |A|2 |x|2−2α dHn

+2 M ∩{|x|>1}

|xn+1 |α |A|2 |x|2−n−2 dHn

≤ (1 + − α)2

M ∩B1 (0)

|xn+1 |α |A|2 |x|2−2α dHn

2

n + 2− − |xn+1 |α |A|2 |x|2−n−2 dHn . 2 M ∩{|x|>1} We want to choose n, , α such that >

α , 2

(1 + − α)2 < 2 and

n 2

2

+−2

< 2.

√ √ These requirements √ are equivalent to −1 −√ 2 + α < < 2 + α − 1 and α/2 < < 2 + 2 − n/2. If α + n < 4 + 2 2 then a suitable choice for is = α/2 + δ where

√ n α >0 δ := k−1 2 + 2 − − 2 2 with k ∈ N large. In this case we may conclude that either M = {xn+1 = 0} or |A|2 ≡ 0, i.e. M is a hyperplane P . Since also H ≡ 0 = α(xn+1 )−1 ν n+1 we must have P ⊥{xn+1 = 0}. Theorem 2 is proved. 3.4.3 Integral Curvature Estimates for Minimal and α-Minimal Hypersurfaces. Further Bernstein Theorems In this subsection we prove two integral curvature estimates for stable minimal and α-minimal surfaces respectively, cp. Theorems 3 and 5. These estimates immediately imply corresponding Bernstein type results which are formulated for the minimal surface equation in Theorem 4 and for the symmetric minimal surface equation in Theorem 6. A basic difference between these equations is that the graph M of some solution u of the minimal surface equation is automatically stable, while this is not true for solutions of the singular or symmetric minimal surface equation. Hence the assumption of stability in Theorem 6 is natural and cannot be dropped. The following integral curvature estimate is due to Schoen, Simon, and Yau [1], and we closely follow their arguments.


181

Theorem3. Suppose that M ⊂ Rn+1 is a stable minimal hypersurface and let q ∈ [0, n2 ). Then we have the estimate

|A|4+2q ξ 4+2q dHn ≤ c(n, q)

|∇M ξ|4+2q dHn

M

M

for every nonnegative function ξ ∈ Cc1 (M, R). Proof. Again (cp. Lemma 1) the crucial step of the proof is an improvement over the obvious inequality |∇M A|2 ≥ |∇M |A||2 , this time by using the fact that the mean curvature satisfies H = 0. Lemma 3. Let M ⊂ Rn+1 be a minimal hypersurface. Then the following estimate, 2 |∇M |A||2 , |∇M A|2 ≥ 1 + n holds Hn -almost everywhere on M . Proof. Pick a point x ∈ M with |A(x)| = 0 and introduce Riemann normal coordinates near x, so that at x, t1 , . . . , tn ∈ Tx M are orthonormal, Γijk = 0, and hij = λi δij for all i, j, k ∈ {1, . . . , n}. Then we compute at x, (24)

|∇M A|2 − |∇M |A||2 ≥

h2ij,k −

i,j,k

=

i,k

h2ij,k ≥

i,j,k i=j

= 2

h2ii,k

h2ij,i +

i =j

h2ij,j = 2

i =j

h2ij,j

i =j

h2jj,i

i =j

where we have used the Codazzi equations. On the other hand, (25) |∇M |A||2 ≤ (hii,k )2 = h2ii,k + h2ii,i i,k

i

i =k

=

h2ii,k +

i

i =k

≤

h2ii,k

i =j

+ (n − 1)

i =k

2 hjj,i

h2jj,i = n

i =j

h2jj,i ,

i =j

where we have used that hii,i = −

j =i

hjj,i

since H =

n

hjj = 0.

j=1

Combining (24) and (25) we obtain the assertion of Lemma 3.

182


The Simons’s identity (10) and Lemma 3 now imply for any minimal hypersurface the estimate 2 1 ΔM |A|2 = |∇M A|2 − |A|4 ≥ 1 + |∇M |A||2 − |A|4 . (26) 2 n We test inequality (26) with ξ 2 |A|2q , q ≥ 0, i.e. we multiply (26) with ξ 2 |A|2q and integrate over M :

1 ΔM |A|2 ξ 2 |A|2q dHn 2 M

2 ≥ 1+ |∇M |A||2 ξ 2 |A|2q dHn − |A|4+2q ξ 2 dHn . n M M An integration by parts yields

1 (∇M |A|2 · ∇M ξ)ξ|A|2q dHn + (∇M |A|2 · ∇M |A|2q )ξ 2 dHn 2 M M

2 ≤ |A|4+2q ξ 2 dHn − 1 + |∇M |A||2 |A|2q ξ 2 dHn , n M M which is equivalent to

2 n 2 2q 2 |∇M |A|| |A| ξ dH + 2 (∇M |A| · ∇M ξ)|A|1+2q ξ dHn 1+ n M M

n 2 2q 2 + 2q |∇M |A|| |A| ξ dH ≤ |A|4+2q ξ 2 dHn , M

hence

M

2 |∇M |A||2 |A|2q ξ 2 dHn 1 + + 2q n M

|A|4+2q ξ 2 dHn − 2 (∇M |A| · ∇M ξ)|A|1+2q ξ dHn . ≤

(27)

M

M

Up to now we have not used the stability of M ; in the stability inequality (1) we replace ξ by |A|1+q ξ and ∇M ξ by (1 + q)|A|q ξ∇M |A| + |A|1+q ∇M ξ and get

|A|4+2q ξ 2 dHn

(28) M

{(1 + q)2 |A|2q |∇M |A||2 ξ 2

≤ M

+ |A|2+2q |∇M ξ|2 + 2(1 + q)|A|1+2q ξ(∇M |A| · ∇M ξ)} dHn .


183

Combining inequalities (27) and (28) and using Young’s inequality in the form 2q|A|1+2q ξ(∇M |A| · ∇M ξ) ≤ q 2 ξ 2 |A|2q |∇M |A||2 + −1 |A|2+2q |∇M ξ|2 ,

> 0,

we arrive at

2 2 − (1 + )q |A|2q |∇M |A||2 ξ 2 dHn n M

−1 ≤ (1 + ) |A|2+2q |∇M ξ|2 dHn . M

If 0 ≤ q < case

(29)

2 n,

> 0 may be chosen with

− (1 + )q 2 > 0, whence in this

2 n

|A|2q |∇M |A||2 ξ 2 dHn ≤ c(n, q)

M

|A|2+2q |∇M ξ|2 dHn . M

Returning to the stability inequality (28) we use Schwarz’s inequality in the form |A|1+2q ξ(∇M |A| · ∇M ξ) ≤

1 2 2q 1 ξ |A| |∇M |A||2 + |A|2+2q |∇M ξ|2 2 2

and infer from (28) and (29)

n 4+2q 2 |A| ξ dH ≤ c(n, q) M

|A|2+2q |∇M ξ|2 dHn M

with some constant c depending only on n and q. For each > 0 we conclude from Young’s inequality |A|2+2q |∇M ξ|2 = ξ 2 |A|2+2q whence

|∇M ξ|2 |∇M ξ|4+2q ≤ ξ 2 |A|4+2q + c() , 2 ξ ξ 2+2q

|A|

ξ dH ≤ c(n, q)

4+2q 2

n

M

M ∩{ξ =0}

|∇M ξ|4+2q ξ −2−2q dHn .

On replacing ξ by ξ 2+q , the final estimate

n 4+2q 4+2q |A| ξ dH ≤ c(n, q) (30) M

follows easily for non-negative q
0 BR (x0 ). Furthermore, the geodesic distance r(x) = distM (x0 , x) satisfies |∇M r| ≤ 1. Consider functions ξ(x) := γ(r(x)) where γ : R → R is a Lipschitz function with γ(t) = 1 for t ≤ R/2, γ(t) = 0 for t ≥ R and with γ(t) decreasing linearly for t ∈ (R/2, R). It follows that |∇M ξ| ≤ 2/R a.e. and (30) implies for all q ∈ [0,

BR/2 (x0 )

2 n)

the estimate

|A|4+2q dHn ≤ cR−4−2q Hn (BR (x0 )).

Now we let R → ∞ and use assumption (31). It follows that |A| ≡ 0, i.e. M must be a plane. Alternatively let M ⊂ Rn+1 be properly embedded and consider the extrinsic distance function r(x) := |x − x0 | for x0 , x ∈ M . Suppose that BR (x0 ) ∩ M M and (32)

lim (R−4−2q Hn (BR (x0 ) ∩ M )) = 0

R→∞

holds for some q ∈ [0, R

2 n)

where BR (x0 ) denotes the Euclidean ball {x ∈

: |x − x0 | < R}. Then again it follows that M is a hyperplane. For example, (32) is satisfied if M = ∂E is the boundary of a set E ⊂ Rn+1 with minimal perimeter in Rn+1 and n ≤ 5, cp. the monograph of Giusti [4]. In this case (32) is proved by simple area comparison, namely we get n+1

Hn (BR (x0 ) ∩ M ) ≤ cHn (∂BR (x0 )) ≤ cRn , so that (32) holds for all n with n < 4 + 2 n2 , i.e. n ≤ 5 as claimed. It follows that every smooth boundary of minimal perimeter in Rn+1 must be a hyperplane if n ≤ 5. Now we can prove rigorously Bernstein’s theorem for minimal graphs in Rn+1 , n ≤ 5.


185

Theorem 4. Let u : Rn → R be a C 2 -solution of the minimal surface equation Du (33) div = 0 in Rn 1 + |Du|2 and suppose that n ≤ 5. Then u is an affine linear function. Proof. By virtue of (32) and Theorem 3 it suffices to show (i) Hn (BR (X0 ) ∩ M ) ≤ cRn holds for all R > 0, some X0 ∈ M and some constant c, where BR (X0 ) = {X ∈ Rn+1 : |X − X0 | < R}, and (ii) M = graph u is stable in the sense of inequality (1). Step (i) follows with a simple test function argument, see for example Gilbarg and Trudinger [1], Chapter 16.2. The argument runs as follows: Without loss of generality we assume X0 = (x0 , u(x0 )) = 0 ∈ Rn × R and define the truncated function uR by ⎧ R for x ∈ {y ∈ Rn : u(y) ≥ R}, ⎪ ⎪ ⎨ uR (x) := u(x) for x ∈ {y ∈ Rn : −R < u(y) < R}, ⎪ ⎪ ⎩ −R for x ∈ {y ∈ Rn : u(y) ≤ −R}. Next we multiply the minimal surface equation (33) with ϕ := ηuR , where η : Rn → R is a Lipschitz continuous function satisfying η = 1 for |x| < R, η = 0 for |x| > 2R and |Dη| ≤ R−1 on Rn , and integrate by parts. We thus obtain

DuDϕ dx = 0, 1 + |Du|2 |x| 0, and hence there is a constant c = c(n, α, q) depending only on n, α, q such that

|xn+1 |α T q−1 |∇M T |2 ξ 2 dHn ≤ c |xn+1 |α T q+1 |∇M ξ|2 dHn (46) M

M

holds true. In view of estimate (45) we apply Schwarz’s inequality in the form T q (∇M T · ∇M ξ)ξ ≤

1 2 q−1 1 ξ T |∇M T |2 + T q+1 |∇M ξ|2 2 2

and infer from (46) and (45) the existence of a further constant c = c(n, α, q) such that

|xn+1 |α T 2+q ξ 2 dHn ≤ c(n, α, q) |xn+1 |α T 1+q |∇M ξ|2 dHn . M

M

By Young’s inequality we have T 1+q |∇M ξ|2 = ξ 2 T 1+q

|∇M ξ|2 |∇M ξ|4+2q ≤ ξ 2 T 2+q + c() . 2 ξ ξ 2+2q

If we select = (n, α, q) > 0 small enough we conclude from the last estimate

|∇M ξ|4+2q |xn+1 |α T 2+q ξ 2 dHn ≤ c |xn+1 |α dHn ξ 2+2q M M ∩{ξ =0} from which the final estimate follows if we replace ξ by ξ 2+q . Theorem 5 is proved. We now conclude from Theorem 5 the following Bernstein type result: Corollary 2. Let M ⊂ Rn+1 be a properly embedded α-stable hypersurface of class C 4 and suppose that the asymptotic area growth condition

(47) lim R−4−2q |xn+1 |α dHn = 0 R→∞

BR (X0 )∩M

2 ) with X0 ∈ M and BR (X0 ) = {x ∈ Rn+1 : holds true for some q ∈ [0, n+α |x − X0 | < R}. Then M must be a hyperplane which is perpendicular to the coordinate plane {xn+1 = 0} or M coincides with {xn+1 = 0}.


195

The proof is similar to the one of Corollary 1. We choose a function ξ(x) = ξ(|x − X0 |) = λ(|x − X0 |) ∈ Cc1 (M, R) with λ : R → R given by ⎧ for |t| ≤ R, ⎪ ⎨1 −1 λ(t) := R (2R − t), R ≤ t ≤ 2R, ⎪ ⎩ 0 for t ≥ 2R. Then ξ is admissible in Theorem 5, |∇M ξ| ≤ R−1 and we obtain the estimate

|xn+1 |α T 2+q dHn ≤ c(n, α, q)R−4−2q |xn+1 |α dHn M ∩BR (X0 )

M ∩B2R (X0 )

2 for all R > 0 and non-negative q < n+α . By virtue of the growth condition (47) and letting R → ∞ we infer the alternative xn+1 = 0 or

T = H = |A| = 0 = α(xn+1 )−1 ν n+1 .

Since M is smooth we conclude that either M is a hyperplane perpendicular to {xn+1 = 0}, or M = {xn+1 = 0}. For example, the asymptotic area growth (47) is fulfilled if M = ∂E is the boundary of a set E ⊂ Rn+1 which locally minimizes the weighted “αperimeter”

|xn+1 |α |DϕE | in Rn+1 . Again it follows from an area comparison argument that

|xn+1 |α dHn ≤ cRn+α BR (x0 )∩M

for some constant c independent of R and all R > 1. Hence (47) holds if 2 or n + α < 5.236 . . . . By Corollary 2 it follows that n + α < 4 + 2 n+α every smooth boundary M = ∂E of minimal α-perimeter in Rn+1 must be a hyperplane perpendicular to or identical with the coordinate plane {xn+1 = 0} if we assume n + α < 5.236 . . . . Consider now a nonparametric smooth solution u : Rn → R of the “symmetric minimal surface equation” α Du = (48) div 2 1 + |Du| u 1 + |Du|2 i.e. the Euler equation of the functional

Eα (u) = uα 1 + |Du|2 dx,

for u > 0.

196


There is a variety of entire non-trivial C ∞ -solutions of (48) (see e.g. Keiper [1], Dierkes [5,8]), and there are also globally defined (= entire) weak Lipschitz solutions of the singular equation (48), namely the cones cα n (x) =

α [(x1 )2 + · · · + (xn )2 ]1/2 . n−1

∞ This follows easily from the fact that the functions cα n are classical C n solutions of (48) on R \{0}, together with a suitable “removability” argument. Nevertheless we can prove a Bernstein type result for the stable solutions of (48). Here we call a solution u ∈ C 2 (Rn ) “α-stable” if M := graph u is α-stable in the sense of condition (2), i.e.

√ 1 |xn+1 |α √ H2 + α|A|2 ξ 2 dHn ≤ |xn+1 |α |∇M ξ|2 dHn α M M

holds for all ξ ∈ Cc1 (M, R). Theorem 6. Suppose n + α < 4 + 2

2 n+α

and α > 0. Then there is no entire

α-stable solution u ∈ C (R ) of the symmetric minimal surface equation (48). 2

n

Proof. First of all, note that there can not exist a point x0 ∈ Rn with u(x0 ) = 0, since u ∈ C 2 and the left hand side of (48) is continuous (and hence necessarily |Du(x0 )| = ∞ which is impossible since u ∈ C 2 ). Without loss of generality assume u > 0. By elliptic regularity theory we then conclude that u ∈ C ω (Rn ). In view of Corollary 2 it is sufficient to show that

|xn+1 |α dHn ≤ cRn+α (49) M ∩BR (X0 )

holds for all R 1, some X0 = (x0 , u(x0 )) ∈ M = graph u and some constant c independent of R. This is a standard argument and quite similar to the one in the proof of Theorem 4, so we only sketch it. Put X0 = (0, z0 ), z0 = u(x0 ) > 0 and define uR (x) by ⎧ ⎪ ⎨ z0 + R, if u(x) ≥ z0 + R, if z0 − R < u(x) < z0 + R, uR (x) := u(x), ⎪ ⎩ z0 − R, if u(x) ≤ z0 − R, and ϕ := η · uR , where ⎧ ⎪ ⎨1 η(x) := R−1 (2R − |x|) ⎪ ⎩ 0

if |x| ≤ R, if R < |x| < 2R, if |x| ≥ 2R.


197

Multiplying equation (48) with ϕ and integrating by parts one obtains

|Du|2 (1 + |Du|2 )−1/2 dx {|x| 0 such that Bρ (ξ) ⊂ U, Bρ (ξ) = {x ∈ Rn+k : |x − ξ| < ρ}, and let γ = γ(t) be a C 1 -function on R satisfying γ (t) ≤ 0 for all t,

γ(t) = 1 for t < ρ/2,

γ(t) = 0

for t > ρ.

By choosing the vector field X(x) := h(x)γ(r)(x − ξ) with r := |x − ξ|, we immediately obtain divM X = (∇M h · (x − ξ))γ(r) + h(x) · (∇M γ(r) · (x − ξ)) + h(x)γ(r) divM (x − ξ), and using the Definitions in Section 4.2, Vol. 2, it follows divM (x − ξ) =

n+k

ej ∇M (xj − ξ j ) =

j=1

=

n+k

n+k

ej (ej )T

j=1

ej (P ej ) =

j=1

n+k j,=1

ej pj e =

n+k

p

=1

= trace P = n where P : Rn+k → Tx M stands for the orthogonal projection with matrix (pij )i,j=1,...,n+k . Similarly we find

200


x−ξ ∇M γ(r) = γ (r)∇M |x − ξ| = γ (r) |x − ξ| ⊥ x − ξ x − ξ − , = γ (r) |x − ξ| |x − ξ|

T

denoting again by η ⊥ the orthogonal projection of the vector η onto the normal space (Tx M )⊥ . Thus we get (x − ξ) (x − ξ)⊥ |x − ξ| |x − ξ| 2 (x − ξ)⊥ = rγ (r) − rγ (r) |x − ξ|

∇M γ(r) · (x − ξ) = rγ (r) − rγ (r)

= rγ (r) − rγ (r)|(Dr)⊥ |2 , x−ξ stands for the Euclidean gradient of r = |x − ξ|. Collecting where Dr = |x−ξ| terms and using the divergence theorem (1) we infer

n (2) γ(r)∇M h · (x − ξ) dH + h[rγ (r) − rγ (r)|Dr⊥ |2 ] dHn M ∩U

+n M ∩U

hγ(r) dHn = −

M ∩U

M ∩U

hγ(r)H · (x − ξ) dHn .

Next we take a C -function φ = φ(t) with φ(t) = 1 for t ≤ 1/2, φ(t) = 0 for t ≥ 1 and φ (t) ≤ 0 for all t and put r . γ(r) := φ ρ 1

Then rγ (r) = rρ−1 φ

r r ∂ = −ρ φ , ρ ∂ρ ρ

and we obtain from (2) the identity

r r d n ∇M h · (x − ξ) dH − ρ dHn φ hφ ρ dρ M ∩U ρ M ∩U

r dHn +n hφ ρ M ∩U

r =− hφ H(x − ξ) dHn ρ M ∩U

r d h|Dr⊥ |2 dHn . − ρ φ dρ ρ M ∩U

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

201

Therefore, abbreviating

I(ρ) :=

r r n h dH = h dHn φ φ ρ ρ M ∩U M

and rearranging terms, we have r h|Dr⊥ |2 dHn φ ρ M ∩U r hφ H · (x − ξ) dHn ρ M ∩U

r ∇M h · (x − ξ) dHn . − φ ρ M ∩U

d d nI(ρ) − ρ I(ρ) = −ρ dρ dρ

−

Equivalently, multiplying by −ρ−n−1 , this yields

! r d d −n ρ I(ρ) = ρ−n h|Dr⊥ |2 dHn (3) φ dρ dρ M ρ

r −n−1 dHn . (x − ξ)[hH + ∇M h]φ +ρ ρ M In particular, if we take h = 1 then (3) implies

! r d −n −n d ρ I(ρ) = ρ |Dr⊥ |2 dHn (4) φ dρ dρ M ρ

r −n−1 φ H(x − ξ) dHn +ρ ρ M where we have defined

I(ρ) :=

φ M

r dHn . ρ

We need the following consequence of Fubini’s theorem. Lemma 1. Let μ be a measure on the measure space B, α > 0 and f ∈ L1 (B, μ), f ≥ 0. Denote by At the set At = {x ∈ B : f (x) > t}. Then we have for each 0 ≤ t0 ≤ T0 ≤ ∞ the relation

T0

(5) t0

tα−1 μ(At ) dt = α−1

[min(f, T0 )α − tα 0 ] dμ, At 0

202


in particular

∞

tα−1 μ(At ) dt = α−1

t0

At 0

and

[f α − tα 0 ] dμ

∞

t

α−1

μ(At ) dt = α

−1

f α dμ.

0

A0

Proof. Apply Fubini’s theorem on the product space At0 × [t0 , T0 ], see e.g. Hewitt and Stromberg [1], Chapter 6. Next we integrate (4) between 0 ≤ σ ≤ ρ and obtain

ρ ρ

r d |Dr⊥ |2 dHn dτ (6) τ −n φ τ −n I(τ ) = dτ M τ σ σ

ρ

r + τ −n−1 φ H(x − ξ) dHn dτ. τ M σ The first integral on the right yields after an integration by parts

ρ τ =ρ

r

r |Dr⊥ |2 dHn |Dr⊥ |2 dHn dτ, φ +n τ −n−1 φ τ −n τ τ τ =σ σ M M and if we let φ tend to the characteristic function χ(−∞,1) in an appropriate way we obtain, putting Bτ = Bτ (ξ), that

ρ τ =ρ n n −n ⊥ 2 −n−1 ⊥ 2 |Dr | dH +n τ |Dr | dH dτ τ τ =σ

M ∩Bτ

= τ −n

M ∩Bτ

ρ |Dr⊥ |2 dHn +

= M ∩(Bρ −Bσ )

M ∩Bτ

σ

σ

n 1 1 1 , |Dr⊥ |2 min − n dHn r σ ρ M ∩Bρ

r−n |Dr⊥ |2 dHn

where we have used Lemma 1 with the choice f (x) = |x − ξ|−1 and μ := |Dr⊥ |2 Hn M i.e.

μ(Aτ −1 ) = |Dr⊥ |2 dHn ,

M ∩Bτ

whence

ρ

τ −n−1 μ(Aτ −1 ) dτ = − σ

1/ρ

tn−1 μ(At ) dt 1/σ

1/σ

tn−1 μ(At ) dt

= 1/ρ

1 = n

M ∩A1/ρ

⊥ 2

|Dr |

min

1 1 , r σ

n

1 − n dHn ρ


203

as required. Similarly we find for the second integral in (6), after letting φ tend to the characteristic function of the interval (−∞, 1) in an appropriate way,

ρ τ −n−1 H · (x − ξ) dHn dτ σ

M ∩Bτ

ρ

=

τ −n−1 μ(A1/τ ) dτ

σ

1/σ

tn−1 μ(At ) dt

= 1/ρ

n 1 1 1 , min − n H · (x − ξ) dHn r σ ρ M ∩A1/ρ n

1 1 1 1 , = H · (x − ξ) min − n dHn . n M ∩Bρ r σ ρ

1 = n

Collecting terms we infer from (6) the important monotonicity formula (7)

ρ−n Hn (M ∩ Bρ (ξ)) − σ −n Hn (M ∩ Bσ (ξ))

r−n |Dr⊥ |2 dHn = M ∩(Bρ −Bρ )

1 + n

M ∩Bρ

H · (x − ξ) min

1 1 , r σ

n

1 − n ρ

dHn

for all 0 < σ ≤ ρ ≤ ρ0 , where ρ0 is such that Bρ0 (ξ) ⊂ U and Bρ = Bρ (ξ) etc. Thus we have proved Theorem 1 (Monotonicity Formula). Let M ⊂ Rn+k be an n-dimensional C 2 -submanifold. Then for any point ξ ∈ U and balls Bσ (ξ) ⊂ Bρ (ξ) ⊂ Bρ0 (ξ) ⊂ U the identity (7) is valid. In particular, if M is minimal we have

r−n |Dr⊥ |2 dHn (8) σ −n Hn (M ∩ Bσ ) = ρ−n Hn (M ∩ Bρ (ξ)) − M ∩(Bρ \Bσ )

for all 0 < σ ≤ ρ ≤ ρ0 . Furthermore, if we choose ξ ∈ M we obtain from (8) the estimate (9)

Hn (M ∩ Bρ (ξ)) ≥ ωn ρn

for all ρ ≤ ρ0 .

Proof. Equation (9) follows immediately from (8) since ξ ∈ M and M is a smooth surface, in particular Hn (M ∩ Bσ (ξ)) = 1. σ→0 ωn σ n lim

204


Next we turn our attention to equation (3):

r ! d d τ −n I(τ ) = τ −n h|Dr⊥ |2 dHn φ dτ dτ M τ

r dHn + τ −n−1 (x − ξ) · [Hh + ∇M h]φ τ M

where r = |x − ξ| and

φ

I(τ ) = M

r τ

h dHn .

Here we proceed similarly as above, i.e. we integrate between 0 < σ ≤ ρ ≤ ρ0 and let φ tend to the characteristic function χ(−∞,1) in a monotonic way. Invoking Lemma 1 we obtain as before

n −n −n (10) h dH − σ h dHn ρ M ∩Bρ

M ∩Bσ

=

M ∩(Bρ −Bσ )

ρ

+

τ −n−1

hr−n |Dr⊥ |2 dHn M ∩Bτ

σ

(x − ξ)[Hh + ∇M h] dHn dτ.

A first consequence of (10) is the estimate

(11) σ −n h dHn M ∩Bσ −n

≤ρ

ρ

n

h dH + M ∩Bρ

τ σ

−n

M ∩Bτ

[h|H| + |∇M h]|] dHn dτ,

which will later be used in the proof of the Michael–Simon inequality. Secondly we note that 1 (x − ξ) · ∇M h = (x − ξ)T · ∇M h = − ∇M [τ 2 − |x − ξ|2 ] · ∇M h, 2 and an integration by parts yields the relation

(12) (x − ξ) · ∇M h dHn M ∩Bτ

1 =− ∇M [τ 2 − |x − ξ|2 ] · ∇M h dHn 2 M ∩Bτ

1 = [τ 2 − |x − ξ|2 ]ΔM h dHn . 2 M ∩Bτ

Hence, applying Lemma 1 in a similar way as above and using identity (12) we obtain from (10) the following mean value formula


(13)

ρ−n

M ∩Bρ

h dHn − σ −n

=

M ∩(Bρ −Bσ )

205

h dHn M ∩Bσ

hr−n |Dr⊥ |2 dHn

n 1 1 1 , (x − ξ) · H min − n dHn r σ ρ M ∩Bρ n

1 1 1 1 2 2 , + [ρ − |x − ξ| ]ΔM h min − n dHn . 2n M ∩Bρ r σ ρ +

1 n

Thus we have proved Theorem 2 (Mean value formula). Let M ⊂ Rn+k be an n-dimensional C 2 -submanifold and U ⊂ Rn+k be an open set with U ∩ M = ∅, U ∩ ∂M = ∅. Fix some point ξ ∈ U and suppose h ∈ C 2 (U ) is arbitrary. Then the identity (13) is valid for all 0 < σ ≤ ρ ≤ ρ0 , where ρ0 is such that Bρ0 (ξ) ⊂ U . Furthermore, if M is minimal and h is subharmonic on M , i.e. ΔM h ≥ 0, then we have the estimate

h dHn ≤ ρ−n h dHn − hr−n |Dr⊥ |2 dHn σ −n M ∩Bσ

M ∩Bρ

M ∩(Bρ −Bσ )

for all 0 < σ ≤ ρ ≤ ρ0 . Finally, if we assume additionally that ξ ∈ M and h is nonnegative we conclude the mean value inequality

1 h dHn h(ξ) ≤ ωn ρn M ∩Bρ (ξ) for all ρ ∈ (0, ρ0 ). Now we proceed to derive two auxiliary results, the first one of which is an easy calculus lemma. Lemma 2. Let f, g : (0, ∞) → R+ be bounded and nondecreasing functions such that the following inequality is satisfied

ρ (14) σ −n f (σ) ≤ ρ−n f (ρ) + τ −n g(τ ) dτ 0

for all 0 < σ < ρ < ∞ and some n ≥ 2. Define ρ0 := 2(f (∞))1/n and suppose that sup (ρ−n f (ρ)) ≥ 1. (0,ρ0 )

Then there exists a number ρ ∈ (0, ρ0 ) such that (15)

f (5ρ) ≤

1 n · 5 ρ0 g(ρ). 2

206


The proof follows by contradiction. Assume that (15) were not correct for all ρ ∈ (0, ρ0 ). Then (14) would imply the inequality

2 · 5−n ρ0 −n f (ρ ) + ρ f (5ρ) dρ sup (σ −n f (σ)) ≤ ρ−n 0 0 ρ0 0 τ } to denote the set {x ∈ M : h(x) > τ } etc., Hn (M ∩ {h > t + })

n 1 dH ≤ ≤ {h−t>}

≤5

n

ωn−1

{γ(h−t)≥1}

γ(h − t) dH

n

γ(h − t) dHn

n1

M

[γ(h − t)|H| + γ (h − t)|∇M h|] dHn

·

M

≤5

ωn−1

n

n1

dH

[γ(h − t)|H| + γ (h − t)|∇M h|] dHn .

n

M ∩{h>t}

M 1

Multiplying this inequality by (t + ) n−1 and observing (t + )

1 n−1

ωn−1

n1

dH

n

M ∩{h>t}

≤

ωn−1

n1

(h + )

n n−1

dH

n

M ∩{h>t}

we obtain 1

(t + ) n−1 Hn (M ∩ {h > t + }) ≤5

ωn−1

n

n1

(h + )

n n−1

dH

n

M ∩{h>t}

[γ(h − t)|H| + γ (h − t)|∇M h|] dHn

· M

≤5

n

·

ωn−1

d − dt

(h + )

n n−1

dH

n

n1

M ∩supp h

γ(h − t)|∇M h| dH +

n

M

{h>t}

|H| dH

n

.

Integrating this inequality over t ∈ (0, ∞) and using Lemma 1 with α = μ := Hn M , t0 = and T0 = ∞ yields the estimate

n n−1 ,

210


∞

1

(t + ) n−1 Hn (M ∩ {h > t + }) dt

0

=

n−1 n

n

M ∩{h>}

n

[h n−1 − n−1 ] dHn

≤ 5n ωn−1

n

(h + ) n−1 dHn

n1

M ∩supp h

·

|∇M h| dH + n

∞

0

M

= 5n ωn−1

M ∩{h>t} n

(h + ) n−1 dHn

|H| dH dt n

n1

M ∩supp h

[|∇M h| + h|H|] dHn , M

where we have again used Lemma 1 with α = 1, μ := |H|Hn M and t0 = 0, T0 = ∞. Finally Theorem 3 follows by letting tend to zero. Although the Michael–Simon inequality (16) can be applied to α-stationary hypersurfaces with bounded mean curvature H = α(xn+1 )−1 ν n+1 it turns out that a somewhat different form of (16) is more appropriate for later applications. We have the following variant of Theorem 3. Theorem 4. Let M ⊂ Rn ×R+ be an α-stationary C 1 -hypersurface in an open set U ⊂ Rn × R+ with U ∩ ∂M = ∅. Furthermore let h ∈ Cc1 (U ) have compact support supp h contained in the closed half space H+ = {x ∈ Rn+1 : xn+1 ≥ } for some > 0. Then we have the inequality n

|xn+1 |α h n−1 dHn

(21)

n−1 n

≤ c(n)− n

α

|xn+1 |α |∇M h| dHn

M

M

provided that supp h fulfills the smallness assumption

−α−n

ωn−1

M ∩supp h

|x

| dH

n+1 α

n

n1

≤

1 . 10α

The proof of Theorem 4 is analogous to that of Theorem 3; so we omit some of the details and refer to the calculations above. The crucial step is again a suitable form of the monotonicity formula. Lemma 4 (Monotonicity inequality). Let M ⊂ Rn × R+ be an αstationary hypersurface of class C 1 and U ⊂ Rn × R+ be open such that U ∩ M = ∅, U ∩ ∂M = ∅. For any non-negative function h ∈ Cc1 (U ) and balls Bσ (ξ) ⊂ Bρ (ξ) ⊂ U we have the inequality


(22)

σ −n

211

|xn+1 |α h dHn

M ∩Bσ (ξ)

≤ ρ−n

M ∩Bρ (ξ)

ρ

+

τ

−n−1

|xn+1 |α h dHn

|x

M ∩Bτ (ξ)

σ

| r[|∇M h| + α(x

n+1 α

n+1 −1

)

h] dH

n

dτ

where r := |x − ξ|. Proof. Since M is α-stationary in U we have for any vector field X(x) = (X 1 (x), . . . , X n+1 (x)) ∈ Cc1 (U, Rn+1 ) the variational identity

|xn+1 |α divM X + α(xn+1 )−1 X n+1 dHn = 0.

(23) M

We take the vector field X(x) = h(x)γ(r)(x − ξ) and obtain as before divM X = γ(r)∇M h · (x − ξ) + hrγ (r)(1 − |Dr⊥ |2 ) + nhγ(r). If we substitute this relation into (23) we get

(24)

|x

n M

|xn+1 |α rγ (r)h[1 − |Dr⊥ |2 ] dHn

| hγ dH + n

n+1 α

M

|xn+1 |α γ∇M h · (x − ξ) dHn

+ M

|xn+1 |α−1 hγ(xn+1 − ξ n+1 ) dHn = 0.

+α M

Consider now a nonincreasing C 1 (R)-function φ(t) with φ(t) =

1 for t ≤ 12 , 0 for t ≥ 1

and of course φ (t) ≤ 0 for all t ∈ R. For ρ > 0 take γ(r) := φ( ρr ), and hence in particular

r r ∂ = −ρ φ ≤0 φ ρ ∂ρ ρ

−1

rγ (r) = rρ

212


and substitute this into (24). We then have

r d r n n+1 α dH − ρ dHn n |x | hφ |x | hφ ρ dρ ρ M M

r ∇M h · (x − ξ) dHn =− |xn+1 |α φ ρ M

r d |Dr⊥ |2 dHn −ρ |xn+1 |α hφ dρ M ρ

r n+1 α−1 (xn+1 − ξ n+1 ) dHn . −α |x | hφ ρ M

n+1 α

If we multiply this identity by −ρ−n−1 we obtain

d r ρ−n dHn |xn+1 |α hφ dρ ρ M

r −n d n+1 α |Dr⊥ |2 dHn |x | hφ =ρ dρ M ρ

r −n−1 n+1 α ∇M h · (x − ξ) dHn +ρ |x | φ ρ M

r (xn+1 − ξ n+1 ) dHn . + αρ−n−1 |xn+1 |α−1 hφ ρ M Letting φ tend to the characteristic function of the interval (−∞, 1) in an appropriate way and using Lemma 1 we obtain similarly as in the proof of Theorem 1 the monotonicity formula (in an almost everywhere sense) d dρ

ρ

−n

M ∩Bρ

d = dρ

|x

| h dH

n+1 α

n

|xn+1 |α hr−n |Dr⊥ |2 dH

M ∩Bρ −n−1

n

+ρ

M ∩Bρ −n−1

|xn+1 |α ∇M h · (x − ξ) dHn

+ αρ

M ∩Bρ

|xn+1 |α−1 h(xn+1 − ξ n+1 ) dHn

where Bρ = Bρ (ξ).


213

Finally we integrate from σ to ρ and obtain

σ −n |xn+1 |α h dHn M ∩Bσ

≤ ρ−n

M ∩Bρ

ρ

+

τ

|xn+1 |α h dHn

−n−1

σ

M ∩Bτ

|xn+1 |α r(|∇M h| + α(xn+1 )−1 h) dHn dτ,

which is the desired estimate (22). The following modification of Lemma 2 is useful

Lemma 5. Let f, g be bounded, nonnegative and nondecreasing functions defined on the interval (0, ∞) and suppose that sup (σ −n f (σ)) ≥ 1,

(25)

0 0 and introduce the regularized function fδ := (|A|2 + δ)p/2 = |A|pδ , where we have put |A|δ := (|A|2 + δ)1/2 . Then ΔM |A|2δ = ΔM |A|2 = 2|∇M |A|δ |2 + 2|A|δ ΔM |A|δ 2 2 |∇M |A||2 − 2|A|4 ≥ 2 1 + |∇M |A|δ |2 − 2|A|4 ≥ 2 1+ n n and in particular |A|δ ΔM |A|δ ≥ −|A|4 ≥ −|A|2 |A|2δ . This implies ΔM fδ = ΔM |A|pδ = divM (p|A|p−1 ∇M |A|δ ) δ |∇M |A|δ |2 + p|A|p−1 ΔM |A|δ ≥ p|A|p−2 |A|δ ΔM |A|δ = p(p − 1)|A|p−2 δ δ δ ≥ −p|A|p−2 |A|2δ |A|2 = −p|A|pδ |A|2 = −pfδ |A|2 . δ

220


Next we multiply this inequality by fδ η 2 = |A|pδ η 2 where η ∈ Cc1 (M ) is nonnegative and integrate over M . Then

fδ ΔM fδ η 2 dHn ≥ −p fδ2 |A|2 η 2 dHn , M

M

and an integration by parts yields

|∇M fδ |2 η 2 dHn − fδ ∇M fδ ∇M η 2 dHn ≥ −p − M

M

fδ2 |A|2 η 2 dHn . M

Using Young’s inequality we get

1 |∇M fδ |2 η 2 dHn ≤ p fδ2 |A|2 η 2 dHn + |∇M fδ |2 η 2 dHn 2 M M M

+2 fδ2 |∇M η|2 dHn , M

which is equivalent to

n 2 2 |∇M fδ | η dH ≤ 2p M

fδ2 |A|2 η 2

fδ2 |∇M η|2 dHn .

n

dH + 4

M

M

Now we let δ → 0+ and obtain the estimate

n n 2 2 2 2 2 |∇M f | η dH ≤ 2p f |A| η dH + 4 (3) M

M

f 2 |∇M η|2 dHn .

M

On the other hand we apply the Michael–Simon Sobolev inequality (Proposition 1, Section 3.5) with exponents p = 1 for dimension n = 2 and p = 2 for n ≥ 3. n = 2 first we have for Considering the case n = 2 with exponent λ = n−1 an absolute constant c1 1/λ

2 2 2 λ (f η ) dH ≤ c1 |∇M (f 2 η 2 )| dH2 M

M

≤ σ

|∇M f |2 η 2 dH2 M

+σ

f 2 |∇M η|2 dH2 + 2c21 σ −1

M

f 2 η 2 dH2 M

where we have again used Young’s inequality. Taking (3) into account we infer 1/λ

(4) (f 2 η 2 )λ dH2 ≤ 2pσ f 2 |A|2 η 2 dH2 M

M

f 2 |∇M η|2 dH2

+ 5σ M

+ c2 σ

−1

f 2 η 2 dH2 M

3.6 Pointwise Curvature Estimates

221

where c2 := 2c21 and λ = 2. Similarly we have for n ≥ 3 and λ =

n n−2

1/λ

2 2 λ

(f η ) dH

by Proposition 1 of Section 3.5

≤ c3 (n)

n

M

|∇M (f η)|2 dHn M

≤ 2c3 (n)

|∇M f |2 η 2 dHn M

f 2 |∇M η|2 dHn ,

+ 2c3 (n) M

which by virtue of (3) yields

1/λ

≤ c4 (n)p

(f 2 η 2 )λ dHn

(5)

f 2 |A|2 η 2 dHn

M

M

f 2 |∇M η|2 dHn

+ c4 (n) M

for a suitable constant c4 depending only on n. Estimation of the term

M

f 2 |A|2 η 2 dHn for n ≤ 5:

We apply Young’s inequality in the following way: ab ≤ δaα + δ −β/α bβ with constants β := 1 + τ, α := 1+τ and δ := (2mp)−1 where τ and m are τ positive numbers which will be determined later. Thus we find |A|2 R2 = (2mp)

−1 |A|2+2τ R2+2τ |A|2 R2 ≤ δ(2mp)1+τ + δ −τ 2mp (2mp)1+τ

= (2mp)τ

−1

+

R2+2τ |A|2+2τ 2mp

and therefore

n 2 2 2 τ −1 −2 f |A| η dH ≤ (2mp) R

f 2 η 2 dHn

M

M

+ (2mp)

−1

R

f 2 |A|2+2τ η 2 dHn .

2τ M

We have to distinguish between n = 2 and n ≥ 3: By Hölder’s inequality,

222


f 2 |A|2 η 2 dHn

(6) M

≤ (2mp)

τ −1

R

−2

f 2 η 2 dHn M

⎧ (2mp)−1 R2τ ( M (f 2 η 2 )λ dH2 )1/λ [ KR/2 |A|4+4τ dH2 ]1/2 ⎪ ⎪ ⎪ ⎪ ⎨ for n = λ = 2, + ⎪ (2mp)−1 R2τ ( M (f 2 η 2 )λ dHn )1/λ [ KR/2 |A|n+τ n dHn ]2/n ⎪ ⎪ ⎪ ⎩ n for n ≥ 3, λ = n−2 , where we have used that supp(η) ⊂ KR/2 . The parameter τ = τ (n) will now be chosen such that 4+4τ < 6 in the two-dimensional case and n+τ n < 4+2 n2 for 3 ≤ n ≤ 5 respectively. Let γ = γ(t) be a cut-off function with 1 for t ≤ R2 , γ(t) = 0 for t ≥ R and |γ (t)| ≤ 3/R for all t ∈ R. Put ξ(x) := γ(r(x)); then |∇M ξ| ≤ 3/R Hn -a.e. on M , and since supp(ξ) ⊂ KR (x0 ) is compact in M , the integral curvature estimate of Theorem 3 in Section 3.4 is applicable. For n = 2 we infer the estimate

c5 (n)K 2 4+4τ |A| dH ≤ |A|4+4τ ξ 4+4τ dH2 ≤ 2+4τ , R KR/2 KR while in the remaining cases n = 3, 4, 5 we get

c5 (n)K n n+τ n |A| dH ≤ |A|n+τ n ξ n+τ n dHn ≤ , Rτ n KR/2 KR where we have used the area-growth hypotheses Hn (KR (x0 )) ≤ KRn . Inserting these estimates in (6) we obtain

(7) f 2 |A|2 η 2 dHn M

≤ (2mp)τ

−1

R−2

f 2 η 2 dHn M

⎧ - .1/λ ⎨(2mp)−1 R−1 (c5 K)1/2 (f 2 η 2 )λ dH2 , for n = 2 = λ, M + - . 1/λ ⎩(2mp)−1 (c K)2/n n (f 2 η 2 )λ dHn , for λ = n−2 , n ≥ 3. 5 M √ Choosing the parameter m ≥ 2 c5 K we conclude from inequality (4) and (7) for n = 2 the estimate


1/λ 2 2 λ

(f η ) dH

≤ 4pσR

2

−2

M

(2mp)

τ −1

f 2 η 2 dH2 M

f |∇M η| dH + 2c2 σ 2

+ 10σ

223

2

2

−1

M

f 2 η 2 dH2 M

where λ = 2. If η is a cut-off function as described in Lemma 1, i.e. |∇M η| ≤ cσ −1 , η = 1 on Kρ−σ we conclude (with λ = 2) 1/λ

f

(8)

2λ

dH

≤ Cp1+τ

2

Kρ−σ

−1

· σ −1

f 2 dH2 , Kρ

which proves the assertion for n = 2 with := 1 + τ −1 > 1. Finally, if 3 ≤ n ≤ 5, we employ inequalities (5) and (7), choosing m ≥ 2c4 (c5 K)1/2 , and get 1/λ

(f 2 η 2 )λ dHn M

−1 ≤ 2c4 p(2mp)τ R−2 f 2 η 2 dHn M

2 2 + 2c4 f |∇M η| dHn . M

Thus we have for 3 ≤ n ≤ 5, λ =

n n−2 ,

p > 1 the inequality

1/λ

f

(9)

2λ

dH

Kρ−σ

n

≤ Cp1+τ

−1

σ −2

f 2 dHn Kρ

for some constant C = C(n, K) depending only on n and K. This proves Lemma 1 with = 1 + τ −1 in the cases 3 ≤ n ≤ 5. n Proof of Theorem 1. First case: n = 3, 4, 5 and λ = n−2 . We employ a well known iteration scheme, originally due to Moser [1,2]. Put ρ0 := R/2 and R , ρk+1 := ρk − σk+1 where k ∈ N (then ρk → R/4 as k → ∞), also σk := 2k+2 g := |A|2 , p := λk−1 . In relation (9) we replace ρ by ρk−1 , σ by σk , and noting k that f 2 = g p , f 2λ = g λp = g λ we have

1/λ

(10)

g Kρk

λk

dH

n

≤ Cλ(k−1) R−2 22k+4 ≤ ck R−2

gλ

k−1

gλ

k−1

dHn

Kρk−1

dHn ,

Kρk−1

where c depends only on K and n. Thus we obtain with a slightly larger constant which is again denoted by c that

224


(11)

R

−n

g

λk

dH ≤ c R n

Kρk

k

−2λ−n

= c

λ g

λk−1

dH

n

Kρk−1

k

R

−n

λ

g

λk−1

dH

n

Kρk−1

where we have used that −2λ − n = −nλ. Define

R−n

Ik :=

λ−k

k

g λ dH

n

.

Kρk

By raising inequality (11) to the power λ−k we obtain Ik ≤ ckλ

(12)

−k

Ik−1

for all k ≥ 1.

Iterating (12) we conclude Ik ≤ c

k s=0

sλ−s

I0 ≤ CI0 = CR

−n

KR/2

|A|2 dHn

for all k ≥ 1 and some constant C depending only on K and n. On the other hand it is easily seen that sup |A|2 = sup g ≤ lim inf Ik

KR/4

k→∞

KR/4

whence we infer the final result sup |A|2 ≤

KR/4

C , R2

using the integral curvature estimate of Theorem 3 in Section 3.4 and the assumption on area growth. Note that instead of Theorem 3 in Section 3.4, the stability condition would have been sufficient here. Second case: n = 2. The argument is identical, so we skip over the details and mention only the main estimate. Instead of (10) we have as above, using inequality (8), 1/2

≤ ck R−1

k

g 2 dH2

(13) Kρk

g2

R

g Kρk

2k

dH ≤ c 2

k

R

dH2 .

Kρk−1

Thus we get with a slightly larger constant c

−2

k−1

−2

2 g

2k−1

dH

2

.

Kρk−1

The rest of the proof is the same as in case one. Theorem 1 is proved.


225

We now proceed to prove pointwise curvature estimates for α-minimal surfaces. Let BR (ξ), ξ ∈ M , denote a Euclidean ball in Rn+1 . We tacitly assume that BR (ξ) does not intersect the boundary of M . Furthermore, appropriate area-growth conditions are essential for the validity of these estimates. We make the following natural assumptions:

(14) |xn+1 |α dHn KR (ξ)

≤

=

c(n, α)(|ξ n+1 |α + Rα )Rn , c(n, α)(|ξ n+1 |α+1 + Rα+1 )Rn−1

⎧ ⎨ c(n, α)[| ξn+1 |α + 1]Rn+α = c(n, α) · k( ξn+1 )Rn+α , R R

(a)

⎩ c(n, α)[| ξ

(b)

n+1

R

n+1

|α+1 + 1]Rn+α = c(n, α)k( ξ R )Rn+α ,

where we have put k(t) := (1 + |t|α ), k(t) = (1 + |t|α+1 ), KR (ξ) = Br (ξ) ∩ M , and c = c(n, α) denotes some constant depending only on n and α. Note that either (14) (a) or (b) are fulfilled for every R > 0, if M = ∂U , U ⊂ Rn+1 is an α-minimizing boundary in Rn+1 or M = graph u is an entire solution of the singular minimal surface equation respectively, see the proof of Theorem 6 in Section 3.4.3. Theorem 2. Let M ⊂ Rn+1 be a hypersurface of class C 4 which is α-stable in BR (ξ) and suppose that the area growth condition (14) (a) or (b) holds true

2 , ξ ∈ M with ξ n+1 = 0. for all R > 0. Suppose also that α + n < 4 + 2 n+α Then there is some constant 0 = 0 (n, α) ∈ (0, 1) depending on n and α only such that for R ≤ 0 |ξ n+1 | we have the estimate ξ n+1 2 2 R−2 , (15) (H + α|A| )(ξ) ≤ C1 n, α, R

while for R ≥ 0 |ξ n+1 | the estimate (16)

(H + α|A| )(ξ) ≤ C2 (n, α) · R 2

2

holds true for all q ∈ [0, n, α and where K =

n+1 k( ξ R )

2 n+α )

−2

R

n+α 2+q

|ξ n+1 |

1

K 2+q

with some constant C2 depending only on n+1

or K = k( ξ R ) respectively.

Remarks. 1. Clearly, Theorem 2 implies the Bernstein results of Corollaries 2 and 3 in Section 3.4 for α-stable hypersurfaces or α-stable entire graphs respectively. However note that the integral curvature estimate in Theorem 5 of Section 3.4 is needed to prove Theorem 2 above; so this result does not provide an independent proof of the Bernstein result mentioned before. Also

226


recall that the stability condition is required even for α-minimal graphs in contrast to the classical situation. In fact, there are nontrivial entire smooth solutions for arbitrary n ≥ 2 and α > 0 of the singular or symmetric minimal surface equation (50) or (48), Section 3.4, respectively. Theorem 2 shows that these solutions can not be (globally) α-stable. Smooth entire solutions of the singular minimal surface equation can be obtained from a careful analysis of the corresponding axially symmetric problem. This leads to a first order system of ordinary differential equations, see Keiper [1] and Dierkes [5,8]. 2. The number 0 = 0 (n, α) is determined by the requirement 1/n c1/n (n, α)(1 − 0 )−1−α/n (1 + α · 0 ≤ 0)

1 10α

n−1

1 )1/n 0 n ≤ 10α in the respective case or, c1/n (n, α)(1 − 0 )−1−α/n (1 + 1+α 0 (14) (a) or (b), where c(n, α) denotes the constant in (14) (up to a constant factor).

We split the proof of Theorem 2 into several auxiliary results. Without loss of generality we may assume that ξ n+1 > 0. p 2 2 Lemma 2. Put f := [T 2+q ] , T := (H + α|A| ) where p, q ∈ R denote num-

2 . If η ∈ Cc1 (M, R) is an arbitrary non-negative bers with p > 1, 0 ≤ q < n+α function we obtain the estimate

4p(2 + q) n n+1 α 2 2 (17) |x | |∇M f | η dH ≤ |xn+1 |α f 2 T η 2 dHn α M M

|xn+1 |α f 2 |∇M η|2 dHn . +4 M

0 Proof. We apply Lemma 6 of Section 3.4 and get for xn+1 = 0, |A(x)| = the estimate ΔM f = p(2 + q)(p(2 + q) − 1)(H2 + α|A|2 )p(2+q)−2 · |∇M (H2 + α|A|2 )|2 + p(2 + q)(H2 + α|A|2 )p(2+q)−1 ΔM (H2 + α|A|2 ) 2 2 p(2+q)−1 n+1 −1 n+1 − T − α(x ) ∇M x ∇M T ≥ p(2 + q)T α =

−2p(2 + q) p(2+q)+1 T − αp(2 + q)(xn+1 )−1 T p(2+q)−1 ∇M xn+1 ∇M T α

=

−2p(2 + q) f T − α(xn+1 )−1 ∇M xn+1 ∇M f. α

Next we multiply this inequality by |xn+1 |α f η 2 and obtain after integration over M


227

−2p(2 + q) |xn+1 |α f 2 T η 2 dHn α M

− (∇M |xn+1 |α · ∇M f )f η 2 dHn .

|xn+1 |α ΔM f f η 2 dHn ≥ M

M

An integration by parts yields by means of Schwarz’s inequality

4p(2 + q) n n+1 α 2 2 |x | |∇M f | η dH ≤ |xn+1 |α f 2 T η 2 dHn α M M

+4 |xn+1 |α f 2 |∇M η|2 dHn , M

which is the final result.

From Proposition 4, Section 3.5, and estimate (17) we conclude with λ := the inequality

n+α n+α−2

1/λ

|x

(18)

| (f η ) dH

n+1 α

M

2 2 λ

n

≤ C(n, α)

|xn+1 |α |∇M (f η)|2 dHn M

≤ C(n, α)

|xn+1 |α |∇M f |2 η 2 dHn M

|xn+1 |α f 2 |∇M η|2 dHn

+ C(n, α)

M

≤ pc1 (n, α)

|xn+1 |α f 2 T η 2 dHn M

|xn+1 |α f 2 |∇M η|2 dHn

+ c1 (n, α) M

where c1 (n, α) depends only on n and α. Note that (18) holds provided that supp(η) ⊂ {x ∈ Rn+1 : xn+1 ≥ } and

−n−α

ωn−1

1/n

Estimation of the term

M ∩supp(η)

M

|x

| dH

n+1 α

n

≤

1 . 10α

|xn+1 |α f 2 T η 2 dHn :

By virtue of Young’s inequality we have ab ≤ δaα + δ −β/α bβ with β := −1 , where c1 = c1 (n, α) denotes the constant in 1 + τ, α := 1+τ τ , δ := (2c1 κp)

228


(18), p > 1, while κ > 0 is a number which will be determined later. We get the inequality T R2 = (2c1 κp)

1 T R2 δ −τ R2+2τ T 1+τ ≤ δ(2c1 κp)1+τ + 2c1 κp (2c1 κp)1+τ

= (2c1 κp)τ

−1

+

R2+2τ T 1+τ . 2c1 κp

Assuming that supp(η) ⊂ KR/2 (ξ), ξ ∈ M , this yields

|xn+1 |α f 2 T η 2 dHn

pc1

(19)

M

= pc1 R−2

|xn+1 |α f 2 T R2 η 2 dHn M

≤ pc1 R−2

KR/2

|xn+1 |α f 2 η 2 {(2c1 κp)τ

−1

+ (2c1 κp)−1 R2+2τ T 1+τ } dHn

−1 −1 = c1 (2c1 κ)τ p1+τ R−2 |xn+1 |α f 2 η 2 dHn KR/2

+ (2κ)−1 R2τ

|xn+1 |α f 2 η 2 T 1+τ dHn .

KR/2

Applying Hölder’s inequality we have the estimate

(20)

|xn+1 |α f 2 T η 2 dHn

pc1 M

≤ c1 (2c1 κ)

τ −1 1+τ −1

p

+ (2κ)−1 R2τ

R

−2

KR/2

|xn+1 |α f 2 η 2 dHn

1/λ

KR/2

|xn+1 |α (f 2 η 2 )λ dHn

·

KR/2

|x

| T

n+1 α

(1+τ )( n+α 2 )

dH

n

2 n+α

n+α where λ = n+α−2 . Next we choose τ = τ (n, α) > 0 so that (1 + τ )(n + 2 α) < 4 + 2 n+α and apply the integral curvature estimate of Theorem 5 in Section 3.4. Inserting a suitable cut-off function we get


229

2/n+α

KR/2

(1+τ )( n+α 2 )

|xn+1 |α T

≤ c(n, α) R

−(1+τ )(n+α)

dH

n

2/n+α

K3R/4

|x

| dH

n+1 α

n

≤ c(n, α)[KR−τ (n+α) ]2/n+α = c(n, α) · K 2/n+α R−2τ n+1

where we have used the area growth assumption (14) and K = k( ξ R ) or n+1

K = k( ξ R ) respectively. This estimate and inequality (20) imply the relation

pc1 (21) |xn+1 |α f 2 T η 2 dHn M

≤ c1 (2c1 κ)

τ −1 1+τ −1

p

R

−2

KR/2

+ c(2κ)−1 K 2/n+α


1/λ KR/2

|xn+1 |α (f 2 η 2 )λ dHn

,

where supp(η) ⊂ KR/2 (ξ). We choose κ := c(K)2/n+α and conclude from relations (18) and (21) the inequality

1/λ

(22) KR/2

|x

| (f η ) dH

n+1 α

≤ 2c1 (2c1 κ)

2 2 λ

τ −1 1+τ −1

p

R

−2

KR/2

+ 2c1

n

KR/2


|xn+1 |α f 2 |∇M η|2 dHn ,

which holds for every nonnegative function η ∈ Cc1 (M, R) with the properties supp(η) ⊂ KR/2 (ξ) ⊂ {x ∈ Rn+1 : xn+1 ≥ } and (23)

−n−α

ωn−1

1/n

KR/2 (ξ)

|x

| dH

n+1 α

n

Let η be a cut-off function defined by 1 on Kρ−σ (ξ), η= 0 on M \ Kρ (ξ),

0 such that (23) is satisfied; here c3 = c3 (n, α) depends only an n and α and it is assumed that p > 1, κ ≥ 1. Now we employ an iteration procedure as before, i.e. for every j = 1, 2, 3, . . . we put 2p := λj−1 , ρ0 := R/4, σj := R/2j+3 , ρj+1 := ρj − σj+1 and g := j−1 j and f 2λ = g λ . In (H2 + α|A|2 )2+q = T 2+q . Then we have f 2 = g 2p = g λ (24) we replace ρ by ρj−1 and σ by σj and infer

1/λ

Kρj

|x

| g

n+1 α λ

j

dH

≤ c3 κ

n

−1

τ −1 1+τ −1

p

σj−2

Kρj−1

|xn+1 |α g λ

j−1

dHn .

−1

Note that p1+τ ≤ (λj−1 )1+τ ≤ cj4 for every j ∈ N and some constant −2 c4 = c4 (n, α); furthermore σj = 22j+6 R−2 . Then

(25) Kρj

|x

n+1 α λj

| g

dH ≤ n

−1 cj5 κλτ R−2λ

λ |x

Kρj−1

n+1 α λj−1

| g

dH

where c5 = c5 (n, α) and κ = c(K)2/n+α . For j ∈ N we define Ij by

Ij := R−n−α

λ−j j

Kρj

|xn+1 |α g λ dHn

,

multiply (25) by R−n−α and raise it to the power λ−j , noting that −2λ − (n + α) = −λ(n + α). Then we infer

−j

Ij ≤ cjλ κτ 5

−1

λ−(j−1)

Ij−1

which yields after iteration j

Ij ≤ c5 With j → ∞ we obtain

s=0

sλ−s τ −1

κ

j s=1

λ−(s−1)

I0 .

n


(26)

231

sup (H2 + α|A|2 )2+q

KR/8

≤ lim inf Ij j→∞

≤ c6 (κ

n+α 2

≤ c7 K τ

−1

)

τ −1

R

−(n+α)

R−(n+α)

KR/4

KR/4

|xn+1 |α (H2 + α|A|2 )2+q dHn

|xn+1 |α (H2 + α|A|2 )2+q dHn

n+1

n+1

n+1

where c7 = c7 (n, α), K = k( ξ R ) = (1 + | ξ R |α ) or K = k( ξ R ) = (1 + n+1

| ξ R |1+α ) respectively. Additionally we have assumed that R > 0 satisfies the smallness condition (23) and KR/2 ⊂ {xn+1 ≥ }. Now suppose that M is α-stable in the ball B1 (ξ) ⊂ Rn+1 . Then Theorem 5 in Section 3.4 and the growth assumption (14) easily yield the following esti-

mate (cp. the proof of Corollary 2, Section 3.4) for R ≤ 1 and q ∈ [0,

2 n+α )

|xn+1 |α (H2 + α|A|2 )2+q dHn

(27) KR/4

≤

K1/4

|xn+1 |α (H2 + α|A|2 )2+q dHn

≤ c8 (n, α, q)

K1/2

|xn+1 |α dHn

≤ c9 (n, α, q)K(ξ n+1 ), where K(ξ n+1 ) = k(ξ n+1 ) or K(ξ n+1 ) = k(ξ n+1 ) respectively. Assuming that R ≤ 1 and ξ n+1 satisfy condition (23) we find from (26) and (27) the estimate (28)

sup (H2 + α|A|2 )2+q ≤ c10 (n, α, q)R−(n+α) K τ

KR/8

−1

· K(ξ n+1 ).

On the other hand we infer from the area growth hypotheses (14) that (23) holds provided we require [(ξ n+1 − R)−(n+α) c(n, α)((ξ n+1 )α + Rα )Rn ]1/n ≤

1 10α

or [(ξ n+1 − R)−(n+α) c(n, α)((ξ n+1 )α+1 + Rα+1 )Rn−1 ]1/n ≤

1 10α

232


in the respective cases (14) (a) or (b). These inequalities are satisfied for all R ≤ 0 (n, α) · ξ n+1 , where 0 = 0 (n, α) < 1 is chosen to fulfill 1/n 0 ≤ c1/n (n, α)(1 − 0 )−1−α/n (1 + α 0)

or

1 10α

1 10α respectively. Hence we may apply inequality (28) with R := 0 (n, α)ξ n+1 assuming 0 (n, α)ξ n+1 ≤ 1 to conclude n−1

c1/n (n, α)(1 − 0 )−1−α/n (1 + 1+α )1/n 0 n ≤ 0

(29)

(H2 + α|A|2 )2+q (ξ) ≤ c11 (ξ n+1 )−(n+α) K(ξ n+1 )

for some constant c11 = c11 (n, α) depending only on n and α, K = k or k respectively and for all q ∈ [0,

2 n+α ).

Note that (29) is satisfied for all hypersur-

faces M ⊂ R which are α-stable in B1 (ξ) and such that 0 (n, α)ξ n+1 ≤ 1. (Recall that we have w.l.o.g. assumed that ξ n+1 > 0.) The proof of Theorem 2 is now completed by invoking a suitable scaling argument. By assumption we have that M ⊂ Rn+1 is α-stable in BR (ξ). Then the scaled surface Mλ := {λx ∈ Rn+1 : x ∈ M } n+1

with λ = R−1 is α-stable in B1 ( Rξ ). If 0 (n, α)ξ n+1 ≤ R we may apply (29) to the surface Mλ , obtaining the estimate n+1 ξ , (Hλ2 + α|Aλ |2 )2+q (λξ) ≤ c11 Rn+α (ξ n+1 )−(n+α) K R where now Hλ , Aλ denote the curvature functions of the scaled surface Mλ . Since on the other hand we have (Hλ2 + α(|Aλ |2 )(λξ) = R2 (H2 + α|A|2 )(ξ), it follows for the original surface M that (H2 + α|A|2 )2+q (ξ) ≤ c11 R−4−2q+n+α (ξ n+1 )−n−α K 2 for all q ∈ [0, n+α ), some constant c11 = c11 (n, α), K = k(ξ n+1 /R) or k(ξ n+1 /R) respectively and all R ≥ 0 (n, α)ξ n+1 . This proves estimate (16). Otherwise R < 0 (n, α)ξ n+1 and we apply inequality (26) putting q = 0. Theorem 5 in Section 3.4 and the area growth assumption yield the result

−1 |xn+1 |α (H2 + α|A|2 )2 dHn (H2 + α|A|2 )2 (ξ) ≤ c7 K τ R−n−α KR/4

≤ c7 K

τ −1

R

−n−α−4

≤ c12 (n, α)K 1+τ

KR/4 −1

R−4

|xn+1 |α dHn


233

where K = k(ξ n+1 /R) or K = k(ξ n+1 /R) respectively. This proves assertion (15) upon choosing −1 ξ n+1 := c12 (n, α)K 1+τ . C12 n, α, R

Theorem 2 is proved.

Finally, we discuss a curvature estimate which leads to a generalization of Moser’s Bernstein theorem, see Corollary 3 of Section 3.1. Theorem 3. Let u be a C 2 -solution of the minimal surface equation in n BR (0) ⊂ R , and set v := 1 + |Du|2 . Then we have (30)

|A|v(0) ≤ C(n)R−1 sup v. KR (0)

If u is an entire solution of the minimal surface equation with |Du(x)| = o(|x| + |u(x)|)

as |x| → ∞

then u is an affine linear function. Proof. Set M = graph u. By Jacobi’s field equation for the unit normal N , Proposition 2 in Section 3.3, ΔM N + |A|2 N = 0 and since N n+1 = v −1 =

1 1 + |Du|2

we have (31)

ΔM v = 2v −1 |∇M v|2 + v|A|2 .

In addition we employ Simons’s identity, Proposition 1 in Section 3.3, in the form (32)

ΔM |A|2 = 2|∇M A|2 − 2|A|4 .

Furthermore, since M is minimal we have by Lemma 3 in Section 3.4 the inequality 2 2 |∇M |A||2 − 2|A|4 ΔM |A| ≥ 2 1 + n or, equivalently (33)

|A|ΔM |A| ≥

2 |∇M |A||2 − |A|4 . n

234


Using (31) and (33) we compute for p ≥ 2 and q > 0 (34) ΔM (|A|p v q ) = divM (p|A|p−1 v q ∇M |A| + q|A|p v q−1 ∇M v) = p(p − 1)|A|p−2 |∇M |A||2 v q + p|A|p−1 ΔM |A|v q + 2pq|A|p−1 v q−1 (∇M |A|∇M v) + q(q − 1)|A|p v q−2 |∇M v|2 + q|A|p v q−1 ΔM v ≥ p(p − 1)|A|p−2 v q |∇M |A||2 +

2p p−2 q |A| v |∇M |A||2 − p|A|p+2 v q n

+ 2pq|A|p−1 v q−1 (∇M |A|∇M v) + q(q − 1)|A|p v q−2 |∇M v|2 + 2q|A|p v q−2 |∇M v|2 + q|A|p+2 v q 2 |A|p−2 v q |∇M |A||2 = (q − p)|A|p+2 v q + p p − 1 + n + q(q + 1)|A|p v q−2 |∇M v|2 + 2pq|A|p−1 v q−1 (∇M |A|∇M v). By virtue of Young’s inequality we get for every > 0 2pq|A|p−1 v q−1 (∇M |A|∇M v) ≤ p2 |A|p v q−2 |∇M v|2 + −1 q 2 |A|p−2 v q |∇M |A||2 whence (34) yields ΔM (|A|p v q ) ≥ (q − p)|A|p+2 v q 2 − −1 q 2 |A|p−2 v q |∇M |A||2 + p p−1+ n + [q(q + 1) − p2 ]|A|p v q−2 |∇M v|2 . Put := (35)

q(q+1) p2

then we get ΔM (|A|p v q ) ≥ (q − p)|A|p+2 v q

pq if (p − 1 + n2 ) − q+1 ≥ 0, or p ≥ q(1 − n2 ) + 1 − n2 . In particular we obtain for p = q ≥ n−2 2 the estimate ΔM (|A|p v p ) ≥ 0

(i.e. |A|p v p is subharmonic on M ) if p ≥ n−2 2 . The mean value inequality, Theorem 2 in Section 3.5, applied to |A|2p v 2p yields 1/2 (36)

|A|p v p (0) ≤ ωn−1/2 R−n/2

KR (0)

|A|2p v 2p dHn


235

where KR (0) = M ∩ BR (0) = {(x, u(x)) ∈ Rn+1 : |x|2 + |u(x)|2 < R2 }. Estimation of the integral KR (0) |A|2p v 2p dHn : Replacing p by (p−1) and q by p we infer from inequality (35) ΔM (|A|p−1 v p ) ≥ |A|p+1 v p

(37)

which holds for p ≥ 3 and p ≥ n − 1. Then we multiply (37) by |A|p−1 v p η 2p , where η stands for a usual cut-off function, and obtain after an integration by parts

(38) |A|2p v 2p η 2p dHn M

|A|p−1 v p η 2p ΔM (|A|p−1 v p ) dHn

≤ M

=−

∇M (|A|p−1 v p η 2p )∇M (|A|p−1 v p ) dHn

M

=−

{|∇M (|A|p−1 v p )|2 η 2p M

+ 2pη 2p−1 |A|p−1 v p ∇M η · ∇M (|A|p−1 v p )} dHn . Since 2pη 2p−1 |A|p−1 v p |∇M η · ∇M (|A|p−1 v p )| ≤

1 |∇M (|A|p−1 v p )|2 η 2p + 2p2 η 2(p−1) |A|2(p−1) v 2p |∇M η|2 2

we arrive by virtue of (38) at

n 2p 2p 2p 2 |A| v η dH ≤ 2p M

|A|2(p−1) v 2p η 2(p−1) |∇M η|2 dHn . M

Another application of Young’s inequality in the form −(p−1) p p−1 ap/p−1 + b ab ≤ p p yields 2p2 |A|2(p−1) v 2p η 2(p−1) |∇M η|2 p−1 −(p−1) 2 2p v 2p |A|2p η 2p + 2 p v |∇M η|2p ≤ 2p2 p p

236


and, by choosing > 0 such that 2

2p we obtain

p−1 p

1 , 2

|A| v η

2p 2p 2p

(39)

≤

dH ≤ c1 (p)

v 2p |∇M η|2p dHn

n

M

M

for a suitable constant c1 = c1 (p) depending only on p. Let η denote a standard cut-off function on B2R (0); with η ≡ 1 on BR (0); then we infer from (39)

1/2 |A| v

2p 2p

(40) KR (0)

dH

n

≤ c2 (p, n)Rn/2 R−p sup v p K2R (0)

for a further constant c2 = c2 (p, n). Note that here we have used the fact that the area of a minimal graph M can be estimated by Hn (M ∩BR (0)) ≤ c(n)Rn , cp. the proof of Theorem 4, Section 3.4. Finally, inequalities (40) and (36) yield the estimate |A|p v p (0) ≤ ωn−1/2 c2 (p, n)R−p sup v p K2R (0)

from which we conclude by selecting p = p(n) suitably that |A|v(0) ≤ c3 (n)R−1 sup v. KR (0)

If |Du(x)| = o(|x| + |u(x)|) as |x| → ∞ then the right-hand side tends to zero as R → ∞ and since v ≥ 1 we must have |A|(0) = 0. As this holds for every point Theorem 3 follows.

3.7 Scholia 3.7.1 References to the Literature on Bernstein’s Theorem and Curvature Estimates for n = 2 There are many proofs of the celebrated Bernstein theorem (Theorem 1) in the two-dimensional case, most of which are valid in more general situations and with stronger conclusions, see the contributions by T. Radó [10], E. Hopf [1,4], L. Bers [2,3], E. Heinz [1], R. Finn [4,5], K. Jörgens [1], J.C.C. Nitsche [1,2], R. Osserman [1,2], H. Jenkins [1], W. Fleming [2], Jenkins and Serrin [1], S.S. Chern [6], M. Miranda [2], L. Simon [2,5], F. Sauvigny [8,9], and we also refer to the review articles by J.C.C. Nitsche [11] and L. Simon [14,17] for further information (cf. also Vol. 1, Chapter 2). Bernstein [4] obtained his

3.7 Scholia

237

famous result as an application of a more general theorem on elliptic equations according to which every entire bounded solution v ∈ C 2 (R2 ) of any elliptic equation of the form (1)

avxx + 2bvxy + cvyy = 0

has to be a constant. E. Hopf [2,4] has shown by example that neither a onesided bound on the solutions is sufficient for such a result, nor does the theorem hold in higher dimensions. Now if u ∈ C 2 (R2 ) is an entire solution of the minimal surface equation it can easily be verified (although the calculations are somewhat tedious) that the functions v = arc tan ux

or

v = arc tan uy

are bounded entire solutions of equation (1) where a := (1 + u2y ),

b := −ux uy ,

c := (1 + u2x ),

and hence Bernstein’s theorem follows immediately. Indeed, Bernstein’s original proof contained a gap which was resolved by E. Hopf [5], and the whole proof is revised in E. Hopf [3]. An independent proof of Bernstein’s theorem was at the same time given by E.J. Mickle [1]. In Section 3.1 we have presented the elegant method of J.C.C. Nitsche [2] who reproved Jörgens’s result [1] (cp. Theorem 1, Section 3.1) concerning entire solutions of the Monge-Ampère equation by using a transformation which was invented by H. Lewy [1] in a different context. Bernstein’s theorem follows then by an observation made by E. Heinz (see K. Jörgens [1], p. 133). E. Heinz [1] was also the first to prove a priori curvature estimates like Theorem 2 in the introduction to this chapter, which can be viewed as a quantitative improvement of Bernstein’s theorem. Note that for minimal surfaces we have H = κ1 + κ2 = 0 and

K = κ1 κ2 = −κ21 = −κ22 ,

and hence Heinz’s estimate |K(x0 )| ≤

16 R2

may be rephrased into an estimate for the length of the second fundamental form (2)

|A|2 (x0 ) = κ21 (x0 ) + κ22 (x0 ) ≤

32 . R2

The Heinz estimate was strengthened and improved by E. Hopf [1], J.C.C. Nitsche [11,38], Osserman [2], Finn and Osserman [1]: Let u ∈ C 2 (BR (x0 )) with BR (x0 ) ⊂ R2 , be a solution of the minimal surface equation in the disk BR (x0 ). Then we even have

238

(3)


W 2 (x0 )|K(x0 )| ≤

c , R2

where W (x0 ) = 1 + |Du(x0 )|2 , and c is a constant ≤ 7.678447 . . . , see Finn and Osserman [1] and Nitsche [11] for the details. The curvature estimates (2) or (3) were extended to the larger class of equations of minimal surface type by Jenkins [1], Jenkins and Serrin [1], and the possibly ultimate results were found by L. Simon [5]. We refer to Chapter 16 in Gilbarg and Trudinger [1] for a thorough account on equations of mean curvature type. We also recall the following beautiful result by L. Bers [2] which is related to the issues discussed in Sections 7.3 and 7.4 of Vol. 1: Theorem 1. Suppose u ∈ C 2 (R2 \Br0 (0)) is a solution of the minimal surface equation on the “exterior domain” R2 \Br0 (0), r0 > 0. Then Du(x) has a limit as |x| → ∞. Theorem 1 implies Bernstein’s theorem, if one takes Moser’s result, Corollary 3 of Section 3.1, into account. Again, Theorem 1 holds for exterior solutions of equations of minimal surface type, see L. Simon [5]. We also refer to the Bernstein-type theorem for minimal surfaces in a wedge, discussed in Section 2.10, and the further results in this direction (minimal surfaces bounded by two intersecting lines) which are quoted in the Scholia 2.11 to Chapter 2. 3.7.2 Bernstein Theorems and Curvature Estimates for n ≥ 3 dimensions Moser [2] has shown a Bernstein type theorem, stating that every entire solution of the minimal surface equation in arbitrary dimensions is affine linear, provided that its gradient is bounded, and we have followed his arguments in some detail in Section 3.1. With a suitable Harnack inequality, Moser was able to derive the remarkable result that the limit of Du as x → ∞ exists for an exterior solution u, if the gradient |Du| is bounded, cp. Theorem 5 in Section 3.1. A different Bernstein theorem follows from the fundamental interior gradient estimate due to Bombieri, de Giorgi, and Miranda [1]: Every entire solution of the minimal surface equation which is bounded on one side by a cone has to be an affine linear function. Without any further assumption on the solution the following Bernstein result holds: Theorem 2. Let u be an entire solution of the minimal surface equation Du =0 (4) div 1 + |Du|2 in Rn . Then u is an affine function provided that n ≤ 7.

3.7 Scholia

239

Many authors have contributed to this fascinating result which was in the center of mathematical activities in geometric analysis for several decades. A crucial idea in the proof of this theorem is due to W. Fleming [2] who introduced methods from geometric measure theory into the analysis of the Bernstein problem. The essence of his idea can roughly be described as follows (for a more thorough account and elegant proofs we refer to Giusti’s monograph [4], in particular Chapter 17, and to the book by Massari and Miranda [1]). Without loss of generality, suppose u(0) = 0 and let U := {(x, t) ∈ Rn × R : t < u(x)} denote the subgraph of u. It follows that U (or rather its boundary) minimizes the perimeter (i.e. the area) locally in Rn+1 , and hence also the “blow down” sets Uj := {(x, t) ∈ Rn × R : j(x, t) ∈ U } = {(x, t) ∈ Rn × R : t < j −1 u(jx)} for j ∈ N minimize with respect to compact variations. It can be shown that a subsequence Ujk converges locally in Rn+1 (in an appropriate sense) to a cone C which minimizes the perimeter in Rn+1 . Anticipating J. Simons’s results in [1] on the nonexistence of non-trivial stable cones (cp. Theorem 1, Section 3.4.2), it follows that C must be a half space if we require n ≤ 6. But then

|DϕC | = ρn ωn , ∀ρ > 0 Hn (∂C ∩ Bρ (0)) = Bρ (0)

and therefore—using the monotonicity and the convergence—one also has

Hn (∂U ∩ Bρ (0)) = |DϕU | = ωn ρn , for all ρ > 0, Bρ (0)

which implies–again by monotonicity (for a C 2 -version, see Theorem 1, Section 3.5)–that U itself must be a cone. This is clearly only possible if u is a linear function and hence the assertion of Theorem 2 follows for n ≤ 6. In short, Fleming [2] has proved the following implication, namely that the nonexistence of nontrivial (or nonplanar) area minimizing hypercones in Rn+1 implies the Bernstein theorem for entire solutions u ∈ C 2 (Rn ). In his paper [2] he could verify the nonexistence of such cones in the case n = 2, thereby providing another proof of the classical Bernstein theorem in two dimensions. The next important observation was made by De Giorgi [2] who could improve Flemings argument to the effect that the nonexistence of nontrivial minimizing cones in Rn is sufficient to imply the assertion of Theorem 2 for entire solutions u ∈ C 2 (Rn ). Indeed, if u ∈ C 2 (Rn ) is a nonlinear solution of equation (4) one can find (see Giusti [4], Chapter 17) a subsequence jk → ∞ such that

240


P := {x ∈ Rn : jk−1 u(jk · x) → ∞

as k → ∞}

is a nontrivial area minimizing cone in Rn , and hence the above observation can be applied. In this way, De Giorgi [2] proved Theorem 2 for n = 3, Almgren [1] for n = 4, and J. Simons [1] for n ≤ 7. The stability of the Simons cone C = {(x, y) ∈ R4 × R4 : |x|2 = |y|2 } cp. Proposition 1 of Section 3.4 and J. Simons original paper [1], already indicated that the Bernstein property of the minimal surface equation might fail for n ≥ 8. This is in fact the case as pointed out by Bombieri, de Giorgi, and Giusti [1] in a celebrated paper. They could not only prove that the Simons cone C minimizes perimeter in R8 (thereby dashing the hope that one might be able to prove smoothness of area minimizing boundaries in Euclidean Rn of arbitrary dimension), but they also constructed a nonlinear entire solution u ∈ C 2 (R8 ) of the minimal surface equation (4). To prove that the Simons cone C is minimizing, Bombieri, De Giorgi and Giusti used an adapted Weierstrass field construction, i.e. a foliation with minimal hypersurfaces which contains the singular cone C. In such a situation, every leaf of the foliation is automatically minimizing. The cone (and the field) are then used to construct a nonlinear entire solution u = u(x1 , . . . , x8 ) of equation (4) in R8 , cp. Giusti’s monograph [4] for a lucid presentation of the method. By putting u(x1 , . . . , xn ) := u(x1 , . . . , x8 ) for arbitrary n ≥ 9 one obtains Theorem 3. For n ≥ 8 there exist nonlinear solutions u ∈ C 2 (Rn ) of the minimal surface equation (4). There are, in fact, many nonlinear entire solutions of (4) which are distinct from the Bombieri–De Giorgi–Giusti examples, as was pointed out by L. Simon [11]. He proved the existence of entire nonlinear solutions of (4) corresponding to each of the codimension-one minimizing cones C in the list of Lawson [8] or Ferus and Karcher [1]. More precisely, Simon [11] proved the following result (for the terminology we refer to Simon’s paper). Theorem 4 (L. Simon [11]). Suppose that C is a strictly minimizing isoparametric cone in Rn with sing C = {0}. Then there exists an entire solution u ∈ C 2 (Rn ) of the minimal surface equation having C × R as the tangent cylinder at ∞. Furthermore graph u inherits all symmetries of C. Simon [10] also proved a strong generalization of Bers’s theorem on exterior solutions of equation (4); cp. also Moser’s variant (Theorem 5, Section 3.1) of this result. Theorem 5 (L. Simon [10]). Let u ∈ C 2 (Rn \ Ω), 3 ≤ n ≤ 7, be a solution of the minimal surface equation in Rn \ Ω where Ω is bounded. Then Du is bounded and has a limit as |x| → ∞.

3.7 Scholia

241

L. Simon proved the more general result, holding for all dimensions n, that either Du has a limit as |x| → ∞ or else all tangent cones of graph u at ∞ are cylinders of the form C × R, where C is an (n − 1)-dimensional minimizing cone in Rn with 0 ∈ sing C (i.e. C is non-planar). Since for n ≤ 7 there are no such cones, Theorem 5 follows immediately. Moser’s [2] Bernstein theorem requires the gradient Du to be a priori bounded (Cor. 3, Section 9.1). Caffarelli, Nirenberg, and Spruck [1] proved the same result under the weaker assumption |Du(x)| = o(|x|1/2 )

as |x| → ∞;

indeed their result applies to a larger class of curvature type equations. For minimal graphs J.C.C. Nitsche [37] extended this to |Du(x)| = O(|x|μ ) for any μ < 1, while Ecker and Huisken [1] merely had to assume that |Du(x)| = o( |x|2 + |u(x)|2 ) as |x| → ∞. We have closely followed their elegant reasoning in the proof of Theorem 3 of Section 3.6. Further improvements of this result are due to Simon [18]. In still another direction, Bombieri and Giusti [1] proved the following generalization of Moser’s result (see Theorem 8 and the following remark in their paper): Theorem 6 (Bombieri–Giusti). Let u ∈ C 2 (Rn ) be an entire solution of the minimal surface equation such that n−7 partial derivatives Di u are uniformly bounded on Rn . Then u is affine linear. In their paper, Bombieri and Giusti generalized Moser’s Harnack inequality to solutions of elliptic equations on minimal hypersurfaces M ⊂ Rn+1 . Since (by Jacobi’s equation) the function g = arc tan

a, ν , νn+1

a ∈ Rn+1 ,

is a bounded solution of an elliptic equation on M , provided that n − 1 partial derivatives are uniformly bounded, this implies that a, ν = const, νn+1 which means that |Du| is bounded on Rn , and hence the conclusion follows by Moser’s result, at least under the stronger assumption that n − 1 partial derivatives of u are bounded. The full result follows by a dimension reduction argument and the results of J. Simons [1]. Concerning integral and pointwise curvature estimates, the paper by Schoen, Simon, and Yau [1] was a major achievement. (In an earlier paper

242


by Miranda [2], somewhat weaker integral estimates were proved.) The three men paper extended the a priori curvature estimate of E. Heinz to higher dimensions and to the general Riemannian situation. We have followed its reasoning in the proofs of Theorems 1 and 3 of Section 3.4. The Bernstein result in Theorem 4 of Section 3.4 for n ≤ 5 is an immediate consequence of the integral curvature estimate of Theorem 3. The pointwise estimate of Theorem 1, Section 3.6 for n ≤ 5 is due to Schoen, Simon, and Yau [1]; our proof differs from their argument and follows the reasoning given in Dierkes [13]. Theorem 2 of Section 3.4, concerning the nonexistence of α-stable cones, is due to Dierkes [10]. The integral curvature estimate of Theorem 5 in Section 3.4 and the Bernstein results, Theorem 6 and Corollary 3, for α-stable solutions of the symmetric or singular minimal surface equation are taken from Dierkes [12]. The pointwise estimate of Theorem 2, Section 3.6, is also due to Dierkes [13]; the Moser iteration technique is used in a similar way as in the proof of Theorem 1, Section 3.6. The pointwise curvature estimate of Theorem 1, Section 3.6 was extended up to dimensions n ≤ 6 for minimal hypersurfaces in Rn+1 by L. Simon [1]. His argument is based on regularity theory for minimizing boundaries (or hypersurfaces) in Rn+1 . This result also holds for n = 7 in the nonparametric case, i.e. when the surface is represented by a graph of some entire solution of the minimal surface equation. Note that a further extension to n = 8 is clearly impossibly since there are nonlinear entire solutions of equation (4) defined on all of R8 , see Bombieri, de Giorgi, and Giusti [1]. Using methods from geometric measure theory, Simon [4] and Schoen and Simon [1] proved curvature estimates for minimal (and stable) solutions of certain parametric functionals. In particular it is proved in Simon [4] that if n = 3, a Bernstein theorem holds for nonparametric solutions of the Euler equation of a parametric functional whose integrand only depends on the normal (and not on the spatial variables), cf. also the references to Section 7.4 of Vol. 1. This result remains true up to dimension n ≤ 7 if the integrand is C 3 -close to the area integrand. Similar results were obtained by Winklmann [2,3] who generalized the method of Schoen, Simon, and Yau [1] to parametric integrals. 3.7.3 Bernstein Theorems in Higher Codimensions The first Bernstein type theorems for minimal surfaces (n = 2) in Euclidean RN , N ≥ 3, were obtained by Chern [5] and Osserman [5]. To explain some of the results for arbitrary dimensions n ≥ 2, let us recall the notation introduced in Scholia 2.9, Vol. 1. An n-dimensional submanifold M ⊂ Rn+k which is the graph of a function u : Ω ⊂ R n → Rk ,

i.e. M = {(x, u(x)) ∈ Ω × Rk },

is minimal (cp. Chapter 4.3 of Vol. 2) if and only if the coordinate functions are harmonic on M , that is, we have

3.7 Scholia

ΔM U = 0,

243

where U = (x, u(x)) ∈ Rn+k for x ∈ Ω ⊂ Rn .

Putting gij

n+k ∂U l ∂U l ∂U ∂U ∂u ∂u := = = δij + ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj l=1

= δij +

k l=1

g := det gij

∂ul ∂ul , ∂xi ∂xj

and

g ij := (gij )−1 ,

this may be reformulated as (cp. Chapter 1 of Vol. 1) 1 ∂ √ ij ∂U = 0, gg (5) √ g ∂xi ∂xj or equivalently we obtain the minimal surface system ⎧ ∂ √ ij ⎪ ⎪ ⎨ ∂xi { gg } = 0, j = 1, . . . , n, (6) ⎪ ∂ √ ij ∂ul ⎪ ⎩ = 0, l = 1, . . . , k. gg ∂xi ∂xj Assuming u ∈ C 2 , one can show that (6) is equivalent to ∂ √ ij ∂ul = 0, l = 1, . . . , k, gg ∂xi ∂xj or

∂ 2 ul = 0 for l = 1, . . . , k, ∂xi ∂xj which reduces to the minimal surface equation (4) in case k = 1. In Chapter 2 of Vol. 1 we have already mentioned a striking example of Lawson and Osserman [1], which is based on the Hopf map η : S 3 → S 2 given by g ij

(z1 , z2 ) → η(z1 , z2 ) = (|z1 |2 − |z2 |2 , 2z1 · z 2 ) ˜ C × C is the unit 3-sphere in R4 and S 2 ⊂ R3 = ˜ R × C is where S 3 ⊂ R4 = the 2-sphere in R3 (considered as R × C). Theorem 7 (Lawson–Osserman). The Lipschitz cone C defined by C := {(x, u(x)) ∈ R4 × R3 } with x 5 4 3 |x|η for x = 0 and u(0) = 0 u : R → R , u(x) := 2 |x| is a (weak) solution to the minimal surface system (6).

244


In particular, and contrary to the case k = 1, there are Lipschitz solutions of the system (6), which are not analytic. In addition, there are nonlinear entire solutions of the minimal surface system which have a bounded gradient on Rn , again in sharp contrast to the codimension one case. Hence a quantitative a priori bound on the gradient of a solution is required for a Bernstein theorem to hold. The first result in this direction is due to Hildebrandt, Jost, and Widman [1], see Chapter 1 of Vol. 1. A more refined reasoning due to Jost and Xin [1] yields the following Theorem 8 (Jost–Xin). Let u : Rn → Rk be a smooth entire solution of the √ minimal surface system (6) such that g ≤ β0 on Rn where β0 is some number which fulfills 2 for k ≥ 2, β0 < ∞ for k = 1. Then u1 , . . . , uk are linear functions on Rn (representing an affine n-plane in Rn+k ). Even more general, Wang [1] proved a Bernstein result for smooth solutions u : Rn → Rk of the minimal surface system which have a bounded gradient (not necessarily small) such that u is an area decreasing map (which is always the case for k = 1); for details we refer to Wang [1] and Giaquinta and Martinazzi [1]. In still another direction, Smoczyk, Wang, and Xin [1] and Fr¨ ohlich and Winklmann [1] investigated minimal graphs and graphs with bounded mean curvature which have flat normal bundle R⊥ = 0. It turns out that in this case a Simons identity similar to Theorem 1 in Section 3.3 can be proved, which together with a Jacobi-type identity due to Fischer-Colbrie [1] for the quantity w = g −1/2 leads to the following integral curvature estimate: Theorem 9 (Fröhlich–Winklmann). If Σ = {(x, u(x)) ∈ Rn ×Rk } is a graph with flat normal bundle then the estimate

|A|p ϕ|∇Σ ϕ|p dHn Σ

p/2 p/3 ≤C {|∇Σ ϕ|p + (|H|p + |∇Σ H|p/2 + K1 + K2 )ϕp } dHn Σ

holds for all p ∈ [4, 4 + 8/n) and all nonnegative test functions ϕ ∈ Cc∞ (Σ). Here C depends only on n and k, and K1 , K2 depend also on ∇Σ H, ∇2Σ H, see Fr¨ ohlich and Winklmann [1] for the details. In particular the pointwise estimate C sup |A|2 ≤ 2 R Σ∩BR follows, provided that 2 ≤ n ≤ 5, Σ ∩ B4R ⊂ Σ and Hn (Σ ∩ B4R ) ≤ cRn . As a corollary of Theorem 9 one obtains

3.7 Scholia

245

Theorem 10 (Smoczyk–Wang–Xin, Wang). Let 2 ≤ n ≤ 5 and Σ ⊂ Rn ×Rk be an entire minimal graph with a flat normal bundle. If Hn (Σ ∩ BR (p)) ≤ cRn for some point p ∈ Σ and some sequence R → ∞, then u must be an affine linear function. By a careful analysis of the Grassmannian manifold G(n, k) of n-dimensional planes in Rn+k , Xin and Yang [1,2] could improve Theorem 8 if the dimension of the minimal graph is less than five. Theorem 11 (Xin and Yang [1]). Let u : Rn → Rk be a smooth entire solution of the system (6) and assume k ≥ 2, n ≤ 4. If √

g 0. This bilinear form μg is called the volume element determined by g and by the orientation of M . It is defined by the equation g(Vp , Vp ) g(Vp , Wp ) μg (p)(Vp , Wp ) := det g(Wp , Vp ) g(Wp , Wp ) if (Vp , Wp ) is an oriented basis for Tp M , and by μg (p)(Vp , Wp ) := −μg (p)(Wp , Vp ) if not. Since g is also nondegenerate, we can for each p ∈ M transform μg (p) into a linear map Φ(g)(p) : Tp M → Tp M via the rule g(p)[Φ(g)(p)Vp , Wp ] = μg (p)(Vp , Wp ). Let J(p) = Φ(g)(p). One then checks that J 2 (p) = −I(p) and that J is an almost complex structure on M . Let us define the map Φ : M → A by g → Φ(g). It is not bijective since we easily see that Φ(ρ · g) = Φ(g) for ρ ∈ P. However, one can check that Φ passes to a bijective map from M/P to A. In each coordinate chart (G, ϕ), the metric g has a local representation gαβ (w) duα duβ for w = (u1 , u2 ) ∈ ϕ(G). It is an obvious and natural question to ask:

4.2 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces

259

Given g, is there an orientation-preserving coordinate mapping ϕ which makes g locally look as nice as possible? In this case as nice as possible means that gαβ (w) = λ(w)δαβ , δαβ = 0 if α = β and δαβ = 1 if α = β, and where λ ∈ P. The answer is yes and is a classical result (see Chapters 1 and 4 of Vol. 1). Such a ϕ is called a conformal coordinate system for g. It is elementary to check that, given any g ∈ M, the set of conformal local coordinates gives a complex structure c(g) for M . One can further see that, as before, c(ρ · g) = c(g) for ρ ∈ P. Thus we obtain a map A → C which is indeed the inverse of the map C → A which we defined earlier. Thus we have sketched a proof of the following result. Theorem 2. For compact two-dimensional manifolds M without boundary there exists a bijective correspondence between C and A and, moreover, between C and M/P. This correspondence is D-equivariant. The quotient space M/P is still a bit too cumbersome to work with. For this reason we shall introduce another model for M/P. With any metric g on M we associate its scalar curvature R(g). We view R : M → C ∞ (M ) as mapping from M into a space C ∞ (M ) of C ∞ -functions on M . In conformal coordinates u1 , u2 we obtain gαβ = λδαβ , and we then define R(g) := λ−1 Δ log λ 2

2

∂ f ∂ f 1 2 where Δf = ∂u 2 + ∂v 2 , (u, v) = (u , u ). According to formula (13) of Section 1.3 in Vol. 1 we have R(g) = 2K where K is the Gauss curvature of a two-dimensional surface in R3 whose first fundamental form is just g. Therefore we define the Gauss curvature K(g) of a metric g on M by

K(g) := −

1 1 Δ log λ = R(g). 2λ 2

Definition 3. Set M−1 := K −1 (−1), i.e., the set M−1 is to consist of all those C ∞ -metrics whose Gauss curvature is the constant function −1. The next proposition is well known. We shall present a proof of a more general version later in this section. Proposition 1. Let M be a compact surface without boundary of genus greater than one. Then, given any g ∈ M, there exists a unique λ ∈ P with K(λg) = −1. This establishes the bijective correspondence between M−1 and M/P, and hence C. In Fischer and Tromba [1] the manifold properties of these spaces are investigated, and it is shown that (as manifolds) they are diffeomorphic. Recall that the group of diffeomorphisms of M also acts on M. The map K : M → C ∞ (M ) has the property that K(f ∗ g) = K(g)◦f . As a consequence

260

4 The General Problem of Plateau: Another Approach

of this basic fact it follows that f ∗ g ∈ M−1 if g ∈ M−1 . Thus D acts on M−1 , and the correspondence between M−1 and C is D-equivariant. Hence we have found as the quintessence of our discussion: In order to understand the quotient spaces C/D and C/D0 , it suffices to understand the quotient spaces M−1 /D and M−1 /D0 . We now proceed to study M−1 /D0 . Let us think of M−1 as an infinite dimensional surface in the linear space S2 . Let g ∈ M−1 . Since D acts on M−1 , it would be useful to known what the tangent space is to the orbit Og (D) of D through g. So let ft , − < t < , be a smooth one-parameter family of diffeomorphisms of M . A tangent vector1 to Og (D) at g is given by d ∗ (f g) (1) = LV g. dt t t=0 t Here V := df dt |t=0 is a vector field on M , and the left-hand side of (1) is the definition of the Lie derivative of the metric g with respect to V which is denoted by LV g. Thus, to paraphrase, all tangent vectors to Og (D) at g ∈ M are of the form LV g. Locally LV g has the following form:

(LV g)αβ = V γ

∂gαβ ∂V γ ∂V γ + g + g . γβ αγ ∂uγ ∂uα ∂uβ

Since we are interested in the quotient space M−1 /D0 we wish to collapse all such orbits Og (D). As a result we are not too much interested in the subspace of the tangent space Tg M−1 to M−1 at g consisting of all symmetric tensors of the form LV g, but in the complement of this subspace if it exists. We have the following basic result, the proof of which is given in Fischer and Tromba [1]. Theorem 3. Let h ∈ Tg M−1 be a symmetric two-tensor. Then h can be uniquely, for some unique V , expressed as a direct sum h = hT T + LV g, where hT T is a symmetric two-tensor on M with the property that, in a conformal coordinate system with respect to the metric g and with local coordinates designated as u + iv, hT T has the local representation hT T = a du2 − a dv 2 − 2b du dv, where a + ib is a holomorphic function. We use quotation marks here because we are working in the category of C ∞ -maps and tensors and thus have no implicit functions theorem. In Fischer and Tromba [1] it is shown that this presents no serious difficulties, and that formal tangent vectors are indeed tangent vectors.

1

4.2 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces

261

Corollary. Every h ∈ Tg M−1 can be expressed uniquely as a direct sum h = Re(ξ(w) dw2 ) + LV g, where ξ(w) dw2 is a holomorphic quadratic differential on M with respect to the complex structure induced by g. Proof of the corollary. Note that a du2 − a dv 2 − 2b du dv = Re{(a + ib)(du + i dv)2 } = Re{ξ(w) dw2 }. Now w → ξ(w) is holomorphic by the theorem, and ξ(w) dw2 is a complex valued two-tensor over M which in any complex (or conformal) coordinate system is holomorphic. Thus ξ(w) dw2 is, by definition, a holomorphic quadratic differential on M . Therefore, formally speaking, the tangent space to the quotient space M−1 /D0 ought to be the set of real parts of holomorphic quadratic differentials. At this point some more historical remarks are in order. As we mentioned earlier, Riemann had conjectured that the space C/D is of dimension 6g − 6 if g > 1, g = genus(M ). Teichm¨ uller later observed that, as a consequence of the Riemann–Roch theorem, the dimension of the space of holomorphic quadratic differentials is as well 6g − 6. This led him to connect, via quasiconformal mappings, the space C/D0 with these differentials. However, the route that Teichm¨ uller chose at that time was more complicated than the approach which we have taken here. Let us proceed with our discussion. Infinitesimally (on the level of tangent spaces) we have that the Teichm¨ uller space (or the Riemann space of moduli) is represented naturally as a finite dimensional space consisting of the real parts of holomorphic quadratic differentials. This is only one step away from actually putting a manifold structure on M−1 /D or on M−1 /D0 . As it turns out, one can put a natural manifold structure on M−1 /D0 but not on M−1 /D, and we shall shortly see the reason why. We want to push down the tangent space Re(ξ(w) dw2 ) onto M−1 to (at least locally) produce a submanifold S of M−1 which is a candidate for a coordinate chart for M−1 /D or M−1 /D0 . This will be carried out as follows. Consider the affine space determined by the set of symmetric two-tensors of the form g + Re(ξ(w) dw2 ) for small ξ. If ξ is small enough then g + Re(ξ(w) dw2 ) is a Riemannian metric. By Proposition 1 there exists a unique λ(ξ) such that the Gauss curvature K(λ(ξ)[g + Re(ξ(w) dw2 )]) = −1. Clearly, since K(g) = −1, we have λ(0) = 1. Moreover, one can verify that the mapping ξ → λ(ξ) is C ∞ -smooth, and that its derivative at 0 is zero. This implies that the mapping

262

(2)


ξ → λ(ξ)[g + Re(ξ(w) dw2 )]

as a map of the holomorphic quadratic differentials into M−1 has a derivative at 0 ∈ Tg M−1 which is the identity map on all tensors of the form Re(ξ(w) dw2 ). An application of the implicit function theorem now implies that, locally, the image of this map is a submanifold S of M−1 .2 We thus have found a candidate for a coordinate chart for M−1 /D or M−1 /D0 , namely the slice S. We want to collapse all orbits onto S, and we must merely check that each point of S corresponds to only one orbit of D or D0 . Precisely at this step the distinction between D and D0 enters the theory. Generally, points of S may represent more than one orbit of D, but each point of S represents only one orbit of the group D0 . The following theorem by Palais and Ebin (see Ebin [1], or Tromba’s ETHLecture Notes [24]) shows that the action of D on M−1 (in fact on M) is proper. Theorem 4 (Ebin–Palais Lemma). Let gn ∈ M be a sequence of metrics converging to some metric g ∈ M, and fn ∈ D be a sequence of diffeomorphism such that fn∗ gn → gˆ ∈ M (where → means convergence in any H s -topology, s > 12 dim M ). Then fn has a convergent subsequence fnj . The next result shows that the action of D0 on M−1 is free. This is not true for the action of D on M. Theorem 5. D0 acts freely on M−1 . Proof (a sketch). To act freely means that if f ∗ g = g and f ∈ D0 , then f = id. Assume that f = id. Then, if c(g) denotes the complex structure associated to g, it follows that f ∗ c(g) = c(g), i.e., that f is a holomorphic self-mapping. Since f is not the identity, the fixed points of f must be isolated, and in fact be non-degenerate. This implies that the Lefschetz fixed point index Λ(f ) is positive. However, since Λ(f ) is a homotopy invariant, we have Λ(f ) = Λ (id), and this is equal to the Euler-characteristic of M , namely (2–2·genus M ) < 0. This gives us the desired contradiction, and hence f must be the identity. Using the Theorems 4 and 5, standard arguments (see Fischer and Tromba [1]) allow us to conclude the following: In a neighbourhood of g we have a bijective correspondence between points in M−1 /D0 (and hence of C/D0 ) and points in S (which is a nice C ∞ submanifold of M−1 ). One checks by elementary techniques that the pieces S constitute a coordinate atlas for M−1 /D0 . We summarize this as 2 M echet space; however S consists of C ∞ -metrics, and by the implicit −1 is again a Fr´ function theorem, it is a submanifold of Ms−1 , consisting of metrics of Sobolev smoothness class H2s , for any s > 2.

4.3 Symmetric Riemann Surfaces and Their Teichm¨ uller Spaces

263

Theorem 6. Let M be a compact oriented surface without boundary and with genus g > 1. Then the spaces M−1 /D0 and C/D0 have the natural structure of a simply connected, finite dimensional C ∞ -manifold of dimension 6g − 6. The tangent space at a point [g] ∈ M−1 /D0 can be naturally identified with Re(ξ(w) dw2 ), the real parts of quadratic differentials which are holomorphic with respect to the complex structure on M induced by the metric g. (Remark. The dimension 6g − 6 follows from the Riemann–Roch theorem.) In a less elementary manner one can show (see Tromba [24]): Theorem 7. The C ∞ -manifold M−1 /D0 is diffeomorphic to Euclidean R6g−6 space. As a consequence of Theorem 7 it follows that the principal D0 -bundle (π, M−1 , M−1 /D0 ) with the natural projection map π : M−1 → M−1 /D0 has smooth sections. This implies that the Teichm¨ uller space T ∼ = M−1 /D0 can be embedded as a (6g − 6)-dimensional submanifold Σ into M−1 , in such a way that π : Σ → M−1 /D0 is onto, and that at every g ∈ Σ we have (3)

Tg Σ ⊕ Tg Og (D0 ) = Tg M−1 .

(A particularly natural and beautiful embedding using harmonic mappings was given by Earle and Eells [1], but, of course, there are many possible such embeddings.) Equality (3) expresses the fact that Σ is transverse to the orbits of D0 , and this will be of fundamental importance in the following approach to Plateau’s problem.

4.3 Symmetric Riemann Surfaces and Their Teichm¨ uller Spaces In attempting to obtain higher-genus solutions of Plateau’s problem via the calculus of variations, the natural domain of definition for our mappings are not closed oriented Riemann surfaces, but oriented Riemann surfaces with boundaries. In this section we shall indicate how to construct the Teichm¨ uller space for surfaces with k boundary components C1 , . . . , Ck , each diffeomorphic to the unit circle. Let M be an oriented, two-dimensional C ∞ -manifold with ∂M = C1 ∪ · · · ∪ Ck which is not the disk or the annulus. We wish to determine the structure of C/D0 where C is the space of all complex coordinate atlasses for M , and D0 is the space of diffeomorphisms which fix the boundary (any f ∈ D0 maps each Cj to itself and is isotopic to the identity). The trick to handle manifolds with

264


Fig. 1. (a) A Riemann surface M with boundary. (b) Its Schottky double

boundary is to reduce the problem to manifolds without boundary by means of a construction due to Schottky. Given a manifold (M, c) with a fixed complex structure c, we can consider ˇ , cˇ). The manifold M ˇ again has k boundary an exact duplicate of it, say, (M components Cˇ1 , . . . , Cˇk and the same coordinate atlas which we now denote ˇ. by cˇ. Moreover, for each point p ∈ M there is a symmetric point pˇ ∈ M ˇ and We construct the double 2M of M by forming the disjoint union M ∪ M identifying each point p ∈ Cj with its symmetric point p ∈ Cˇj , 1 ≤ j ≤ k. Then we have to check that 2M has a complex structure induced by the complex structure c of M . For points away from the curves of transition Cj we / ∂M , define the complex coordinate mappings as follows. If p0 ∈ 2M and p0 ∈ ˚ ∪M ˚ . Suppose that p0 ∈ M ˚ . Let (Gp , ϕ) ∈ c be a coordinate then p0 ∈ M 0 neighbourhood of p0 disjoint from ∂M . Define our new coordinate mapping ˚ , there ψ : Gp0 → C by ψ(p) = ϕ(p), the complex conjugate of ϕ(p). If p0 ∈ M ˇ , cˇ) disjoint from ∂ M ˇ . In this ˇ p , ϕ) ˇ ∈ ( M is a coordinate neighbourhood (G 0 ˇ p → C by ψ(p) = ϕ(p). ˇ If p0 ∈ Cj , then there is a complex case define ψ : G 0 coordinate system ϕ ∈ c which takes a neighbourhood Gp0 of p0 in M to the upper half B + of the unit disk in the complex plane with ϕ(Gp0 ∩Cj ) = (−1, 1) and ϕ(p0 ) = 0. ˇ pˇ , ϕ) ˇ pˇ → C, (G Similarly we can consider the map ϕˇ : G 0 0 ˇ the exact dupli˜ p of p0 in 2M by cate of (Gp0 , ϕ). Define a new coordinate neighbourhood G 0 the disjoint union ˇ pˇ ˜ p = Gp ∪ G G 0 0 0 with identifications made along ∂M as described above. Define a new coordi˜ p → C by setting ψ(p) = ϕ(p) if p ∈ Gp and ψ(p) = ϕ(p) ˇ nate mapping ψ : G 0 0 ˇ if p ∈ Gpˇ0 . Doing this for all p0 ∈ M we obtain a complex coordinate system 2c for 2M .


265

It is, however, important to note that 2c is not just an arbitrary complex structure for 2M , but also a symmetric complex structure. By this we mean that there is a map S of (2M, 2c) into itself which is an antiholomorphic ∂S = 0) diffeomorphism of 2M with S 2 = identity. We just set (locally ∂w S(p) = pˇ, S being the symmetry for (2M, 2c). Since we assumed at the outset that M be not the disk or the annulus, the genus of 2M is greater than one. We shall now consider only compact oriented surfaces 2M without boundary with a C ∞ -involution S, S 2 = id, such that the fixed point set of S consists of k disjoint curves C1 ∪ · · · ∪ Ck . Denote by CS the set of all complex structures on 2M with the property that, for each c ∈ CS , the involution S is an antiholomorphic map of (2M, 2c) to itself. Definition 1. The Teichm¨ uller space T(2M ) is defined to be the quotient space CS /D0 , where D0 are those C ∞ -diffeomorphisms of 2M to itself which are homotopic to the identity, S-symmetric and map each half of 2M to itself. We now want to follow the construction in Section 4.2 in order to show that the Teichm¨ uller space has the structure of a C ∞ -smooth finite dimensional manifold of dimension −3χ(M ), χ(M ) the Euler-characteristic of M . To start we do not consider all metrics on 2M but only those metrics for which S is an isometry. Let us call these symmetric metrics, and denote them by MS . The condition of symmetry immediately implies that the curves of transition are geodesic. In fact, this property characterizes those metrics on M which are the restriction of symmetric metrics on 2M . That is, if g is a metric on M such that ∂M are geodesics, then g can be extended to a symmetric metric on 2M . Then, as before, we obtain an identification between CS and the space MS−1 of all those C ∞ -symmetric metrics whose curvature is the constant −1. Moreover, we can identify MS−1 /D0 with CS /D0 and MS /PS /D0 , where PS is the set of C ∞ -symmetric positive functions on M . To achieve this identification we shall need the symmetric version of Theorem 3 of Section 4.2. Theorem 1. Let M be a smooth closed surface of genus greater than zero, endowed with a smooth metric g. Then there exists a metric G of constant scalar curvature conformal to g. The metric G is uniquely determined and depends smoothly on g if we require that the Gauss curvature K(G) satisfies K(G) = −1 in case of a negative Euler characteristic χ(M ), and that the volume of (M, G) equals that of (M, g) if χ(M ) = 0. If (M, g) is symmetric with respect to the isometry S, then also (M, G) is symmetric with respect to S. Proof. By the Gauss–Bonnet formula we have K(h) dμh = 2πχ(M ) ≤ 0 (1) M

for any metric h on M . Hence the curvature K(G) of the theorem must indeed be negative or zero depending on whether χ(M ) is negative or zero, since

266


K(G) is supposed to be constant. Writing G = ev g we obtain in conformal coordinates xα , xβ that gαβ = λδαβ , K(g) = −

1 Δ log λ, 2λ

Gαβ = ev λδαβ , K(G) = −

e−v Δ log(λev ), 2λ

1 Δv . K(G) = e−v K(g) − 2λ

whence

Requiring that K(G) = −1 if χ(M ) < 0 and K(G) = 0 if χ(M ) = 0 we obtain −Δg v + 2ev = −2K(g)

(2) and (2 )

−Δg v = −2K(g)

respectively, where Δg denotes the Laplace–Beltrami operator of (M, g). Let us first turn to (2 ). As the kernel of Δg consists of the constants, equation (1) expresses the orthogonality of K(g) to this kernel in L2 (μg ). Hence (2 ) is uniquely solvable under the normalization condition v dμg = 0. M

If, therefore, g is symmetric with respect to S, then v = v ◦ S since v ◦ S is also a solution to (2 ). By a simple scaling argument we see that the condition of equal volumes guarantees uniqueness as well. We now turn to the nonlinear equation (2). The uniqueness of G follows immediately from (2) by means of the maximum principle for elliptic equations. We remark that (2) is the Euler equation of the functional {ev + K(g)v} dμg I(v) = E(v, g) + 2 M

1

where E(v, g) = 2 M g(∇v, ∇v) dμg , and ∇v denotes the gradient of v with respect to g. This is, in fact, Dirichlet’s functional evaluated on the pair (v, g). We shall solve (2) by minimizing I in an appropriate set of functions, K. We observe that, if a solution v of (2) attains its maximal value in a point p ∈ M , then necessarily −Δg v(p) ≥ 0, and hence 0 ≤ ev(p) ≤ −K(g)(p), i.e., v(p) ≤ log | min K(g)| =: κ.


267

We therefore choose K := {u ∈ H21 (M ) : u(p) ≤ 1 + κ}. Since the first eigenvalue of −Δg on M is zero, with the corresponding eigenspace consisting of the constant functions, we have the estimate (3) 2E(u, g) ≥ λ2 u2 dμg M

for all u ∈ with u dμg = 0, where λ2 is the second eigenvalue of −Δg . It follows that the expression

1 (4) E(u, g) + u dμg , A := dμg , A M M H21 (M )

is a norm on H21 (M ). Let us now show that I is bounded from below on K, and that any minimizing sequence is bounded in the norm (1). Let u 1∈ K, and decompose u in u dμg . Using (1) and the form u = u0 + m with M u0 dμg = 0 and m = A M (3) we obtain K(g)u0 dμg + 2m K(g) dμg I(u) ≥ E(u, g) + 2 M

1 ≥ E(u, g) − λ2 4 ≥

u20 dμg − 4λ−1 2

M

1 E(u, g) + 4πmχ(M ) − 4λ−1 2 2

M

K(g)2 dμg + 4πmχ(M ) M

K(g)2 dμg . M

In case that m ≥ 0 we have |m| = m ≤ 1 + κ and therefore 1 −1 K(g)2 dμg . I(u) ≥ E(u, g) − 4π(1 + κ)|χ(M )| − 4λ2 2 If, however, m < 0, we obtain 1 I(u) ≥ E(u, g) − 4π|χ(M )||m| − 4λ−1 2 2

K(g)2 dμg .

We conclude immediately from the last two inequalities that I is bounded from below on K and that any minimizing sequence in K is bounded in H21 (M ). Standard compactness and lower semicontinuity arguments then give the existence of a minimizer v ∈ K (see Morrey [8], and also Chapter 4 of Vol. 1). Let us show that v ≤ κ. For arbitrary w ∈ K we obtain I(v) ≤ I((1 − t)v + tw) = I(v + t(w − v)).

268


d Then, using the fact that dt I(v + t(w − v))|t=0 ≥ 0, we clearly obtain the variational inequality {g(∇v, ∇(w − v)) + 2[ev + K(g)](w − v)} dμg ≥ 0 M

for all w ∈ K, where we again use ∇ as an abbreviation for the gradient with respect to the metric g. Inserting w = min{κ, v} we obtain {g(∇v, ∇v) + 2[ev + K(g)](v − κ)} dμg ≤ 0. u>κ

By the choice of κ we have ev +K(g) ≥ 0 for v > κ, and we conclude that v = 0 a.e. on the set {x ∈ M : v(x) > κ}. We thus have shown that v ≤ κ a.e. and, consequently, v lies in the interior of the set K with respect to the L∞ -norm. The Euler equations for v can then be derived in the usual way, proving that v is a weak solution of (2). Since the nonlinear term ev is already bounded, regularity of v follows from regularity theory (see Gilbarg and Trudinger [1]). In the symmetric case we minimize I in the class of symmetric functions KS = {u ∈ H21 (2M ) : u ≤ 1 + κ, u ◦ S = u}, and we find a symmetric minimizer v. By the same reasoning as above, the relation {g(∇v, ∇w) + 2[ev + K(g)]w} dμg = 0 (5) 2M

holds for all symmetric functions w ∈ H21 (2M ). Since g is symmetric and all functions in the integrand of (5) are symmetric, we may conclude that v is a weak solution of (2). The smooth dependence of G from g stated in our theorem follows readily from the implicit function theorem applied to the equation (2) since the linearization of (2) with respect to v is always an isomorphism between suitably chosen Sobolev or C k,α -spaces. The theorem is now completely proved. We now wish to characterize the tangent space to MS−1 at g, as we did earlier, as a direct summand, one of whose terms consists of tensors of the form LV g ∈ Tg Og (D0 ). This follows as in Section 4.2, with the exception that since we are considering only symmetric metrics g, we are imposing a condition on the metrics g along the curves of transition. To see what condition must be required on the subspace of Tg MS−1 complementary to Tg Og (D0 ), let p0 ∈ Cj , and con˜ p → C for 2M which locally flattens Cj . We sider a coordinate system ψ : G 0 ˜ p ) be the open unit disk B in the plane, and that shall require that ψ(G 0 ˜ p ∩ Cj ) = (−1, 1) × {0}, ψ(p0 ) = 0. Moreover, we assume that ψ takes ψ(G 0


269

symmetric points on 2M to symmetric points in B with respect to complex conjugation. In such a coordinate system we may represent g as (6)

g11 du2 + g22 dv 2 + 2g12 du dv.

However, locally, the map (u, v) → (u, −v) is required to be an isometry. Thus we must have equality in expression (6) when we replace (u, v) by (u, −v) and dv by −dv. Thus we find that g21 (u, v) = −g21 (u, −v), g11 (u, v) = g11 (u, −v), g22 (u, v) = g22 (u, −v). In particular we find that g12 (u, 0) = 0 holds on the boundary. We therefore conclude that in any coordinate system for 2M about a point p0 ∈ ∂M which respects symmetry (as above) the local representation of any symmetric metric must have its off-diagonal elements vanish at the boundary.3 We now state the symmetric analogue Theorem 3 of Section 4.2. Theorem 2. Let h ∈ Tg MS−1 be a symmetric two-tensor. Then h will also be S-symmetric4 and can be uniquely written as a direct sum h = hT T + LV g with LV g ∈ Tg Og (D0 ) (= the tangent space to the orbit of O0 through g). The S-symmetric tensor hT T can be expressed in conformal coordinates as hT T = a du2 − a dv 2 − 2b du dv where a + ib is locally a holomorphic function. If this conformal coordinate system taken about a point of the boundary respects symmetry and maps the boundary to the real axis as above, then we have b = 0 on the real axis (thus a + ib is real on the boundary). Corollary. Every h ∈ Tg MS−1 can be expressed uniquely as a direct sum (7)

h = Re(ξ(w) dw2 ) + LV g

where ξ(w) dw2 is a holomorphic quadratic differential which is real on the boundary, and the vector field V is tangent to ∂M .5 Moreover, each holomorphic quadratic differential on M arises as such a sum (7). As before, it follows from the theorem of Riemann–Roch that the dimension of the space of all such holomorphic quadratic differentials real on ∂M has dimension −3χ(M ) = 6g − 6 + 3k, and this therefore must be the dimension of the Teichm¨ uller space of such symmetric surfaces. One now imitates the previous construction to show 3

This fact can be shown to be equivalent to ∂M being a geodesic. The two notions of symmetry should cause no confusion. 5 This last condition on V means that the family of diffeomorphisms generated by V maps each boundary component to itself. 4

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Theorem 3. Let 2M be a symmetric surface with symmetry S as above. Then the Teichm¨ uller spaces CS /D0 and MS−1 /D0 have the structure of a finite dimensional C ∞ -manifold of dimension −3χ(M ). The tangent space at a point [g] ∈ MS−1 /D0 can be identified with Re(ξ(w) dw2 ), the real parts of holomorphic (with respect to the complex structure induced by g) quadratic differentials which are real on ∂M . We should like to point out that T S = MS−1 /D0 may be embedded as a finite dimensional C ∞ -submanifold Σ into MS−1 , in such a way that π : Σ → MS−1 /D0 is onto and that Tg Σ ⊕ Tg Og (D0 ) = Tg MS−1 . It also follows from Fischer and Tromba [4] that MS−1 /D0 is diffeomorphic to the Euclidean space to dimension −3χ(M ), which concludes our discussion of Teichm¨ uller spaces for oriented surfaces. Remark. One can in a similar way consider Teichm¨ uller theory for unoriˇ . The Teichm¨ uller space is ented surfaces M by passing to a Z2 -cover M then obtained as above from those metrics which satisfy an additional Z2 symmetry. This Teichm¨ uller space is needed for the existence theory of nonoriented minimal surfaces. We shall not develop these ideas here. We shall end this section with a discussion of the Weil–Petersson metric on T(M ). First, there is an L2 -metric on M−1 , , : T M−1 × T M−1 → R, given by 1 trace(HK) dμg (8) h, kg = 2 M where h, k ∈ Tg M−1 and H = g −1 h, K = g −1 k are the (1, 1)-tensors on M obtained from h and k via the metric g, i.e., by Hβα = g αγ hγβ , where (g αβ ) denotes the inverse of (gαβ ), and similarly for K. In local coordinates equation (8) can be written as 1 h, kg = {g αβ g γδ hαγ hβδ } dμg . (8 ) 2 M The inner product (8) is D0 -invariant (actually, D-invariant). This invariance is easy to see; namely, since g → f ∗ g is linear in g, D-invariance follows from (9)

F ∗ h, f ∗ kf ∗ g = h, kg .

4.4 The Mumford Compactness Theorem

But F ∗ h, f ∗ kf ∗ g =

1 2

1 = 2

271

tr(f ∗ H, f ∗ K)p dμf ∗ g M ∗

tr(f (HK)p ) dμf ∗ g M

1 = 2

tr(HK)f (p) dμf ∗ g . M

By the change of variables theorem this equals 1 tr(HK)p dμg = h, kg 2 M which proves (9). Thus D0 acts smoothly on M−1 as a group of isometries with respect to this metric, and consequently we have an induced metric on T(M ) in such a way that the projection map π : M−1 → M−1 /D0 becomes a Riemannian submersion. As it turns out (see Fischer and Tromba [3]), this induced metric is precisely the metric originally introduced by Weil, now called the Weil– Petersson metric. Let , be the induced metric on T(M ). We can characterize , as follows. We know that, given g ∈ M−1 , every h ∈ Tg M−1 can be written as (10)

˜ T T + LV g h=h

˜T T where LV g is the Lie derivative of g with respect to some (unique) V , and h is a trace free, divergence free, symmetric tensor. Moreover the decomposition ˜ T T a horizontal tangent vector in Tg M−1 and (10) is L2 -orthogonal. We call h LV g a vertical tangent vector. Let h, k ∈ T[g] T(M ). Then for any g ∈ π −1 [g] there exist unique horizontal ˜ T T ) = h and Dπg (k˜T T ) = k. ˜ T T and k˜T T such that Dπg (h tangent vectors h Then ˜ T T , k˜T T g . h.k[g] = h By D-invariance this is independent of the choice of g ∈ π −1 [g].

4.4 The Mumford Compactness Theorem We now come to the compactness theorem for the moduli space which is fundamental to any existence proof for minimal surfaces of higher genus within a given boundary configuration. In its original form the theorem is due to Mumford [1]. We present another proof, given by Tomi and Tromba [4], using only basic geometric notions instead of the uniformization theorem; this method also works for symmetric surfaces as well as for unorientable ones. For completeness we include the flat case since the proof requires only a few additional comments.

272


Theorem 1. Let M be a closed connected smooth surface, and {g n } be a sequence of smooth metrics of curvature −1 or 0 respectively on M such that all their closed geodesics are bounded below in length by a fixed positive bound. In the flat case we assume furthermore that the areas of the (M, g n ) are independent of n. Then there exist smooth diffeomorphisms f n of M which are orientation preserving if M is oriented, such that a subsequence of {f n∗ g n } converges in C ∞ towards a smooth metric. If M admits a symmetry S which is an isometry for all g n , then the maps f n can also be chosen to be S-symmetric and to map each half of M to itself. Before proceeding with the proof we should note that this is a compactness theorem for Riemann’s moduli space M−1 /D. The corresponding statement for the Teichm¨ uller space would be false. To see this we recall that D/D0 is an infinite discrete group. Let fn be a sequence of diffeomorphism such that its classes fn ∈ D/D0 have no convergent subsequence. Let g be any metric, and consider the orbit class {fn∗ g}. If the Mumford theorem were true for Teichm¨ uller’s moduli space, this would imply the existence of a sequence hn ∈ D0 such that h∗n (fn∗ g) = (fn ◦ hn )∗ g converges in M−1 (say, in the H s topology, s > 2). By Section 4.2, and in particular by the Ebin–Palais lemma, this implies that fn ◦ hn has a convergent subsequence. Thus fn ◦ hn = fn has a convergent subsequence, and we have found a contradiction. We can now proceed to the Proof. Since, on a negatively curved surface, there are no conjugate points along any geodesic, it follows that every geodesic arc is a relative minimizer of the arc length (with fixed end points). Therefore, any two geodesic arcs with common endpoints cannot be homotopic with fixed end-points; otherwise, by a common Morse-theoretic argument (see Milnor [1]), there would exist a nonminimizing geodesic arc joining these endpoints. Hence we may conclude that a lower bound l on the lengths ln of the closed geodesics of g n implies a bound on the injectivity radii ρn of M n = (M, g n ), ρn ≥ ρ ≥ l/2. It follows that on each open disk BR (p), p ∈ M n and R ≤ ρ, one can introduce a geodesic polar coordinate system. By a classical result in differential geometry, which can easily be derived from the results of Chapter 1 in Vol. 1, the metric tensor associated with gn in these coordinates assumes the form

(sinh r)2 if R(g n ) = −1, 1 0 n , f (r) = (1) (gij ) = 0 f (r) r2 if R(g n ) = 0, where r denotes the polar distance. For the area of BR (p) we obtain from (1) the simple estimate area BR (p) ≥ πR2 . The genus of the manifolds M n being fixed, the total area of M n is determined by the Gauss–Bonnet formula if R(g n ) = −1. It follows that there is an upper


273

bound, depending only on R, for the number of disjoint open disks BR (p) in M n . Let us now take R = 14 ρ, and let N (n) be the maximal number of open disjoint disks of radius 14 R in M n . By passing to a subsequence we can assume that N (n) = N holds independently of n. It follows that for each n ∈ N we can find points pnj ∈ M n , j = 1, . . . , N , with the property that the disks B1/4R (pnj ) are disjoint while the disks B1/2R (pnj ) cover M n . Let us now denote by H the Poincaré upper halfplane in the hyperbolic case6 and the Euclidean plane in the flat case. We pick an arbitrary point ζ0 ∈ H, e.g. ζ0 = i, the imaginary unit, and introduce geodesic polar coordinates on B4R (pnj ) ⊂ M n and on B4R (ζ0 ) ⊂ H, respectively. The corresponding metric tensors assume the same form (1) in each of both cases, and we may therefore conclude that there exist isometries ϕnj : B4R (pnj ) → B4R (ζ0 ),

ϕnj (pnj ) = ζ0 .

Let then I n denote the set of all pairs (j, k), 1 ≤ j, k ≤ N , such that B2R (pnj ) ∩ B2R (pnk ) = ∅. By passing to a subsequence we can assume that I n = I is independent of n. For (j, k) ∈ I, the transition mappings n := ϕnj ◦ (ϕnk )−1 : ϕnk [B4R (pnj ) ∩ B4R (pnk )] → ϕnj [B4R (pnj ) ∩ B4R (pnk )] τjk

are well defined local isometries of H. Before proceeding further with the proof we first want to show that any such local isometry in fact extends to a global one. We only consider the hyperbolic case, the flat one being trivial. Lemma 1. Let f : U → H be a C 1 isometry on an open connected subset U of the hyperbolic plane. Then f (w) =

Aw + B , Cw + D

A, B, C, D ∈ R,

and AD − BC = 1. Aw+B Proof. The class of maps w → Cw+D , AD − BC = 1 with real coefficients is the group of isometries of the Poincaré metric. Thus we must show that a local isometry is also a global isometry. It is clear that we can take f to be orientation preserving. Then an easy calculation shows that f must be holomorphic and has to satisfy the nonlinear condition

(2)

|f (w)| =

Im f (w) . Im w

This is the half plane (u, v) ∈ R2 with v ≥ 0 endowed with the metric {(du)2 +(dv)2 }/4v 2 . The scalar curvature of this metric is ≡ −1.

6

274


Aw+B One can check that every map of the form w → Cw+D as above satisfies condition (2) and that the set of maps satisfying (2) from a fixed domain to itself forms a group. Therefore, by composition with an appropriate element of the three-dimensional conformal group of H we may assume that f satisfies the following additional conditions: f is defined in a neighbourhood of i ∈ H, and

f (i) = Im f (i).

(3)

Now, writing w = u + iv and using (2), we have (log f ) = {Re(log f )}w = {log |f |}w =

vw −if i (Im f )w − = + . Im f v Im f v

By (3), (log f ) (i) = 0. Similarly

if (log f ) = − Im f

=

i + v w w

if −if i2 − (if ) + . Im f (Im f )2 v2

Again we see that (log f ) (i) = 0. Proceeding inductively we obtain (log f )n (i) = 0 for all n ∈ N. Thus, since log f is holomorphic in a neighbourhood of i, it follows that log f is constant, and so the mapping w → f (w) is constant; therefore f (w) = w. Since we normalized f by the isometry group of H, this proves that our initial map f must be in this isometry group, and the proof of the lemma is complete. n := ϕnj (pnk ) ∈ B4R (ζ0 ), For (j, k) ∈ I we have pnk ∈ B4R (pnj ), and hence qjk n since ϕj is an isometry. It is obvious from the definition that

(4)

n n n = τjk (qkj ). qjk

We are now going to construct a limit manifold of the sequence M n = (M, g n ). For this purpose we prove n )n∈N is compact for each Lemma 2. The family of transition mappings (τjk (j, k) ∈ I. n Proof. By Lemma 1, each τjk is a global isometry of H and there are a fixed n ∈ K such that (4) holds. From this compact subset K of H and points qjk n decomposes into the assertion follows at once in the flat case, since each τjk a rotation and a bounded translation. In the hyperbolic case, by composition


275

with a conformal map of H onto the unit disk B ⊂ R2 we may assume that n is a conformal map of B onto itself and (suppressing the indices j, k) each τjk that there are points pn strictly staying away from ∂B such that also τ n (pn ) stays away from ∂B. Each τ n is of the form τ n (w) = dn

w − an , 1−a ¯n w

where |an | < 1, |dn | = 1. It suffices to show that |an | stays strictly below 1. If not, we can assume an → a, |a| = 1, and dn → d, |d| = 1. The limit map a ¯w − 1 w−a = ad = −ad τ (w) = d 1−a ¯w 1−a ¯w then collapses the disk onto a point on ∂B, which is a contradiction.

We can now continue with the proof of Mumford’s theorem. Passing to subsequence we can by Lemma 2 assume that n τjk → τjk

(5)

as (n → ∞).

ˆ as the disjoint union of N disks BR (ζ0 ) ⊂ We now define a limiting manifold M H, labelled as B1 , . . . , BN with the identifications p ∈ Bj

equals q ∈ Bk ⇔ (j, k) ∈ I

and

p = τjk (q).

ˆ is a differentiable manifold carrying a natural Riemannian It is clear that M metric which on each Bj coincides with the Poincaré metric or Euclidean ˆ is compact. Assume to the contrary metric, respectively. We claim that M that there were a point q ∈ ∂BR (ζ0 ) such that q ∈ / τjk (BR (ζ0 )) for some j and all k with (j, k) ∈ I. Then it would follow that, for sufficiently large n, we n have q ∈ / τjk [B(3/4)R (ζ0 )], which means that (ϕnj )−1 (q) ∈ / B(1/4)R (pnj ). This, however, would imply that B(1/4)R [(ϕnj )−1 (q)] ∩ B(1/4)R (pnk ) = ∅ for k = 1, . . . , N, contradicting the choice of N as the maximal number of disjoint disks in M n of radius R/4. The remainder of the proof rests upon the following ˆ → M n , f n (Bj ) ⊂ B2R (pn ) Lemma 3. There are diffeomorphisms f n : M j such that (6)

ϕnj ◦ f n → id

in C ∞ on each Bj ,

as n → ∞.

In the symmetric case f n can be chosen to be symmetric, i.e. to commute ˆ and M n respectively, with the symmetries Sˆ and S n on M ˆ S n ◦ f n = f n ◦ S.

276


The proof of this lemma is somewhat technical, and we refer the reader to Tomi and Tromba [4] for a proof. Let us quickly finish the proof of Mumford’s theorem assuming the lemma. Denoting by g the Poincaré metric or the Euclidean metric respectively, we have from (6) that f n∗ ϕn∗ j g → g as n → ∞ on each Bj . Since, however, ϕnj was an isometry between g and g n on M n , this means that f n∗ g n → g as n → ∞ ˆ . Choosing now any (symmetric) diffeomorphism f : M → M ˆ , we obtain on M (f n ◦ f )∗ g n → f ∗ g which proves Mumford’s theorem.

as n → ∞,

4.5 The Variational Problem Given a smooth compact surface M with k boundary components and k disjoint curves Γ1 , . . . , Γk in RN , N ≥ 2, we would like to prove the existence of a minimal surface X : M → RN such that X|∂M parametrizes Γ1 ∪ · · · ∪ Γk . (We do not prescribe the orientation of the Γj because this might lead to degeneration.) The question of whether such a minimal surface is immersed or not is as in the case of disk a separate question (see Gulliver [7]). Also, as in the disk case, the need to control parametrizations in finding surfaces of least area spanning Γ1 , . . . , Γk leads one to the definition of a minimal surface as a conformal harmonic map X : M → RN on a given parameter surface M into RN . The topological type of X is given by M . As M is no longer a planar domain, we have now to define what we understand to be a conformal harmonic mapping X of M into RN . The following is in a way a repetition of what we have said in Section 3.6 of Vol. 1. Suppose that g is a metric on M , and let Δg be the Laplace–Beltrami operator corresponding to g. In local coordinates ϕ : G → R2 on M we have

1 ∂ ∂ −1 αβ (Δg X) ◦ ϕ = √ g det g β X ∂u det g ∂uα where det g = det(gαβ ). The mapping X : (M, g) → RN is said to be harmonic if (1)

Δg X = 0.

It turns out that X : M → RN is harmonic if and only if, for any system of conformal coordinates on M defined by ϕ : G → R2 , the pull-back X := X◦ϕ−1 is harmonic in the classical sense, i.e. ΔX = 0. Note that the definition

4.5 The Variational Problem

277

of harmonicity by means of the equation (1) is intrinsic, i.e., independent of the chosen coordinates (u1 , u2 ) on M . Also, the definition of harmonicity of X by means of the equation ΔX = 0 for X = X ◦ ϕ−1 and a conformal map ϕ : G → R2 is intrinsic since the transition map between two conformal coordinate system is holomorphic, and harmonic mappings of domains in R2 remain harmonic after composition with holomorphic mappings. We call a mapping X : (M, g) → RN conformal if, for any system ϕ : G → 2 R of conformal coordinates w = u1 +iu2 on (M, g), the pull-back X = X◦ϕ−1 satisfies (2)

Xw , Xw = 0,

that is, if X(w) = (X 1 (w), X 2 (w), . . . , X N (w)), then (2 )

1 2 2 2 N 2 (Xw ) + (Xw ) + · · · + (Xw ) = 0.

Clearly, this definition of conformality of a mapping X is intrinsic as well. Let c be a complex structure on M . Then we can equally well view X as a mapping from (M, c) into RN , and we can call X : (M, c) → RN harmonic and conformal if, for any chart (G, ϕ) ∈ c, the function X = X ◦ ϕ−1 is harmonic and satisfies (2). These two definitions mean the same since, because of the one-to-one correspondence C ↔ M/P, for every conformal equivalence class of metrics g on M there is exactly one complex structure c = c(g) generated by g. The precise formulation of this bijective correspondence between complex structures on a manifold M with boundary and symmetric Riemannian metrics on M , genus M > 1, of constant Gauss curvature −1 was given in Section 4.3. This leads us to the following two equivalent formulations (P0 ) and (P) of Plateau’s problem: (P0 ) Given M and Γ1 , . . . , Γk as above, determine a complex (or conformal) ˚ ) and a map X : M → RN , continuous structure c on M (not just on M on M and smooth in the interior, such that (a) X|∂M : ∂M → Γ1 ∪ · · · ∪ Γk is one-to-one and preserves orientations if M is oriented. (b) X is conformal and harmonic with respect to c. (P) Given M and Γ1 , . . . , Γk as above, determine a smooth symmetric metric g on 2M of Gauss curvature −1 and a map X : M → RN , continuous on M and smooth in the interior, such that (a) X|∂M → Γ1 ∪ · · · ∪ Γk is one-to-one and preserves orientations if M is oriented. (b) the induced metric X∗ gN (gN the Euclidean metric on RN ) is conformal to g; i.e., X∗ gN = λg, λ ≥ 0, and X is g-harmonic; i.e. Δg X = 0.

278


Let us now introduce the generalized Dirichlet integral E(X, g) as a functional on mappings X : M → RN and on metrics g. The functional E is defined by 1 g(p)[∇g Xj , ∇g Xj ] dμg (3) E(X, g) = 2 M where g(p) : Tp M × Tp M → R is a Riemannian metric on M, ∇g Xj is the g-gradient of the j-th component function of X, and μg is the classical volume measure induced by the metric g. In local coordinates (u1 , u2 ) on M , the functional E is given by 1 g αβ Xuα , Xuβ dμg (u1 , u2 ) E(X, g) = 2 M

where dμg (u1 , u2 ) = det(gαβ ) du1 du2 , and we are using the standard conventions of differential geometry. With a slight abuse of notation we have written X(w) instead of X(w) for X = X ◦ ϕ−1 and a chart (G, ϕ). Obviously, the functional E(X, g) depends only on the conformal class of g, i.e., E(X, g) = E(X, λg) for any positive function λ on M . One also readily verifies the invariance of E under diffeomorphisms f of M . E(X, g) = E(X ◦ f, f ∗ g). If g is a symmetric metric on the double 2M (equivalently, if ∂M is a geodesic for M ), the conformal invariance of Dirichlet’s energy and the fact that, given any g, there is a unique λ such that the Gauss curvature K(λg) ≡ −1 immediately implies that we may restrict Dirichlet’s energy to symmetric metrics of Gauss curvature −1, i.e., to g ∈ MS−1 . Now let Γ be an oriented Jordan curve (or a collection of Jordan curves Γ1 , . . . , Γk ) in RN , and set ηΓ = {X : M → RN , X ∈ H21 (M, RN ) ∩ C 0 (M, RN ), X : ∂M → Γ monotonically}. Let E : ηΓ × MS−1 → R be Dirichlet’s energy. For fixed g, suppose we have a minimum X0 satisfying E(X0 , g) ≤ E(X, g) for all X ∈ ηΓ . In the case that M is the disk, this implies that X0 is harmonic and conformal, i.e., a minimal surface. However, this is not any longer true in the higher genus case. Instead one also has to vary the metrics g on M . Minimizing over all metrics we shall be able to produce a conformal map.


279

Suppose now that we had a minimum pair (X0 , g0 ) for E, that is, E(X0 , g0 ) ≤ E(X, g) for all (X, g) ∈ ηΓ × In general such a minimum will not exist as we shall presently see. For the moment we would like to understand why a minimum is a harmonic conformal map. The relation E(X0 , g0 ) ≤ E(X, g0 ) MS−1 .

for all X ∈ ηΓ immediately implies as in the disk case that X0 is harmonic, ΔX0 = 0, ˚ , RN ). whence X0 ∈ C ∞ (M If we knew that the minimum X0 would be smooth up to the boundary ∂M provided that Γ were smooth, say, of class C ∞ , then we would be able to deduce the additional boundary condition ∂X0 ∂X0 , ≡ 0 on ∂M (4) ∂N ∂T i.e. the normal derivative would be perpendicular to the tangential derivative. In a local conformal coordinate system flattening ∂M this would be equivalent ∂X to ∂X ∂u , ∂v ≡ 0. In the case of the disk, equation (4) suffices to insure that ∂X0 ∂X0 , ≡ 0, (5) ∂z ∂z i.e. that X0 is a minimal surface. Interestingly, one can show that (4) holds up to the boundary ∂M even though X0 may not a priori be known to be smooth up to the boundary. We have the following result. Theorem 1. Fix some metric g and let X0 be a minimum for X → E(X, g), X ∈ ηΓ . Then the expression 2 N ∂Xj0 (6) ξ= dw2 ∂w j=1 is a holomorphic quadratic differential of class C ∞ (up to the boundary) which 0 ∂X0 is real on ∂M ; i.e. ∂X ∂u , ∂v ≡ 0 in any local conformal coordinate system flattening ∂M . ˚ follows Proof. That ξ is a holomorphic quadratic differential of class C ∞ on M ∞ ˚ immediately from the fact that X0 is C smooth on M . Moreover, ξ is of class C ∞ up to the boundary and real on ∂M as was proved in the disk case in Tromba [16], and since this is only a local result, the same proof applies with only superficial alterations. We refer the reader to this source for a proof.

280


We would now like to prove Theorem 2. If (X0 , g0 ) is a minimum for the functional (X, g) → E(X, g),

(X, g) ∈ ηΓ × MS−1 ,

then X0 : (M, g0 ) → RN is harmonic and conformal, i.e., 2 N ∂Xj0 ξ= dw2 ≡ 0. ∂w j=1 Proof. We have already established harmonicity. Let us calculate the derivative of g → E(X0 , g) with respect to the variable g. Recall (cf. Section 4.3) that Tg MS−1 can be decomposed as an L2 -orthogonal direct sum, with h ∈ Tg MS−1 expressed as h = LV g + hT T where hT T is the real part of a holomorphic quadratic differential vanishing on ∂M ; hT T = Re ζ. Again suppose for the moment that X0 is smooth up to ∂M and let ft be a one parameter family of diffeomorphisms with f0 = id, df dt |t=0 = V . Then d ∗ (f g0 ) = LV g0 dt t

(7) is the derivative of g0 , and

d ∗ (f X0 ) = dX0 (V ) dt t

(8)

is the derivative of X0 in the direction V . By the invariance of Dirichlet’s energy (3) we obtain (9)

E(ft∗ X0 , ft∗ g0 ) = E(X0 , g0 ).

Therefore it follows that

d E(ft∗ X0 , ft∗ g0 ) = 0, dt t=0

and we arrive at the differential relation (10)

∂E ∂E (X0 , g0 )[dX0 (V )] + (X0 , g0 )[LV g0 ] = 0. ∂X ∂g

∂E Here we have used the notations ∂E ∂X and ∂g for the first variations of E with respect to the first and the second arguments respectively, i.e.,


∂E d ∗ (X0 , g0 )[dX0 (V )] := E(ft X0 , g0 ) ∂X dt

281

, t=0

∂E d (X0 , g0 )[LV g0 ] := E(X0 , ft∗ g0 ) . ∂g dt t=0 Now, if X0 is a minimum (or even merely a critical point) for the mapping X → E(X, g0 ), it follows that (11)

∂E (X0 , g0 ) = 0 ∂X

which implies that (12)

∂E (X0 , g0 )[LV g0 ] = 0 ∂g

for all vector fields V arising as infinitesimal symmetric diffeomorphisms. This says that, formally, at a minimum (or more generally at any critical point) derivatives of Dirichlet’s energy with respect to g in the vertical directions LV g are zero and therefore yield no additional information. Therefore, if we are to conclude that ξ = 0 we should consider derivatives of E with respect to g only in horizontal directions (cf. formula (10) of Section 4.3). Lemma 1. Let (X, g) ∈ ηΓ × MS−1 , and let ρ = hT T be a trace free divergence free (0, 2)-tensor, ρ = Re{ν(w) dw2 } where ν is a holomorphic quadratic differential that is real on ∂M . Then (13)

1 ∂E (X, g)ρ = − Re ξ, ρg . ∂g 2

Here , denotes the L2 -inner product introduced at the end of Section 4.3, and 2 N

∂Xj (dw)2 ξ= ∂w j=1 is a (0, 2)-tensor of class L2 on M . Proof of Lemma 1. In local coordinates 1 g αβ Xuα , Xuβ dμg (u1 , u2 ). E(X, g) = 2 M Now the derivative of the function g → μg in the direction h can easily be calculated to be 1 trg h μg h→ 2 where trg h = g αβ hαβ .

282


Therefore, if h is trace free, the derivative of this term vanishes. Just as easily one computes that the derivative of g → g αβ in the direction h is h → −hαβ , where hαβ = g αγ g βδ hγδ . Then we obtain for dE(X, g)ρ := dE(X, g)ρ = −

(14)

1

∂E ∂g (X, g)ρ

1 2

the formula

ραβ Xuα , Xuβ dμg . M

2

In conformal coordinates (u , u ) = (u, v) we have ραβ = λ−2 ραβ ,

gαβ = λδαβ ,

and we see that (14) is equal to dE(X, g)ρ = −

1 2

dμg = λ du dv,

λ−2 ραβ lαβ λ du dv, M

where lαβ := Xuα , Xuβ . Since ρ is trace free, we have ρ11 = −ρ22 , and we obtain 1 λ−2 {ρ11 (l11 − l22 ) + 2ρ12 l12 }λ du dv. dE(X, g)ρ = − 2 As

⎤ N Re ξ = Re ⎣ (Xjw )2 (dw)2 ⎦ = (l11 − l22 )[(du)2 − (dv)2 ] + 4l12 du dv ⎡

j=1

we obtain (Re ξ)11 = −(Re ξ)22 = l11 − l22 , Therefore, 1 (Re ξ), ρ = 2 = =

1 2 1 2

(Re ξ)12 = (Re ξ)21 = 2l12 .

g αβ g γδ (Re ξ)αγ ρβδ dμg M

λ−2 (Re ξ)αδ ραδ dμg

M

λ−2 {2ρ11 (l11 − l22 ) + 4ρ12 l12 }λ du dv

M

= −2 dE(X, g)ρ, which yields Lemma 1.


283

We are now in a position to complete the Proof of Theorem 2. By Theorem 1 the C ∞ -quadratic differential ξ is holomorphic and real on ∂M . By Lemma 1 we have 1 dE(X0 , g0 )ρ = − Re ξ, ρg0 = 0 2 for all ρ = Re(ν(w) dw2 ), where ν is any quadratic differential which is holomorphic and real on ∂M . This immediately implies that ξ = 0. As we have already observed, the second variable of Dirichlet’s energy belongs to MS−1 . Fortunately it is possible to reduce the infinite dimensional space MS−1 to a finite-dimensional C ∞ -manifold, without changing the critical points, in considering Dirichlet’s energy as follows. The bundle π : MS−1 → MS−1 /D0 = T(M ) is a trivial principal fibre bundle of class C ∞ and admits a C ∞ -smooth section τ : T(M ) → MS−1 (see Fischer and Tromba [1]). Let Σ ⊂ MS−1 be the image of such a section. Then Σ is a smooth finite-dimensional C ∞ -submanifold of MS−1 , everywhere transverse to the group of symmetric diffeomorphisms homotopic to the identity, and whose dimension is equal to that of T(M ). The reader may now verify the following Theorem 3. Consider Dirichlet’s energy restricted to ηΓ × Σ, (15)

E : ηΓ × Σ → R.

Then the critical points of E (among them the minimizers) are precisely the minimal surfaces spanning Γ . Now that we know that the critical points of (15) are minimal surfaces, it seems natural to try to generate minimal surfaces of a given genus spanning a curve Γ by finding a minimizer of (15). In general there will not exist a minimizer of prescribed topological type. For example, a plane circle cannot bound any minimal surface apart from a disk. In order to generate a minimal surface, one would take a minimizing sequence, say, (Xn , gn ) ∈ ηΓ × Σ and attempt to find a convergent subsequence, again denoted by (Xn , gn ), such that (Xn , gn ) → (X0 , g0 ) and E(X0 , g0 ) ≤ lim inf E(Xn , gn ). n→∞

To see what can go wrong in attempting to produce such a minimizing sequence let us consider the simple case of two coaxial circles Γ1 , Γ2 of the same radius R, although this case does not fit into the picture we have developed so far. Here M is an annulus, and 2M is a torus whose genus is one. In this case the Teichm¨ uller space consists of symmetric metrics of zero curvature

284


Fig. 1. Four terms of a degenerating minimizing sequence with shrinking necks. This phenomenon is excluded by the condition of cohesion

and fixed volume modulo the group of symmetric diffeomorphisms homotopic to the identity. Thus there is still a Σ, but it is not a submanifold of MS−1 but of the manifold of symmetric flat metrics of prescribed volume. Yet this example is instructive as it will show what may go wrong in the general case. If the distance d is small enough then the boundary configuration Γ1 , Γ2 bounds two catenoids of revolution. If, however, d exceeds a critical number d∗ , then no minimal surface of the type of the annulus can span Γ1 , Γ2 ; cf. Section 4.1 of Vol. 2. Suppose d > d∗ , and choose a minimizing sequence {Xn , gn } for E. It will develop a narrower and narrower neck, and the images of the Xn will be degenerating in topological type by tending to two disjoint disks spanning Γ1 and Γ2 . Notice that, as the Xn degenerate, the images of homotopically nontrivial loops shrink to zero in length. This observation is the basis of Courant’s condition of cohesion described below. The first to cope with minimal surfaces of higher topological type (than the disk) and with the phenomenon of degeneracy was Douglas. He formulated the following result. Theorem of Douglas. Let α denote the infimum of the Dirichlet integrals of all oriented connected surfaces of genus g spanning the given curves Γ1 , . . . , Γk and let α∗ be the corresponding infimum over all oriented connected surfaces of genus less than g and all oriented, disconnected surfaces of total genus g consisting of two or more components spanning proper, non-empty, disjoint subsets of Γ1 ∪ · · · ∪ Γk whose union equals Γ1 ∪ · · · ∪ Γk . If α < α∗ then there exists an oriented minimal surface of genus g and having Γ1 ∪ · · · ∪ Γk as boundary. Since in any topological class the infimum of Dirichlet’s integral coincides with the infimum of area (see Chapters 4 and 8 of Vol. 1 for disks and schlicht domains, and Tomi and Tromba [4] for the general case), the above theorem can be rephrased replacing Dirichlet’s integral by area. Another approach to the general Plateau problem is due to Courant and Shiffman. They attacked the problem by staying in a class of surfaces of fixed topological type. Courant noted that degeneration of minimizing sequences can be excluded by imposing a certain additional condition, the so-called condition of cohesion.

4.6 Existence Results for the General Problem of Plateau in R3

285

Courant’s condition of cohesion. A family F of (differentiable) mappings X : M → RN satisfies the condition of cohesion if there is a positive lower bound for the length of the images under any X ∈ F of all homotopically non-trivial closed loops on M . In their work on minimal surfaces in Riemannian manifolds, Schoen and Yau [2] used the same condition under the name of incompressibility condition. While Courant first applied his method only to surfaces with schlicht parameter domains (the general case is sketched in Courant [15], Section 4.3), Shiffman [3] later extended the method to the higher-genus case. He essentially proved the following result. Theorem (Shiffman). If there is a minimizing sequence for Dirichlet’s integral in the class of all surfaces of genus g spanning Γ1 , . . . , Γk which satisfies the condition of cohesion, then there is a minimizing minimal surface in this class. Douglas’ proof of his theorem was quite ingenious, but not entirely satisfactory; in fact, some parts are not established at all. The proofs of Courant and Shiffman are completely stringent; however, Shiffman did not explicitly derive Douglas’s theorem from his result; possibly he considered this more or less evident. A presentation of the Courant-Shiffman method can be found in Courant’s treatise [15]. A modified version of Courant’s approach to the Douglas problem for surfaces defined on schlicht domains was described in Chapter 8 of Vol. 1, assuming the sufficient condition of Douglas. Modern and complete proofs of the results of Douglas and Shiffman are given in Jost [6] and Tomi and Tromba [5]. In the following section we want to present sufficient geometric-topological criteria for a system of rectifiable Jordan curves Γ1 , . . . , Γk in R3 which guarantees the existence of a minimal surface of prescribed topological type spanning the configuration Γ1 , . . . , Γk . This result will not be as general as the theorems of Douglas–Courant–Shiffman. On the other hand it is rather satisfactory as it replaces the sufficient condition of Douglas or Courant’s condition of cohesion by concrete geometric condition that can, in principle, be checked.

4.6 Existence Results for the General Problem of Plateau in R3 We want to derive a sufficient condition ensuring the existence of a minimal surface of prescribed topological type within a configuration Γ1 , . . . , Γk of rectifiable Jordan curves Γj .

286


We first set up our variational problem from a different point of view which allows the introduction of an artificial constraint, or obstacle, in form of a 3dimensional submanifold T with boundary which is of sufficient topological complexity depending on the topological type of surfaces we wish to produce. We will then restrict the class of admissible surfaces to those surface contained in the submanifold T . By a topological condition on the position of Γ in T we can ensure that homotopically nontrivial loops on an admissible surface are also homotopically nontrivial in T and therefore bounded below in length. This will guarantee (via the condition of cohesion) the existence of a minimizing surface which in general, however, need not be a minimal surface in R3 since it can touch the boundary of T along portions of arbitrary size. If, however, T is H-convex, i.e. if the inward mean curvature of ∂T is nonnegative then we may apply a maximum principle and conclude that a minimizing surface in T is actually contained in the interior of T and hence is a minimal surface in R3 (Vol. 2, Chapter 4). We treat the case of one boundary curve and oriented genus-one minimal surfaces in complete detail and shall later indicate the modifications necessary for tackling higher genus, several contours, and unoriented surfaces. We begin with a few preliminaries. Again let X be of class H21 (M, RN ), and let us consider Dirichlet’s energy, where M is as before and 1 g αβ Xuα , Xuβ dμg . E(X, g) = 2 Lemma 1. Let X be a map of Sobolev class H21 (M, RN ) and g a smooth metric on M . (i) If the pair (X, g) is stationary for E with respect to all smooth variations ˚ = M \ ∂M , then the metric X∗ gN induced of g compactly supported in M by the Euclidean metric gN is conformal to g, i.e. X∗ gN = λg a.e. for some λ ≥ 0. (ii) If the pair (X, g) is stationary for E with respect to all smooth variations ˚ , then X is smooth and g-harmonic. of X compactly supported in M Proof. We need only prove (i) since again (ii) is a standard result from the calculus of variation (see also Chapter 4 of Vol. 1). With the abbreviations γ = det(gαβ ),

lαβ = Xuα , Xuβ

we compute for any smooth symmetric 2-tensor h = (hαβ ) on M that 1 1 √ dg E(X, g)h = −g αν hνσ g σβ + g νσ hνσ g αβ lαβ γ du1 du2 2 M 2 1 1 √ −g να lαβ g βσ + g αβ lαβ g νσ hνσ γ du1 du2 . = 2 M 2


287

From our hypothesis it therefore follows that g να lαβ g βσ = λg νσ

a.e. on M with λ =

1 αβ g lαβ , 2

which can easily be rewritten as lαβ = λgαβ .

It is now obvious that each critical point of E on a suitable space of pairs (X, g) will furnish a solution to Plateau’s problem, and vice versa. In what follows we shall solely be concerned with absolute minima of E. The next theorem, a basic result of this section, illustrates the importance of the cohesion condition formulated in Section 4.5. Theorem 1. Let M be a compact smooth surface which is not simply connected and has k ≥ 1 boundary components C1 , . . . , Ck with the genus of the Schottky double 2M greater than one (this excludes the annulus), and let Γ1 , . . . , Γk be pairwise disjoint rectifiable Jordan curves in RN . Furthermore, let a sequence (Xn , gn )n ∈ N, be given where each Xn is a mapping of class C 0 ∩ H21 (M, RN ) which maps Cj monotonically onto Γj (j = 1, . . . , k), and where the gn are smooth metrics on M . We suppose that the Dirichlet integrals E(Xn , gn ) as well as sup |Xn | are uniformly bounded and that the family {Xn } satisfies the condition of cohesion. Then there exists a smooth metric g on M and a map X ∈ H21 (M, Rn ) such that X |∂M is continuous and maps each Cj monotonically onto Γj and such that E(X, g) ≤ lim inf E(Xn , gn ). n→∞

In case that M is oriented and that all Xn map Cj onto Γj with a prescribed fixed orientation, then X can be chosen to map Cj onto Γj in the same orientation. The proof of Theorem 1 will be carried out in several steps. The following theorem links the condition of cohesion with the hypothesis of Mumford’s compactness theorem (see Section 4.4) on a lower bound of the length of closed geodesics on (M, g). This result is a simple consequence of the collar theorem of Halpern [1] and Keen [1]; its usefulness for minimal surfaces was observed by Schoen and Yau [2], and in what follows we present their argument. Theorem 2. Let (M, g) be a closed oriented surface with R(g) = −1 and let X : M → RN be a map of class C 0 (M, RN ) ∩ H21 (M, RN ) such that for all homotopically non-trivial closed C 1 -loops α on M the length of X ◦ α is bounded below by δ > 0. Then the length l of any closed geodesic γ on (M, g) can be estimated by √ 1 2 5 −1 E(X, g) l ≥ min 1, δ π − 2 arctg . 2 4

288


Remark. Corresponding estimates are obtained for unoriented surfaces by passing to the oriented cover. Proof. By standard results in differential geometry (cf. Section 4.4), there is an isometry Φ of a neighbourhood U of γ with a region T in the Poincaré half plane, T = {reiθ : 1 ≤ r ≤ el , θ0 < θ < π − θ0 } where 0 < θ0
0 of any two of them. For δ > 0 denote by σ(δ) the supremum of the lengths of all shortest subarcs of Γ1 ∪ · · · ∪ Γk joining any two points at spatial distance not exceeding δ. (Clearly, σ(δ) → 0 as δ → 0.) Furthermore, let X : 2M → RN be a symmetric continuous map which maps Cj monotonically onto Γj , j = 1, . . . , k, and let α be a homotopically nontrivial loop on 2M such that the length L(X ◦ α) < ρ. Then there exists a homotopically non-trivial loop α0 on M with L(X ◦ α0 ) ≤ L(X ◦ α) + σ(L(X ◦ α)).


289

Proof. If α is totally contained either in M or in S(M ), we may set α0 = α or α0 = S ◦ α, respectively. In the remaining case let α1 , α2 , . . . be the maximal (open) subarcs of α contained in S(M ) with endpoints pj , qj on M . We infer from |X(pj ) − X(qj )| ≤ L(X ◦ αj ) ≤ L(X ◦ α) < ρ that pj and qj are always contained in the same component of ∂M . We can therefore connect pj and qj by some arc βj in ∂M with the property that X ◦ βj is the shortest subarc of Γ1 ∪ · · · ∪ Γk joining X(pj ) and X(qj ). If the arcs αj and βj are homotopic in S(M ) with fixed endpoints for all j, then also Sαj and Sβj = βj are homotopic in M with fixed endpoints, and it follows that α on α−1 (M ), α0 := Sαj on αj−1 (S(M )) (j = 1, 2, . . .) is a loop in M homotopic to α which satisfies L(X ◦ α0 ) = L(X ◦ α). In the other case there is some j such that αj and βj are not homotopic in S(M ) with fixed endpoints. Then αj βj−1 is not contractible in S(M ), and hence α0 := S(αj βj−1 ) = (Sαj )βj−1 is not contractible in M . We obtain the trivial estimate L(X ◦ α0 ) ≤ L(X ◦ αj ) + L(X ◦ βj ) ≤ L(X ◦ α) + σ(L(X ◦ α)), thereby proving the lemma also in this case.

We shall also need Lemma 3. Let (M, g) be a smooth compact surface with boundary. Then there is a smooth closed surface (2M, gS ) together with an involutive isometry S = id such that there is a natural inclusion (M, g) → (2M, gS ) which is conformal. Moreover, we have 2M = M ∪ S(M ), and ∂M is precisely the fixed point set of S. Proof. Let A be an atlas of M such that in local coordinates the metric tensor g is in isothermal form, gαβ = λδαβ and, moreover, that the range of each boundary chart is the upper half unit disk {(u1 , u2 ) ∈ R2 : (u1 )2 + (u2 )2 < 1, u2 ≥ 0}. Let M be a second copy of M , endowed with an atlas A in which each chart of A is replaced by its complex conjugate. Then let 2M be the disjoint union of M and M with the boundary points identified. The charts for points of ∂M = ∂M are constructed by glueing together each boundary chart of M with its complex conjugate in the obvious way. We thus produce an atlas AS on 2M , the transition maps of which are conformal maps in the plane and therefore real analytic. The map S assigns to each point of M the same point in M , leaving ∂M fixed. The metric g is extended from M onto 2M such that S ∗ g = g. This extended metric, however, need not be smooth. In order to construct a smooth metric on M having S as isometry we cover ∂M = ∂M with coordinate neighbourhoods G2 , . . . , Gn from the atlas AS . In

290


k local coordinates of each Gk we define a 2-tensor g k by gαβ = δαβ . Consider ∞ functions ηk of class C0 (Gk ), 1 ≤ k ≤ n, such that ηk = ηk ◦ S, and Σηk ≤ 1, Σηk = 1 in a neighbourhood of ∂M . We then define the metric gS on 2M by n n gS = 1 − ηk g + ηk g k . k=1

k=1

It is obvious from the construction that on M the metric gw differs from g only by a conformal factor. This concludes the proof of the lemma. We can now turn to the Proof of Theorem 1. Replacing (Xn , gn ) by a subsequence we can assume that limn→∞ E(Xn , gn ) exists. By the preceding lemma we can conformally embed (M, gn ) into a closed symmetric surface (2M, gnS ) of genus greater than one. Therefore we can apply Theorem 1 of Section 4.3 and replace gnS by a symmetric metric GSn of Gauss curvature −1 that is conformality equivalent to gnS . Due to the conformal invariance of Dirichlet’s integral we have E(X, gn ) = E(X, gnS ) = E(X, GSn ). We now extend Xn to a symmetric mapping XSn : 2M → RN . We know from Lemma 2 that the family {XSn } again satisfies the condition of cohesion on 2M . By means of Mumford’s Theorem 1 of Section 4.4, we obtain the existence of symmetric diffeomorphisms fn : 2M → 2M satisfying fn (M ) = M , which are orientation preserving if M is oriented such that (after passing to a subsequence) fn∗ GSn converges in C ∞ to a smooth symmetric metric GS . We observe that E(Xn , GSn ) = E(Xn ◦ fn , fn∗ GSn ). If we therefore replace Xn by Xn ◦fn and gn by fn∗ GSn , we may assume that the sequence {gn } converges in C ∞ to a smooth metric g and, after passing to a further subsequence, that {Xn } converges weakly in H21 (M, RN ) to some X ∈ H21 (M, RN ). Since any nonnegative continuous quadratic form on a Hilbert space is weakly lower semicontinuous, we have E(X, g) ≤ lim inf E(Xn , g) = lim E(Xn , gn ). n→∞

n→∞

It remains to be shown that the boundary values of X are continuous and map Cj monotonically onto Γj , j = 1, . . . , k. For this purpose it suffices to show that the boundary values of the sequence Xn ◦ fn (again denoted by Xn ) are equicontinuous. As in Lemma 2 we denote by σ(δ) the supremum of the lengths of all shortest subarcs of Γ1 ∪ · · · ∪ Γk joining any two points at a spatial distance not exceeding δ. Let then p0 ∈ Cj ⊂ ∂M be arbitrary, and fix some coordinate chart (G(p0 ), ϕ) where ϕ(p0 ) = 0 and ϕ : G(p0 ) → ϕ(G(p0 )) = {w = u1 + iu2 : |w| < 1, u2 ≥ 0}.


291

For 0 < ρ < 1 let γ(ρ) denote the arc in G(p0 ) with endpoints p(ρ), q(ρ) ∈ ∂M which corresponds to a halfcircle of radius ρ in ϕ(G(p0 )) centered at 0. Since we can assume that the metrics gn converge uniformly to a smooth metric g, the Dirichlet integrals (with respect to the Euclidean metric) of Xn ◦ ϕ−1 are uniformly bounded. We can therefore apply the Courant–Lebesgue lemma and obtain sequences ρn,ν > 0 such that (2)

1 1 < ρn,ν < √ ν ν

(n, ν ∈ N),

and (3)

length Xn (γ(ρn,ν )) ≤ δ(ρn,ν )

where δ(ρ) → for ρ → 0. In particular, we have (4)

|Xn (p(ρn,ν )) − Xn (q(ρn,ν ))| ≤ δ(ρn,ν ).

The points p(pn,ν ) and q(ρn,ν ) divide Cj into two subarcs Cn,ν , Cn,ν where ) were Cn,ν is contained in G(p0 ). If, for some pair (n, ν), the arc Xn (Cn,ν shorter than Xn (Cn,ν ), then we have by (4) length Xn (Cn,ν ) ≤ σ(δ(ρn,ν )) and the sum of the two arcs Cn,ν and γ(ρn,ν ) would form a closed loop homotopic to Cj (and hence nontrivial) whose image under Xn has a length not exceeding δ(ρn,ν ) + σ(δ(ρn,ν )). This contradicts the condition of cohesion provided that ν is sufficiently large. We can therefore conclude that, for ν ≥ ν0 , the curve Xn (Cn,ν ) is always shorter than Xn (Cn,ν ), and hence length Xn (Cn,ν ) ≤ σ(δ(ρn,ν )).

In view of (2) this shows the equicontinuity of (Xn |∂M ) at the point p0 ∈ M , and the theorem is proved. Observing the compact embedding of H21 (M ) into L2 (M ) and Lemma 1 we immediately obtain the following generalization of Shiffman’s theorem. Corollary 1. Let M, Γ1 , . . . , Γk be as in Theorem 1, and let K be some compact subset of RN . We consider the variational problem E(X, g) → min in the set of all pairs (X, g) where g is a smooth metric on M and X a map of Sobolev class H21 (M, RN ) with continuous boundary values mapping Cj monotonically onto Γj , j = 1, . . . , k, and with X(p) ∈ K for almost all p ∈ M . If this problem admits a minimizing sequence (Xn , gn ) where all Xn are continuous and satisfy the condition of cohesion, then the problem possesses a solution (X, g) such that X is conformal with respect to g.

292


Remark 1. It follows from the proofs of Theorem 2, Lemma 2 and Theorem 1 that the condition of cohesion needs only to be checked for simple, piecewise smooth curves on M with finitely many interior and boundary edges meeting under acute angles. Remark 2. In order to avoid technical complications we have used the condition of cohesion only in connection with continuous maps. We are now in a position to use our previous results to obtain existence theorems for higher genus minimal surfaces in R3 . Let us start with a topological consideration based on the fact that the fundamental group of an oriented surface of genus g ≥ 1 with one boundary component is a free group on 2g generators, and that the fundamental group of a solid 2g torus is also a free group on 2g generators, see Massey [1]. The following theorem then is an easy consequence of algebraic results due to Zieschang [1,2]. Theorem 3. Let M be an oriented surface of genus 1 with one boundary component and let T be a solid 2-torus (i.e. the connected sum of two tori). Furthermore let Γ be a Jordan curve in T and X : M → T a continuous map which maps ∂M monotonically onto Γ . Finally, suppose a base point p0 ∈ ∂M is fixed. Then in order that the induced map X∗ : π1 (M, p0 ) → π1 (T, x0 ),

x0 = X(p0 ),

be an isomorphism it is necessary and sufficient that there are two generators of π1 (M, p0 ) and two corresponding generators of π1 (T, x0 ) such that the classes of ∂M in π1 (M, p0 ) and of Γ in π1 (T, x0 ) respectively are represented by the same word with respect to the above sets of generators. Proof. Necessity: Let c1 , c2 be free generators of π1 (M, p0 ). If X∗ is an isomorphism, then clearly X∗ (c1 ), X∗ (c2 ) generate π1 (T, x0 ) freely, and obviously the word for [∂M ] with respect to c1 , c2 is the same as for [Γ ] = X∗ ([∂M ]) with respect to X∗ (c1 ), X∗ (c2 ). Sufficiency: Let us suppose that the hypothesis on Γ is fulfilled for generators c1 , c2 of π1 (M, p0 ) and generators γ1 , γ2 of π1 (T, x0 ). Then we can define an isomorphism ϕ : π1 (T, x0 ) → π1 (M, p0 ) such that ϕ(γj ) = cj for j = 1, 2. By hypothesis we have ϕ([Γ ]) = [∂M ]. From the canonical models of surfaces (see Massey [1]) we see that in the oriented case we can now choose generators A, B, of π1 (M, p0 ) such that [∂M ] = AB A−1 B −1 .


293

That is, ∂M is the commutator with respect to the basis A, B. If follows that ϕX∗ (AB A−1 B −1 ) = ϕ([X∗ (∂M )]) = ϕ([Γ ]) = [∂M ] = AB A−1 B −1 , i.e., ϕX∗ fixes the commutator AB A−1 B −1 . It now follows from Zieschang’s results [1,2] that ϕX∗ is an isomorphism of π1 (M ). We now have the following existence result. Theorem 4. Let M be an oriented surface of genus 1 with one boundary component and let Γ be a rectifiable Jordan curve in R3 . We assume that (i) there exists a H-convex7 solid 2-torus T of class C 3 in R3 such that Γ ⊂ T , and that (ii) with respect to suitable chosen base points and generators, the class of Γ in π1 (T ) is represented by the same word as the class of ∂M in π1 (M ), respectively. Then there exists a minimal surface X : M → R3 mapping ∂M topologically onto Γ and such that X(M ) is contained in T . Proof. We consider the class of all pairs (X, g) where g is a smooth metric on M and X ∈ H21 (M, R3 ), X(M ) ⊂ T , and X is mapping ∂M onto Γ continuously and monotonically. We must show that this class is not empty. By approximation it clearly suffices to produce a continuous map X : M → T satisfying the boundary condition. By the classification of surfaces (see Massey [1]) we obtain a topological model of M in form of an annulus a whose inner boundary circle (of radius 1 2 ) corresponds to ∂M , whereas the outer circle (of radius 1) is broken up into 4 consecutive segments C1 , . . . , C4 such that all vertices are identified with one point a, and the segments are identified according to the rule C3 = C1−1 , C4 = C2−1 . It follows that A := C1 , B := C2 generate the fundamental group of M with base point a and that ∂M is freely homotopic to AB A−1 B −1 . Connecting an arbitrary base point b on ∂M with a by means of some simple ˜ := γBγ −1 , we arc γ, and replacing A, B by their conjugate A˜ := γAγ −1 , B ˜ A˜−1 B ˜ −1 is the class of ∂M in ˜ B ˜ generate π1 (M, b) and that A˜B see that A, π1 (M, b). It follows therefore from hypothesis (ii) of the theorem that there are closed loops α, β based at some point c ∈ Γ generating π1 (T, c) and such that [Γ ] = αβα−1 β −1 . Let h : [0, 1] × S 1 → T be a homotopy between Γ and γ = αβα−1 β −1 . We can clearly choose the parametrization of γ in such a way that γ|C1 = α, γ|C2 = β, γ|C3 = α−1 , γ|C4 = β −1 . Then we define a map X : a → T by X(reiθ ) = h(r − 1, eiθ ). It is clear from the construction that X respects the identifications of a which make a a topological equivalent of M , and therefore X can be considered as a map X : M → T with X(∂M ) = Γ . As the class of admissible maps is not empty, 7

That is, the mean curvature of ∂T with respect to the inward normal to ∂T is nonnegative.

294


we can now pick an energy minimizing sequence {Xn , gn }. By approximation we may assume all Xn to be continuous. It then follows from Theorem 3 that any homotopically nontrivial loop on M is mapped by Xn into a homotopically nontrivial loop in T whose length is therefore bounded below by a positive constant depending only on T . Thus Xn satisfies Courant’s condition of cohesion. Corollary 1 now provides the existence of minimizing conformal pair (X, g). From the regularity theory of variational inequalities, cf. Tomi [4], Hildebrandt [12,13], and Vol. 2, Section 2.12, No. 15, and Section 4.7, we obtain that X ∈ C 1 (M \ ∂M ) and X ∈ C ∞ (M \ (∂M ∪ X−1 (∂T ))). If both sets (M \ ∂M ) ∩ X−1 (∂T ) and M \ (∂M ∪ X−1 (∂T )) were non-empty, then there would exist a point where the surface X touches ∂T tangentially. This, however, contradicts E. Hopf’s strong maximum principle if T is H-convex (see Hildebrandt [11] and Vol. 2, Section 4.7). Let us next consider the case when X(M ) ⊂ ∂T . Then X is a harmonic map from (M, g) into the submanifold ∂T of R3 which, by assumption, is of class C 3 . It follows that X ∈ C 3 (M \ ∂M ) and that, apart from isolated branch points, X is immersed, and its mean curvature H with respect to the interior normal ν equals that of ∂T whence H ≥ 0. On the other hand, since X is area minimizing among all surfaces in T with boundary Γ , we obtain from the first variation of area that Y, νH dvol ≥ 0 dA(X)Y = −2 M

for all compactly supported smooth variations Y with Y, ν ≥ 0. It follows that H ≤ 0, and hence H ≡ 0, i.e., X is a minimal surface. In the last case when X(M \ ∂M ) ⊂ T \ ∂T it is clear anyway that X is minimal. Thus Theorem 4 is proved. Theorem 4 gives the existence of a surface whose boundary is the same as a classically known physical example (see Fig. 1 in the introduction to Chapter 4 of Vol. 1) since the boundary of this curve can be considered as the commutator with respect to a basis of a solid 2-torus which is the same as [∂M ], M an oriented surface of genus one. This example is embedded. For our existence theorem it can be deduced from results of R. Gulliver [7] that the minimal surfaces resulting from existence Theorem 4 are immersed on the interior. In fact, J. Jost [9,17] has shown in the situation of Theorem 4 and under the additional hypothesis that the curve Γ lies on the boundary of the torus T that Γ spans an embedded minimal surface. These remarks apply to a generalization of Theorem 4 concerning the question of existence for oriented and unoriented minimal surfaces spanning one or several contours. For the case of several contours we refer the reader to Tomi and Tromba [4]. These cases may require the Teichm¨ uller theory for orientable and nonorientable symmetric surfaces of genus one (which we have not developed here) and of genus greater than one.


295

Fig. 1. (a)–(c) Three types of H-convex surfaces as building blocks lead to minimal q-tori of arbitrary genus g, e.g. to the classical minimal surface in the configuration (d)

In the case of one boundary curve Γ one can also prove the following existence result (see Tomi and Tromba [4], p. 70). Theorem 5. Let M be a surface of genus g ≥ 1 with one boundary component and let T be a solid mean convex q-torus (that is, the connected sum of q-solid tori) of class C 3 with q = 2g in case M is orientable and q = g if not. Then both fundamental groups will have the structure of a free group on q generators. Assume either of the following hypotheses on Γ and M . (i) M g oriented of genus g and Γ homotopic in T to the commutator product π1 (T ); j=1 [αj , βj ] for some set α1 , . . . , βg of free generators of g (ii) M non-orientable of genus g and Γ homotopic in T to j=1 αj2 where α1 , . . . , αg are free generators of π1 (T ); (iii) M non-orientable of genus g = 2k + 1 and Γ homotopic in T to ⎤ ⎡ k ⎣ [αj , βj ]⎦ γ 2 j=1

where α1 , . . . , βk , γ are free generators of π1 (T ); (iv) M non-orientable of genus g = 2k, k ≥ 1, and Γ homotopic in T to k−1 [ j=1 [αj , βj ]]αk βk αk−1 βk , for generators α1 , . . . , βk of π1 (T ).

296


Then there is a minimal surface X of the topological type of M spanning Γ , i.e., a minimal surface X : M → R3 mapping ∂M topologically onto Γ such that X(M ) is contained in T and X restricted to the interior of M is an immersion. Analogously to Theorems 4 and 5, Tomi and Tromba [4] have proved the following result for boundaries Γ consisting of several components. Theorem 6. Let M be a surface of genus g and of k = m + 1 boundary components, and let Γ = Γ0 , . . . , Γm be a boundary configuration consisting of m+1disjoint Jordan curves contained in some solid, mean-convex, (q +m)torus T of R3 where q = 2g if M is orientable and q = g if not. Choose some base point x ∈ T and paths δj connecting x with points xi ∈ Γi , and define the loops σj = δj Γj δj−1 , j = 0, 1, . . . , m. Assume also that there are elements α1 , β1 , . . . , αg , βg ∈ π1 (T, x) in the oriented case and α1 , . . . , αg ∈ π1 (T, x) in the unoriented case such that σ1 , . . . , σm , α1 , . . . , βg (or σ1 , . . . , σm , α1 , . . . , αg respectively) generate π1 (T, x) and that the relations σ0 . . . σm

g j=1

[αj , βj ] = 1

or

σ0 . . . σm

g

αj2 = 1

j=1

hold in the respective cases. Then there exists a minimal surface X : M → T such that X(∂M ) = Γ0 ∪ Γ1 ∪ · · · ∪ Γm .

4.7 Scholia 1. If one wants to study the classes of all conformally equivalent structures on a general topological 2-manifold M , one obviously has to study the moduli space R(M ). Unfortunately, the topology of the moduli space is fairly complicated; in particular, it does not have the structure of a manifold. This led Teichm¨ uller to introduce another set which has become known as Teichm¨ uller space T(M ). This space has much better properties than the moduli space and is accessible to the methods of real and complex analysis. Classical tools of Teichm¨ uller theory are quasiconformal mappings whose study was initiated by Grötzsch, Ahlfors and, in a completely different way, by Morrey, and the theory of Fuchsian and Kleinian groups. For a presentation of Teichm¨ uller theory we refer to the monographs of Lehto [1] and Kra [1]. More recently, harmonic diffeomorphisms have proved to be an at least equally valuable tool which admits a more geometric approach to Teichm¨ uller theory. We refer the reader to presentations of this approach in the monographs of Tromba [24] and Jost [17]. These ideas will certainly play an important role in the future development of global analysis as they show a way to handle bifurcation processes when geometric objects change their topological type. 2. Chapter 4 is a minor revision of the joint work of A. Tromba with F. Tomi [5]. For a more penetrating study of the subject, the reader should

4.7 Scholia

297

consult Tromba’s original papers as well as his joint work with A. Fischer and F. Tomi. 3. The first to study general Plateau problems for minimal surfaces of higher topological type was Jesse Douglas; his work was truly pioneering, and his ideas and insights are as exciting and important nowadays as at the time when they were published, more than half a century ago. It seems that Douglas was the first to grasp the idea that a minimizing sequence could be degenerating in topological type, and he interpreted such a conceivable degeneration as a change in the conformal structure. He based his notion of degeneration on the representation of Riemann surfaces as branched coverings of the sphere. Then degeneration meant disappearance of branch cuts. The intuitive meaning of degeneration is the shrinking of handles and the tendency to separate the Riemann surface into several components. Since degeneration is unavoidable in general, Douglas had the idea of minimizing not over surfaces of a fixed topological type, but also over all possible reductions of the given type. In this set of Riemann surfaces of varying topological type Douglas introduced a notion of convergence as convergence of branch points in the representation of the surfaces as branched coverings of the sphere. The compactness of this set of Riemann surfaces seemed to be a trivial matter to him since his whole argument reads: This is because the set can be referred to a finite number of parameters, e.g. the position of the branch points . . . . This reasoning is, however, rather inaccurate since the position of branch points alone does not determine the structure of the surface. Douglas also argued on a rather intuitive level when it came to the lower semicontinuity of Dirichlet’s integral with respect to the convergence of surfaces. Taking the compactness of the above set of Riemann surfaces and the lower semicontinuity of Dirichlet’s integral for granted, it is then obvious that an absolute minimum of Dirichlet’s integral in the class of surfaces considered by Douglas must be achieved, either in a surface of desired (highest) topological type or in one of reduced type. In this way Douglas was led to his celebrated theorem that we stated in Section 4.6. Courant and Shiffman gave the general results of Douglas a solid basis by solving the variational problem within a class of surfaces of fixed topological type. In Chapter 8 of Vol. 1 we have given an exposition of Courant’s method [15] for “schlicht domains” whose boundary consist of k closed curves, supposing the sufficient Douglas condition. The case k = 2 is also treated in Nitsche’s Vorlesungen [28]. As we have remarked before, modern and complete proofs of the results of Douglas and Shiffman were given by Tomi and Tromba [5] and Jost [6]. We also refer to Jost [14] and [17], and to F. Bernatzki [1], who treated the Douglas problem for nonorientable surfaces.

Chapter 5 The Index Theorems for Minimal Surfaces of Zero and Higher Genus

5.1 Introduction In 1981 R. B¨ ohme and A. Tromba [2] proved an index theorem for branched minimal surfaces of disk type in Euclidean space Rn . A consequence of their result is the finiteness of the number of branched minimal disks spanning a contour in general position. Later Tomi and Tromba [6] proved an index theorem for minimal surfaces of higher topological type spanning one boundary contour. In the present chapter we prove these two index theorems. Let E and F be Banach spaces. A linear map L : E → F is said to be Fredholm if the dimensions of its kernel and cokernel are finite, i.e. if (a) dim ker L < ∞, (b) dim coker L < ∞. This implies that the range of L is closed. The Fredholm index of L is defined by: index L := dim ker L − dim coker L. The Atiyah–Singer formula (see Palais [3]) gives a topological formula for this index in the case of linear elliptic operators L defined on sections of vector bundles. Smale [1] defined the notion of a nonlinear differentiable Fredholm map f between two Banach manifolds N and A. A map f is said to be Fredholm if the derivative Df (p) : Tp N → Tf (p) A between tangent spaces is a linear Fredholm map for all p ∈ N. The Fredholm index of f is defined by index f := dim ker Df (p) − dim coker Df (p) for any p ∈ N. If N is connected this is well defined, otherwise it is assumed that the index is the same on all components. U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften 341, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11706-0 5,

299

300

5 The Index Theorems for Minimal Surfaces of Zero and Higher Genus

A point y ∈ A is said to be a regular value of f if either f −1 (y) is empty

(1) or (2)

if x ∈ f −1 (y) then Df (x) : Tx N → Tf (x) A is surjective.

A map f is said to be proper if the inverse image of any compact set is compact. In practice, verification of properness involves elliptic, a priori estimates. The Smale–Sard theorem [1] is the following result: Theorem 1. If f : N → A is C r and Fredholm where r > max(index f, 0), and if N is second countable, then the set of regular values is of first category in A. If f is proper, this set is open and dense in A. Proof. The proof argues locally by a change of variables and applying the classical Sard–Brown theorem. Let p ∈ N. Then, in a coordinate chart we have f : U → F , U open in a Banach space E. We may assume that p is the origin and that f (0) = 0. Suppose that E1 = ker Df (0), F0 = range Df (0). Identify E1 with Rm and a closed complement of F0 in F with Rn . If E0 is a closed complement of E1 , the mapping Df (0) : E0 → F0 is an isomorphism. With respect to the splitting F = F0 × Rn , f may be written as f (x, y) = (f0 (x, y), f1 (x, y)). Define F by F(x, y) = (f0 (x, y), y) ∈ F0 × Rn . Then DF(0) is an isomorphism of E onto F0 × Rn . Let G be its local inverse. Hence F ◦ G(z, w) = (z, w) = (f0 ◦ G(z, w), w) and thus f0 ◦ G(z, w) = z, implying that f ◦ G(z, w) = (z, f1 ◦ G(z, w)).

Before proceeding with the proof of the Smale–Sard theorem we state a useful local representation of Fredholm maps which is obtained by setting ψ := f1 ◦ G. Theorem 2. Let f : N → A be a Fredholm map of index i, where N and A are Banach manifolds modelled on E and F respectively. Then, for arbitrary p ∈ N, there is a closed subspace F0 ⊂ F and a coordinate chart G about p, locally diffeomorphic to an open neighbourhood of 0 ∈ F0 ×Rm , m = dim coker Df (p), such that with respect to a given coordinate chart about f (p), we have (3)

If f is C r then ψ is C r .

f ◦ G(z, w) = (z, ψ(z, w)), z ∈ F0 , ψ(z, w) ∈ Rn .

5.1 Introduction

301

Theorem 2 yields the following Corollary 1. Fredholm maps are locally proper, therefore (locally) the images of closed sets are closed. Proof. Suppose f (pj ) → y. Then, by (3), pj = G(zj , wj ) with (zj , ψ(zj , wj )) converging to a point y = (z, v). Thus zj → z and hence {wj } ∈ Rm can be assumed bounded, {wj } has a convergent subsequence, and the corollary is proved. A point p0 ∈ N is said to be a critical point of f if Df (p0 ) is not surjective. Let C ⊂ N be the set of critical points; then f (C) is called the set of critical values. The set of regular values simply is the complement, A \ f (C), of the set of critical values. Before completing the proof of the Smale–Sard theorem we state the classical Sard–Brown Theorem: Let U ⊂ Rm be open and ψ : U → Rn a C r map, r > max(m − n, 0). Then the set of critical values is of first category. Remark 1. Here the assumption r > max(m − n, 0) is essential as counterexamples are known for r ≤ max(m − n, 0). Now we proceed with the proof of the Smale–Sard Theorem: By the second countability of N we need only show that, locally, the image of the critical set is nowhere dense. Let y0 ∈ A. Since f is locally proper the image of the (closed) critical set is closed. Therefore it is enough to prove that, near y0 , there is a y which is a regular value for f . By (3) we may assume y0 = (z0 , ψ(z0 , w0 )). Thus it suffices to find a y1 := (z0 , ψ(z0 , w1 )) which is a regular value for f . Applying the classical Sard–Brown theorem to g : w → ψ(z0 , w) we know that there is a regular value of g near g(w0 ), and so the Smale–Sard theorem is proved. Remark 2. Counterexamples to the Smale–Sard Theorem are known if f is not Fredholm. For A = R and N being modelled on an infinite-dimensional Banach space, there is a counterexample due to I. Kupka. In Chapter 6 we state conditions under which the Sard theorem holds for smooth real-valued maps on Banach manifolds and apply such a result to Plateau’s problem. We now state the Smale transversality theorem, which is crucial to the development of a Morse theory in the next chapter. Let f : N → A be a Fredholm map and g : M → A be a C 1 embedding of a finite dimensional manifold M . We say that f is transversal to g (or g is transversal to f ) if, whenever f (x) = g(y), the subspaces Df (x)(Tx N) and Dg(y)(Ty M ) of Tf (x) A span Tf (x) A. Then Smale’s result can be stated as follows: Theorem 3. Let f : N → A be a C r Fredholm map with N being second countable, and g : M → A be a C 1 embedding with r > max(index f + dim M, 0). Then there exists a C 1 -approximation g of g such that g is an embedding transversal to f . Furthermore, if f is transversal to the restriction of g to a closed set C ⊂ M , then g may be chosen so that g = g on C.

302


Proof. Since N and M have a countable base of open sets, a standard argument reduces the proof to the local Lemma 1. Suppose f (x) = g(y). Then there is a neighbourhood U of y such that for any ε > 0 there is a C 1 -approximation g to g which is ε-close to g in C 1 and such that f is transversal to g|U . Proof. Working in coordinate systems as in the proof of Sard’s theorem, we may find coordinate systems about y and g(y) and a splitting F = F0 × Rk , F being the model space of A, such that locally (4)

g(y) = (0, y)

where k = dim M . The existence of such a chart follows immediately from the fact that g(M ) is a submanifold of A. Let π0 : F → F0 be the projection onto the first factor. Let U ⊂ V be neighbourhoods of y such that (4) holds, and ϕ a real valued C ∞ -function which is 1 on U and 0 outside of V . Now consider π0 ◦ f . By Smale–Sard there is a regular value for π0 ◦ f (which is still locally Fredholm), say z, close to 0. Define g1 (y) = (z, y) and g by g(y) = ϕ(y)g1 (y) + [1 − ϕ(y)]g(y). It is clear that g˜ is transversal to f on U , and for z sufficiently close to 0, g˜ is C 1 -close to g. We shall apply this last result in the next chapter; for the present we have developed the necessary terminology in order to be able to state the main results of this chapter. The index theorems presented below are the first non-trivial examples of the calculation of a Fredholm index for an intrinsically global problem, that is, for a problem truly defined on manifolds, as opposed to open sets of Banach spaces.

5.2 The Statement of the Index Theorem of Genus Zero For integers r and s with r s ≥ 7 define D as D := Ds = {u : deg u = 1 and u ∈ H s (S 1 , C)}, where H s denotes the Sobolev space of s-times differentiable (in the distribution sense) functions with values in C. Set A := {α ∈ H r (S 1 , Rn ) : α an embedding (i.e. α is one-to-one and α (p) = 0 for all p ∈ S 1 ) and the total curvature of α is less than π(s − 2)}.

5.2 The Statement of the Index Theorem of Genus Zero

303

Let π : A × D → A denote the projection map onto the first factor. A minimal surface X : B → Rn spanning α ∈ A can be viewed as an element of A×D, since X is harmonic and therefore determined by its boundary values X|∂B = X|S 1 = α ◦ ξ,

where (α, ξ) ∈ A × D.

The classical approach to minimal surfaces was to understand the set of minimal surfaces spanning a given fixed wire α, that is, the set of minimal surfaces in π −1 (α). The index approach is first to understand the structure of the set of minimal surfaces as a subset of the bundle N = A × D as a fibre bundle over A and then to look at the question of characterizing the set of minimal surfaces in the fibre π −1 (α) in terms of the singularities of the projection map π restricted to a suitable subvariety of N. Let us say that a minimal surface X ∈ A × D has branching type (λ, ν), λ = (λ1 , . . . , λp ) ∈ Zp , ν = (ν1 , . . . , νq ) ∈ Zq , λi , νi ≥ 0, if X has p distinct but arbitrarily located interior branch points w1 , . . . , wp in B of integer orders λ1 , . . . , λp and q distinct boundary branch points ζ1 , . . . , ζp in S 1 of (even) integer orders ν1 , . . . , νq . In a formal sense the subset M of minimal surfaces in N is an algebraic subvariety of N and is a stratified set, stratified by branching types. To be more precise, let Mλν denote the minimal surfaces of branching type (λ, ν). We show that for ν = 0, Mλ0 is a submanifold of N, and the restriction π λ of π to Mλ0 is Fredholm of index I(λ) = 2(2 − n)|λ| + 2p + 3 where |λ| := Σλi . Moreover, locally, for ν = 0, Mλν ⊂ Wλν , such that Wλν is a submanifold of N, where the restriction πλν of π to Wλν is Fredholm of index I(λ, ν) = 2(2 − n)|λ| + (2 − n)|ν| + 2p + q + 3. The number 3 comes from the equivariance of the problem under the action of the 3-dimensional conformal group of the disk. This index measures (in some sense) the stability of minimal surfaces of branching type (λ, ν) in Rn and the likelihood of finding such surfaces; the more negative the index of πνλ , the less likely it is to find a wire admitting minimal surfaces of branching type (λ, ν) which span it. Applying Smale’s version of Sard’s theorem these stratification and index results are then the basis of proving the generic (open-dense) finiteness and stability of minimal surfaces of the type of disk, i.e. there exists an open dense ⊂ A such that if α ∈ A, there exists only a finite number of minimal subset A surfaces bounded by α, and these minimal surfaces are stable with respect to perturbations of α. This open dense set will be the set of regular values of the map π. In addition, we are led to an appropriate definition of non-degeneracy. A minimal surface X ∈ Mλν spanning a wire α is non-degenerate if I(λ, ν) = 0 (hence ν = 0) and the map π0λ restricted to a neighbourhood of X in Mλ0 is a local diffeomorphism onto a neighbourhood of α in A.

304


Moreover, if n > 3 these minimal surfaces are all immersed up to the boundary, and if n = 3 they are simply branched. We would like to emphasize that we are considering not only minimal surfaces bounded by α ∈ A minimizing Dirichlet’s integral, but all critical points of Dirichlet’s functional defined on the space of surfaces spanning α which satisfy the classical monotonicity condition along the boundary. There are some surprising consequences of the Index Theorem. First we know that minimal surfaces in R3 are free of interior branch points if they minimize area (see Vol. 1, Chapter 4, No. 2, and Vol. 2, Chapter 6), whereas we shall see that most minimal surfaces with simple interior branch points are stable with respect to perturbations of the boundary. Second, for n ≥ 4, minimal surfaces in Rn may have branch points even if they are area minimizing, but for such n no such minimal surface in Rn is stable as a branched surface under perturbation of the boundary. Remark. In this chapter we shall not distinguish between a mapping X : ˆ : B → Rn which is harmonic in B ∂B = S 1 → Rn and its extension X and agrees with X on ∂B; both mappings will here be denoted by X. This differs from the custom of Chapters 6 in Vols. 2 and 3 where it is essential ˆ However, the reader will easily perform the to distinguish between X and X. necessary adjustments when connecting the pertinent chapters.

5.3 Stratification of Harmonic Surfaces by Singularity Type Since our stated goal is to stratify the set of minimal surfaces according to branching type we shall start by first stratifying harmonic surfaces in Rn according to singularity type. If B ⊂ R2 denotes the open unit disk, then a harmonic mapping X : B → Rn is said to have an interior branch point of order λ0 at w0 ∈ B if the complex gradient Xw = 12 (Xu − iXv ), as a mapping Xw : B → Cn , has a zero of the order λ0 at w0 . On the other hand if the harmonic mapping X : B → Rn is sufficiently smooth up to the boundary, we can define in the same way the notion of a boundary branch point of X at ζ0 ∈ ∂D = S 1 of order ν0 . So any harmonic and any minimal surface X : B → Rn , which is sufficiently smooth up to the boundary, may have a (possibly empty) set {w1 , . . . , wp } of interior branch points with the multiplicities λ1 , . . . , λp and a (possibly empty) set of boundary branch points {ζ1 , . . . , ζq } with multiplicities {ν1 , . . . , νq } respectively. Here we conveniently assume that X ∈ C N (B, Rn ), where N > supj=1,...,q νj . A fundamental consequence of the index theorem is that the character and stability behavior of a minimal surface strongly depends on the number and multiplicities of interior as well as of boundary branch points. Therefore we will separately study the sets Mnλν of all minimal surfaces in Rn (the regularity class being fixed suitably) with exactly p interior branch points of multiplicities λ = (λ1 , . . . , λp ) ∈ Np and with exactly q boundary branch points of multiplicities ν = (ν1 , . . . , νq ) ∈ Nq respectively.

5.3 Stratification of Harmonic Surfaces by Singularity Type

305

n In order to do that it is necessary first to study the much larger set Hλν of all n harmonic surfaces in R (the regularity class being fixed suitably) with exactly p interior branch points of multiplicities λ = (λ1 , . . . , λp ) and with exactly q boundary branch points of multiplicities ν = (ν1 , . . . , νq ). It is obvious that n . Mnλν ⊂ Hλν The main result of this section is Theorem 3, which essentially states that n Hλν is a smooth submanifold in the space Hn of all harmonic surfaces in Rn (the regularity class being fixed as above) with a finite codimension

(1)

n codimR (Hλν ; Hn ) = 2n|λ| + 2n|ν| − 2p − q,

where |λ| = λ1 + · · · + λp and |ν| = ν1 + · · · + νq . Clearly, if p and q are both n ⊂ Hn is an open subset, and our formula gives codimension zero then H00 zero. The meaning of (1) can be explained easily. If h0 : B → Rn is a harmonic surface in Hn which has exactly p interior branch points of orders p λ = (λ1 , . . . , λp ) at t = (t1 , . . . , tp ) ∈ B× · · · ×B and exactly q boundary q branch points of orders ν = (ν1 , . . . , νq ) at s = (s1 , . . . , sq ) ∈ S 1 × · · · ×S 1 , n n (s; t) as the set of all h in Hλν such that the branch then we may define Hλν points are identically located at t = (t1 , . . . , tp ) in B and s = (s1 , . . . , sq ) on ∂B = S 1 , as are those of h0 . It is not hard to see that (2)

n codimR (Hλν (s; t); Hn ) = 2n|λ| + 2n|ν|.

n The additional term −(2p + q) in (1) just indicates that in Hλν one is allowed to let the branch points move, where clearly any interior branch point has 2 degrees of freedom and a boundary branch point has only one. This adds up to 2p + q additional parameters. Therefore Theorem 5 should be nearly obvious, but its proof is technical and somewhat lengthy. One also has to be careful about the topologies of the different function spaces which are involved. In the present chapter we use the notation H s for the Hilbert space H2s . Let us start now with some definitions.

Definition 1. Let s ∈ N be fixed, s ≥ 2. If n ∈ N, we define C[w] := {complex polynomials in the variable w}, Hn := {h ∈ H s (S 1 , Rn )} = {h ∈ H s+1/2 (B, Rn ) : Δh = 0}, Zn := {h ∈ H s−1 (S 1 , Cn )} ∼ = {h ∈ H s−1/2 (B, Cn ) : Δh = 0}, An := {h ∈ Zn : h holomorphic on B}. Here H s (S 1 , Rn ) denotes the Sobolev space of functions with distributional derivatives of orders 0 ≤ j ≤ s which are in L2 (S 1 ). If n = 1, we simply write H, Z and A. It is clear that Hn , Zn and An are real or complex Hilbert spaces respectively with the standard H s or H s−1 inner product.

306


Definition 2. If λ = (λ1 , . . . , λp ) ∈ Np and ν = (ν1 , . . . , νq ) ∈ Nq we define θνλ := {P ∈ C[w] : degree(P ) = |λ| + |ν|, p q (w − wj )λj Πk=1 (w − ζk )νk , where all P (w) = Πj=1

wj ∈ B, and wj = wl if j = l, and all ζk ∈ ∂B = S 1 , ζk = ζl if k = l}. If p or q happens to be zero, then we write θ0λ or θν0 , and θ00 contains only the polynomial P ≡ 1. Lemma 1. θνλ has the structure of a smooth real manifold of dimension 2p+q. p q (w − wj )λj Πk=1 (w − ζk )νk ∈ θνλ , then the tangent space Tp θνλ If P (w) = Πj=1 at P can be described as the space of all complex polynomials H ∈ C[w] such that (i) degree (H) ≤ |λ| + |ν| − 1; (ii) for any j the polynomial H vanishes at wj of order at least (λj − 1); (iii) for any k the polynomial H vanishes at ζk of order at least (νk − 1); (iv) for any k = 1, . . . , q,

d dw

νk −1

w=ζk

H = iμk ζk ·

d dw

νk

P

w=ζk

holds for some real constant μk . Proof. θνλ is a subset of the vector space F|λ|+|ν| of polynomials of degree at most |λ| + |ν|. Let t = (t1 , . . . , tp ) ∈ B× . p. . ×B and s = (s1 , . . . , sq ) ∈ Rq ; then we define a map Φλν : B p × Rq → F|λ|+|ν| by p q (w − tj )λj Πk=1 (w − eisk )νk = R(w) Φλν (t, s) = Πj=1

where Φλν is smooth, locally injective, and its image set is θνλ . If τ = (0, . . . , τr , 0, . . . , 0) ∈ Cp and σ = (0, . . . , σl , 0, . . . , 0) ∈ Rq , then the differential of Φλν at (t, s) is p q λj Φλν∗ |(t,s) [τ, σ] = (−τr )λr Πj=1,j · Πk=1 (w − eisk )νk =r · (w − tj ) p · (w − tr )λr −1 + νl (iσl eisl ) · Πj=1 (w − tj )λj q isk νk · Πk=1k ) · (w − eisl )νl −1 = H(w). =l (w − e

It is apparent that H satisfies (i), (ii) and (iii), and, with −σl νl = μl , (iv) also. Sometimes it is convenient to use a different but clearly equivalent condition instead of (iv). If we restrict H and P to S 1 ⊂ C and introduce the


307

angular variable φ, then (iv) holds if and only if there exist real constants μ1 , . . . , μq such that dνk −1 H dν k P (ζk ) = μk ν (ζk ) ν −1 k dφ dφ k

for k = 1, . . . , q.

Definition 3. If p ∈ N, λ = (λ1 , . . . , λp ) ∈ Np , and t = (t1 , . . . , tp ) is a p-tuple of distinct points in B, we define n Hλ0 (t) := h ∈ Hn : f = hw satisfies f = P fˆ, where P ∈ θ0λ , vanishing at tj of order λj , and fˆ is smooth and vanishes nowhere in B and n := Hλ0

n Hλ0 (t),

t∈T

where T is a set of admissible p-tuples. 1 Theorem 1. Hλ0 is a C ∞ -smooth submanifold of H = H1 of codimension 2|λ| − 2p. If h ∈ Hλ0 (t) ⊂ Hλ0 then η ∈ H is in the tangent space Th Hλ0 if and only if (d/dw)i η(tj ) = 0 for 1 ≤ i ≤ λj − 1.

Proof. We define Aλ0 := {f ∈ A : f = P fˆ, P ∈ θ0λ , fˆ smooth and without zeros in B}. The h is in Hλ0 if and only if f = hw is in Aλ0 , and it suffices to show that Aλ0 is a submanifold of A of codimension 2|λ| − 2p. Assume that f = P fˆ is in Aλ0 , P ∈ θ0λ , fˆ without zeros in B. Fix a neighbourhood U of fˆ in A such that no g ∈ U vanishes in B, and a neighbourhood V of P in θ0λ . Define a mapping ψ : U × V → A by ψ(g, R) = g · R. Then ψ maps U × V into Aλ,0 , and ψ is C ∞ smooth and injective. The range of ψ covers a neighbourhood of f in Aλ0 . We compute the derivative of ψ at (fˆ, P ): Dψ|(fˆ, P ) : A × Tp θ0λ → A

satisfies Dψ|(fˆ, P )[k, R] = fˆR + kP.

Clearly Dψ|(fˆ, P ) is injective, its range R is closed, and Lemma 1, (ii) implies that g ∈ A is in R if and only if

d dw

k g = 0 for 0 ≤ k ≤ λj − 2. tj

Therefore R ⊂ A has complex codimension |λ| − p and real codimension 2|λ| − 2p. So ψ is locally a smooth embedding of U × V onto a piece of Aλ0 ,

308


and Aλ0 is a submanifold as stated above. Moreover R is the space Tf Aλ0 . This implies Theorem 1. One can think of the previous result in the following way. If f = P fˆ ∈ Aλ0 , p (w − tj )λj , then f ∈ Aλ0 (t), and on the tangent space level and P (w) = Πj=1 we have Tf Aλ0 = Tf Aλ0 (t) ⊕ TP θ0λ · fˆ. This describes Tf Aλ0 in terms of a subspace of holomorphic functions with fixed branch points and a complementary space of holomorphic functions whose branch points can vary. We can easily generalize Theorem 1 to a theorem for harmonic functions with values in Rn . n Theorem 2. Hλ0 is a C ∞ -smooth submanifold of Hn of codimension 2n|λ| − n n if and only if 2p. If h ∈ Hλ0 (t) ⊂ Hλ0 , then η is in the tangent space Th Hλ0 for j = 1, . . . , p the following two conditions hold:

(i) (d/dw)i η(tj ) = 0 for 1 ≤ i ≤ λj − 1, and (ii) (d/dw)λj η(tj ) = βj · (d/dw)λj +1 h(tj ) = βj · fˆ(tj ); here βj ∈ C is a fixed common multiplier, and dh/dw = f = fˆ · P , P ∈ θ0λ , where fˆ has no zeros in B. Proof. The proof of Theorem 1 can be carried over word by word. We define Anλ0 , and we will show that Anλ0 ⊂ An is a submanifold of real codimension 2n|λ| − 2p. We factor f = dh/dw in the form f = P · fˆ, where P ∈ θ0λ and the Cn -valued function fˆ has no zeros in B. Fix U ⊂ An and V ⊂ θ0λ as above and define for G ∈ U , R ∈ V again ψ(G, R) := G · R. The derivative at (fˆ, P ) is Dψ(fˆ,P ) [H, R] = fˆ · R + H · P. We know that the derivatives of R vanish at tj up to order λj − 2 and those of P at tj up to order λj − 1. This implies that any W in the range of Dψ has all its derivatives vanishing at tj up to order λj − 2. Moreover dλj −1 W = βj · fˆ(tj ). dwλj −1 tj It is not hard to see that these two conditions uniquely characterize the range of Dψ. This finishes the proof of Theorem 2. We must emphasize at this point that with the above methods one cannot handle a situation where the number and multiplicity of boundary branch points are prescribed. The reason is essentially the following: If f ∈ A, f = P fˆ, P ∈ θ0λ , then fˆ is also holomorphic and is as smooth as f . But an equivalent statement is not true in the case of boundary branch points. If f ∈ A, f |∂B ∈ H s−1 (S 1 , C), and if f , for example, vanishes at w = 1, then the function g = (w − 1)−1 · f |∂B is in general a mapping of regularity class


309

H s−2 (S 1 , C) only. Thus much more effort is needed to prove boundary branch point analogues to Theorems 1 and 2. We shall now proceed to do precisely this. Definition 4. If q ∈ N, ν = (ν1 , . . . , νq ) ∈ Nq , |ν| < s − k, k ≥ 1 and s = (s1 , . . . , sq ) is a q-tuple of distinct points of ∂B = S 1 , we define

dh n satisfies f = Q · fˆ, where Q ∈ θν0 , Hν (s) = h ∈ Hn : f = dw and f vanishes nowhere on ∂B but possibly on B and Hνn =

Hνn (s),

s∈S

where S is the set of all admissible q-tuples s = (s1 , . . . , sq ). Theorem 3. Hν = Hνn=1 is a C k -smooth submanifold of H = H1 of codimension 2|ν| − q. If h ∈ Hν (s) ⊂ Hν , then η ∈ Th Hν if and only if (i) (d/dw)j η(sk ) = 0 for 1 ≤ j ≤ νk − 1 and (ii) (d/dw)νk η(sk ) = iμk · sk (d/dw)νk +1 h(sk ). It is obvious that Hν (s) ⊂ H is a submanifold of H of codimension 2|ν|. So the description of Th Hν means that Th Hν = Th Hν (s) ⊕ Wh , where Wh is a q-dimensional subspace of H. This corresponds to the possibility that the location of the q branch points may vary along S 1 . The proof of Theorem 3 depends on a series of lemmata. We start with Lemma 2. Let Y denote the space

dj η dj η s−1 Y := η ∈ H ([−π, +π], C) : j (−π) = j (+π), if 0 ≤ j ≤ s − 2 . dt dt If ν ∈ N, ν < s − k, k ≥ 1, then we define for any θ ∈ (−π, π]:

di η dν η Yν (θ) := η ∈ Y : i (θ) = 0, if 0 ≤ i < ν, ν (θ) = 0, and dt dt η(t) = 0 if t = θ and Yν =

θ∈[−π,π]

Yν (θ).

310


Then Yν is a smooth submanifold of Y of real codimension 2ν − 1. If X ∈ Yν (θ) ⊂ Yν , then η ∈ Y is in TX Yν if and only if dj η (θ) = 0 dtj

for 0 ≤ j ≤ ν − 2,

dν−1 η dν X (θ) = μ (θ) dtν−1 dtν

for some μ ∈ R.

Proof. We assume that θ = 0 and without loss of generality that Re(dν X/ dtν )(0) > 0. If δ > 0 is small, there exists a c > 0 such that |X(t)| ≥ 2c if δ ≤ |t| ≤ 2π − δ and Re(dν X/dtν )(t) ≥ 2c if |t| ≤ 2δ. Fix a C ∞ -function ψ on R, periodic mod 2π, such that ψ(t) = 0 if 2δ ≤ |t| ≤ 2π − 2δ,

ψ(t) = 1 if 0 ≤ |t| ≤ δ.

Let U be a sufficiently small neighbourhood of X in Y . Let J denote the interval (−δ, +δ), fix y ∈ U and θ ∈ J, and let αν (y, θ) ∈ C ∞ (R, C) be defined by αν (y, θ)(t) := y(θ) +

dν−1 y dy (t − θ)ν−1 (θ)(t − θ) + · · · + ν−1 (θ) . dt dt (ν − 1)!

Using αν we define F(y, θ) ∈ Y by F(y, θ)(t) := y(t) − ψ(t)αν (y, θ)(t). F(y, θ) is defined as a periodic function in an obvious way. We observe the following properties of F: (i) F : U × J → Y is C k -smooth. This is clear if one distinguishes just the derivatives with respect to y and to θ. But F definitely is not C ∞ , which makes an important difference from the situation of interior branch points. (ii) If U is sufficiently small, then for any y ∈ U we have |DF(X,0) (t)| ≥ c if δ ≤ |t| ≤ 2π − δ. If in addition |θ| is sufficiently small, then the constants y(θ), (dy/dt)(θ), . . . , (dν−1 y/dtν−1 )(θ) are very small, and so the function ψ · αν (y, θ) is very small on 0 ≤ |t| ≤ 2δ. So we have also that |F(y, θ)(t)| ≥ c if δ ≤ |t| ≤ 2π − δ. (iii) The function z(·) = z(t) := F(y, θ)(t) obviously has a zero of order ν at t = θ. We claim that there are no other zeros in 0 ≤ |t| ≤ δ. Since we have z(θ) = · · · = (dν−1 z/dtν−1 )(θ) = 0,

z(t) = 0

1

(1 − τ )ν−1 (ν − 1)!

dν z (θ + τ (t − θ))dτ · (t − θ)ν . dtν


311

We can assume that Re(dν z/dtν )(t) ≥ c if 0 ≤ |t| ≤ δ. So we get

1 (1 − τ )ν−1 dτ |t − θ|ν , |z(t)| ≥ c (ν − 1)! 0 which proves (iii). Now (ii) and (iii) imply that for any (y, θ) ∈ U × J the function F(y, θ) is in Yν (θ). On the other hand, if y ∈ Yν (θ) then F(y, θ) = y. We compute the differential of F at (X, 0) ∈ Yν × J.

dν−1 h dh tν−1 (3) DF(X,0) [h, σ] = h − ψ h(0) + (0)t + · · · + ν−1 (0) dt dt (ν − 1)! ν ν−1 d X t + σψ ν (0) (4) . dt (ν − 1)! It is clear that the null space or kernel of DF(X,0) is isomorphic to the set of

dν−1 h tν−1 . h = ψ h(0) + · · · + ν−1 (0) dt (ν − 1)! Therefore dimC ker DF(X,0) = ν and dimR ker DF(X,0) = 2ν. On the other hand, η ∈ Y is in the range of DF(X,0) if and only if (dj η/dtj )(0) = 0 for 0 ≤ j ≤ ν−2 and (dν−1 /dtν−1 )η(0) = μ(dν X/dtν )(0) for some real constant μ. Since X ∈ Yν was arbitrary it follows from the rank theorem for Fredholm maps that Yν = range F is a submanifold of Y of real codimension 2ν − 1. Lemma 3. Again let Y denote the space

dj η dj η Y := η ∈ H s−1 ([−π, +π], C) : j (−π) = j (+π), 0 ≤ j ≤ s − 2 . dt dt Generalizing the above notation, we define for (5)

θ ∈ Rq ,

−π < θ1 < θ2 < · · · < θq ≤ π

ν ∈ Nq ,

νj < s − k, k ≥ 1

and for (6)

for any j,

the space

Yν (θ) :=

η∈Y :

di η (θj ) = 0 if 0 ≤ i < νj , j = 1, . . . , q, dti dνj η (θj ) = 0, j = 1, . . . , q, dtνj and η(τ ) = 0 if τ = θ1 , . . . , θq .

312


If T is the set of admissible θ ∈ Rq , as above, let Yν := θ∈T Yν (θ). Then Yν is q a smooth C k -submanifold of Y of real codimension j=1 (2νj − 1) = 2|ν| − q. The tangent space TX Yν for X ∈ Yν (θ) can be described as the set of η in Y with di η (θj ) = 0, dti

0 ≤ i ≤ νj − 2

and

dνj −1 dX νj η(θ ) = μ (θj ). j j dtνj −1 dtνj

Proof. Clearly, Lemma 3 is only a slight generalization of Lemma 2. Fix a δ sufficiently small such that the intervals [θj − 2δ, 2δ + θj ] do not overlap. Assume ψj ∈ C ∞ (R) periodic mod 2π, identically 1 on [θj − δ, θj + δ] for some j, and identically zero in [−π, +π] outside of [θj − 2δ, θj + 2δ]. With the notation of Lemma 2 we define F(y; θ1 , . . . , θq ) = y −

q

ψj · ανj (y, θj ).

j=1

Clearly all further arguments do not have to be changed to get Lemma 3. The next lemma is again analogous to the last two. Lemma 4. Again let Z := {h ∈ H s−1 (S 1 , C)}. Fix q ∈ N, and ν = (ν1 , . . . , νq ) ∈ Nq , and with νj < s − k, k ≥ 1, for any j, let s = (s1 , . . . , sq ) denote a finite set of points on S 1 ordered along the natural orientation of S 1 . We define

∂kh ∂ νj h (sj ) = 0 for 0 ≤ k < νj , νj (sj ) = 0 Zν (s) := h ∈ Z : k ∂φ ∂φ if j = 1, . . . , q, and h(ζ) = 0 if ζ ∈ / {s1 , . . . , sq } . If S denotes the set of all admissible s ∈ (S 1 )q as above, let Zν := s∈S Zν (s). Then Zν is a smooth C k -submanifold of Z of real codimension 2|ν| − q, |ν| = νj . If h ∈ Zν (s) and η ∈ Z, then η ∈ Th Zν if and only if ∂kη (sj ) = 0 ∂φk

if 0 ≤ k ≤ νj − 2

and

∂ νj −1 η ∂ νj h (s ) = μ (sj ) j j ∂φνj −1 ∂φνj

with some real constant μj , for j = 1, . . . , q. Lemma 5. We consider the space A := {h ∈ Z : h : S 1 → C is the boundary value of a holomorphic function on the disk B} as a subspace in Z. Let ν ∈ Nq and Aν ⊂ A be defined by Aν := Zν ∩ A and Aν (s) := Zν (s)∩A. Then Aν is a real C k -submanifold of A of real codimension 2|ν| − q. If X ∈ Aν and v ∈ A then v ∈ TX Aν if and only if v ∈ TX Zν ∩ A.


313

Proof. We just have to show that the submanifold Zν in Z and the subspace A in Z intersect transversally at any point of Aν . This follows if we can show that for X ∈ Aν the space TX Zν has a complement V (X) such that Z = TX Zν ⊕ V (X), dimR V (X) = 2|ν| − q and V (X) is contained in A. But clearly there exists a (2|ν| − q) real dimensional space of polynomials P in C[w] such that if s ∈ (S 1 )q and ν ∈ Nq and X ∈ Aν (s) are given, then the equations for k = 1, . . . , q, ∂rP (sk ) = λkr ∂φr

for 0 ≤ r ≤ νk − 2,

∂ νk −1 P ∂ νk X (sk ) = isk ρk (sk ), ν −1 k ∂φ ∂φνk

i=

√

−1,

are solvable by some P ∈ C[w], if the constant λkr ∈ C, and ρk ∈ R are arbitrarily prescribed. This implies transversal intersection of A and Zν at X, and Lemma 5 follows. If X ∈ Aν (s), then by the Lemmata 4 and 5 we know that η ∈ A is in TX Aν if k ∂ η(sj ) = 0 for 0 ≤ k ≤ νj − 2 ∂φ and

∂ ∂φ

νj −1 η(sj ) = μj

∂ νj X(sj ) ∂φνj

for some real μj and j = 1, . . . , q.

As noted in Lemma 1 and the Remark following we have: If η ∈ A, an equivalent of conditions characterizing the tangent space is

d dz

k η(sj ) = 0

for 0 ≤ k ≤ νj − 2

and

d dz

νj −1

η(sj ) = iμj sj

d dz

νj X(sj ),

for real μj , j = 1, . . . , q.

After these preliminaries we obtain a proof of Theorem 3 immediately: Proof of Theorem 3. By definition, if h ∈ Hν and if c is a real constant then (h + c) ∈ Hν , also. The linear mapping d/dw : H → Z is continuous and closed, the range of d/dw being exactly the space A. Any h ∈ H is in the set Hν if and only if (d/dw)h ∈ Aν . So the manifold structure of Aν can be pulled back to H, and Hν is a submanifold of H of codimension 2|ν| − q. If X ∈ Hν then the description of TX Hν as given the statement of Theorem 3 follows from Lemma 5.

314


The next result is the n-dimensional version of Theorem 3. Theorem 4. If n ∈ N, q ∈ N, ν = (ν1 , . . . , νq ) ∈ Nq , νj < s − k, k ≥ 1, and if s = (s1 , . . . , sq ) is a q-tuple in ∂D = S 1 , then Hνn is a C k -submanifold of Hn of codimension 2n|ν| − q. If X ∈ Hνn (s) ⊂ Hνn then η ∈ TX Hνn provided that j d (i) η(sk ) = 0 for 1 ≤ j ≤ νk − 1 dw and (ii)

d dw

νk

η(sk ) = iμk sk

d dw

νk +1 X(sk ),

μk ∈ R.

Proof. Clearly one has to go over the Lemmata 2–5 again. All arguments continue to hold. The key to Condition (ii) is formula (3) which remains valid if Y is replaced by Y n , considering Cn -valued, 2π-periodic, smooth functions

dν−1 h tν−1 dh (0)t + · · · + ν−1 (0) DF(X, 0)[h, σ] = h − ψ h(0) + dt dt (ν − 1)! + σψ

dν X tν−1 . (0) ν dt (ν − 1)!

Therefore η ∈ Y n is in TX Yνn if and only if dj v (0) = 0 dtj and

for 0 ≤ j ≤ ν − 2

dν−1 η dν X (0) = μ · (0) dtν−1 dtν

for some μ ∈ R.

In Theorems 1 and 2 we treated harmonic surfaces with prescribed branch points in the interior, and in Theorems 3 and 4 we considered such surfaces with prescribed branch points on the boundary; so we may now proceed to consider both types of restrictions simultaneously. We start with a definition. Definition 5. If p ≥ 0, q ≥ 0, λ ∈ Np , ν ∈ Nq , where for all j, νj < s − k, k ≥ 1, if s is a q-tuple of distinct points on S 1 , and t is a p-tuple of distinct points in B, then we define

dh n satisfies f = Rfˆ, where R ∈ θνλ , (t; s) := h ∈ Hn : f = Hλν dw vanishing at tj of order λj and at sk of order νk ,

and fˆ : B → Cn is continuous and has no zeros at all in B ,


and again n Hλν :=

315

n Hλν (t, s),

t∈T,s∈S

where T and S denote the admissible index sets. n n (if q = 0 we write Hλ0 , these spaces being obviously If p = 0, we write H0ν n n different from the spaces Hν (or Hλ ), since in Hνn (or Hλn ) no considerations about interior (respectively boundary) branch points are taken. We now have: n Theorem 5. Hλν is a C k -smooth submanifold of Hn of codimension 2n|λ| + n n n (t, s) ⊂ Hλν then η ∈ Hn is in TX Hλν if and only 2n|ν| − 2p − q. If X ∈ Hλν if

(i) (d/dw)i η(tj ) = 0 for 1 ≤ i ≤ λj − 1, (ii) (d/dw)λj η(tj ) = βj · (d/dw)λj +1 X(tj ), where βj ∈ C, (iii) (d/dw)l η(sk ) = 0 for 1 ≤ l ≤ νk − 1, √ (iv) (d/dw)νk η(sk ) = iμk sk (d/dw)νk +1 X(sk ), where μk ∈ R, i = −1, and j = 1, . . . , p, k = 1, . . . , q. Proof. As before define An := {g ∈ H s+(1/2) (B, Cn ) : g is holomorphic} and Anλν := {g ∈ An : g = Rfˆ, R ∈ θνλ , fˆ has no zeros on B}. We must show that Anλν is a C k -submanifold of An . Let g0 ∈ Anλν . Then g0 = P0 F0 , F0 ∈ Anν = {g ∈ An : g = Qf, Q ∈ θν and f has no zeros or poles on B}. In Lemma 5 and Theorem 4 we have proved that Anν is a submanifold of An . Fix neighbourhoods U of P0 ∈ θλ and V of F0 ∈ Anν such that F ∈ V implies that F has no zeros or poles other than on ∂B. Define ψ : U ×V → An by ψ(P, F ) = P ·F . Then ψ provides a differentiable embedding of U × V onto a neighbourhood W of P0 F0 ∈ Anλν . It is clear that all g ∈ Anλν , g sufficiently close to P0 F0 , must be in the range of ψ. Since ψ is the restriction of a C ∞ map to a C k -manifold, its image is a C k -submanifold of An . The characterization of the tangent space follows as before. For technical reasons which will become clear in the next section we have to study another space of harmonic functions in Hn which contains Hνn , but is not identical to Hνn if ν = 0. Definition 6. If q ∈ N, ν ∈ Nq , |ν| < s − k, k ≥ 1, and if s = (s1 , . . . , sq ) is a q-tuple of distinct points of S 1 , then

316


n H∗,ν (s)

:=

h∈H : n

d dφ d dφ

i h(sk ) = 0 for 1 ≤ i ≤ νk , k = 1, . . . , q, νk +1

and n = H∗,ν

h(sk ) = 0

n H∗,ν (s).

s∈S n Theorem 6. H∗,ν is a C k -smooth submanifold of Hn of codimension n|ν|−q. n n if and only if both If X ∈ H∗,ν (s), then η ∈ TX H∗,ν

(i) (d/dφ)i η(sk ) = 0 for 1 ≤ i ≤ νk − 1, (ii) (d/dφ)νk η(sk ) = μk · (d/dφ)νk +1 X(sk ) for 1 ≤ k ≤ q. Proof. The proof is very similar to that of Theorem 3, so we leave it to the reader. n . Since Hνn is a submanifold It is obvious that Hνn is contained in H∗,ν n n is a submanifold of H of codimension 2n|ν| − q by Theorem 1, and H∗,ν n of H of codimension n|ν| − q by Theorem 3, we conclude that Hνn is a n submanifold of H∗,ν of codimension n|ν|. Thus we know that there are n|ν| n whose kernels determine TX Hνn . linearly independent functionals on TX H∗,ν For later purposes it will be useful to have a description of these functionals which we now proceed to give. Since in the next section we shall be considering the case that for all j either νj ≡ 0(mod 4) or νj ≡ 2(mod 4) we shall for simplicity make this assumption here too. If X ∈ Hνn has critical points {ζj }qj=1 on S 1 of orders {νj }qj=1 , then the tangent space TX Hνn can be described as the set of all h ∈ Hn such that

∂ k h = 0, ∂wk ζj and

k = 1, . . . , νj − 1, j = 1, . . . , q

∂ νj h = iμj ζj f (ζj ), ∂wνj ζj

μj ∈ R,

n , and the tangent where ∂X/∂w = Qf , Q ∈ θν0 . If X ∈ Hνn then X ∈ H∗,ν n n space TX H∗,ν can be described as the set of h ∈ H such that for ζj = eiθj we have

ˇ (k) (θj ) = 0, h

k = 1, . . . , νj − 1 and

ˇ (νj ) (θj ) = μj Fˇ (θj ) h

ˇ where h(eiθ ) = h(θ) and Fˇ is defined by n X (θ) = Πj=1 (θ − θj )νj Fˇ (θ),

Fˇ (θ) = 0.


317

Now the question is: How do we describe n|ν| linearly independent funcn necessary to determine TX Hνn ? Clearly we must have then tionals on TX H∗,ν n(|ν| − q) additional relations (∂ k h/∂wk )|ζj = 0, k = 1, . . . , νj − 1. A direct computation shows that the Fˇ and f above are related at ζj by the equation ν +1

±Fˇ (θj ) = Re(iζj j

ν +1

) Re f (ζj ) − Im(iζj j

) Im f (ζj ),

where the sign depends on νj ≡ 2 or νj ≡ 0 mod 4. Assuming that h already satisfies the n(|ν|−q) additional relations, we have that the following equations hold (if evaluated at ζj ): ν

±ζj j

∂ νj h ∂ νj ˜ = νj (h + ih), ν j ∂w ∂θ

˜ denotes the harmonic conjugate of h. If h ∈ TX Hn , we have where h ∗,ν ∂ νj h ν +1 = μj Fˇ (θj ) = ± Re(μj ζj j if (ζj )). ∂θνj ν

ν +1

Thus if Im(ζj j (∂ νj h/∂wνj )) = ± Im(μj ζj j

if (ζj )) then necessarily

∂ νj h ν +1 = μj ζj j if (ζj ), ∂wνj ∂ νj h = μj ζj if (ζj ); ∂wνj ζj ν

ζj j

i.e. h ∈ TX Hνn . To avoid confusion about signs, we assume νj ≡ 0(4), the other case being treated similarly. If we assume that h ∈ TX Hνn and (∂ νj h/∂θνj )(ζj ) = 0, the scalar multiple μj is determined. Therefore, given μj , we have to satisfy the additional nq relations νj ν ∂ h ν +1 Im ζj j νj = μj Im ζj j if (ζj ) . ∂z This situation can be described in the language of linear functionals as follows. The subspace of TX Hνn satisfying νj νj ∂ h νj +1 = ρj Im ζj if (ζj ) Im ζj ∂wνj ζj for some scalar ρj ∈ R is an (n − 1)q dimensional subspace. But for each h ∈ TX Hνn , we have νj νj ∂ h ν +1 = μj Re(ζj j if (ζj )) = μj Fˇ (θj ), μj ∈ R. Re ζj ∂wνj We can assume without loss of generality that the first component Fˇ1 (θj ) of the vector Fˇ (θj ) satisfies Fˇ1 (θj ) = 0. Then in order to guarantee that μj = ρj we need the q additional linear functionals

318


νj ν ∂ h1 if1 (ζj )) Im ζj j ∂wνj ζj νj ν ∂ h1 ν +1 Im(ζj j if1 (ζj )) = 0, − Re ζj j ∂wνj ζj ν +1

Re(ζj j

j = 1, . . . , q which concludes this discussion. Theorem 5, suitably understood, means that, except for the restriction on ν, and for the condition of p, q being finite, we have divided the set Hn into submanifolds, the indexing set of the partition being the branching type. This provides an example of a stratification of a function space with respect to a discrete parameter of great geometric significance.

5.4 Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type The purpose of the present section is a refinement of Theorem 5 in the last section, which will be very important for our approach to the Plateau problem, namely: Given a smooth regular Jordan curve Γ in n-space, one asks questions about the set of minimal surfaces among the harmonic surfaces in Rn bounded by Γ . However, it is not always true that the image of the restriction to S 1 of every smooth harmonic surface h ∈ Hn is actually a smooth regular curve; this essentially depends on the occurrence of boundary branch points. Therefore it is one goal to restrict ourselves to harmonic surfaces h such that the image of h|S 1 is a smooth curve and to construct in this set manifolds analogous to n Hλν . We now begin this task. Definition 1. Let us once and for all fix positive integers r, s such that r ≥ 2s + 4, and s ≥ 7. Let A denote the set A := {α ∈ H r (S 1 , Rn ) : α is a regular differentiable embedding of S 1 with total curvature less than π(s − 2)}. Furthermore, let D denote the set D := {ξ ∈ H s (S 1 , S 1 ) : the degree of ξ equals 1}. It is convenient to introduce the symbol N := A × D for the cartesian product of A and D. A is an open subset and thus a submanifold of H r (S 1 , Rn ). As we saw in Chapter 6 of Vol. 2, D has a natural structure of a C ∞ -Hilbert manifold, and if ξ ∈ D, then the tangent space Tξ D can be identified with the space of real valued H s -functions on S 1 . Therefore the product space A × D has the

5.4 Stratification of Harmonic Surfaces with Regular Boundaries

319

structure of a C ∞ -Hilbert manifold, and is in fact a C ∞ -Hilbert fibre bundle with total space A×D and base space A. This point of view will be important at a later point. Definition 2. A function ξ ∈ D is said to have a critical point of order ˇ := ξ(eit ) has a q < s − 1 at ζ0 = eit0 ∈ S 1 if and only if the function ξ(t) critical point of order q at t0 ; i.e. if

d dt

j

ˇ 0) = 0 ξ(t

for 1 ≤ j ≤ q

and

d dt

q+1 ˇ 0 ) = 0. ξ(t

Clearly a function ξ ∈ D having only critical points of orders < (s − 1) is topological (i.e., a homeomorphism) if and only if all critical points have even orders. In what follows we consider only critical points of order strictly less than s − 1. Theorem 1. There exists a natural mapping ω : H r (S 1 , Rn ) × D → H s (S 1 , Rn ), ω being defined by ω(f, ξ) = f ◦ ξ. The mapping ω is linear in f and C r−s -smooth in both variables. Proof. This follows from Section 6.3 of Vol. 2.

Corollary 1. The mapping ω induces a natural C r−s -smooth map, again denoted by ω, namely ω : A × D → H s (S 1 , Rn ). This mapping ω will be used to pull back the submanifold structure of the n set Hλν ⊂ Hn into the product space A × D. In order to do that, we need several results on the range of ω and the range of its differential. Theorem 2. If ξ ∈ D has no critical points at all, then its inverse ξ −1 also belongs to D. In this case the linear mapping ω(·, ξ) : H r (S 1 , Rn ) → H s (S 1 , Rn ) has dense range. Proof. The first statement follows from the chain rule, when applied to the equation ξ −1 ◦ ξ = id|S 1 . For the second we assume g ∈ H s (S 1 , Rn ). Then h := g ◦ ξ −1 is in H s (S 1 , Rn ), too. We approximate h in H s -norm by a function h1 ∈ H r (S 1 , Rn ). Clearly h1 ◦ ξ and h ◦ ξ will be H s -close too. But since h1 ◦ ξ = g1 is the range of ω(·, ξ) and h ◦ ξ = g, the theorem follows. As indicated in the introduction to this section there are new problems if ξ ∈ D has critical points. It is convenient and sufficient for our purpose to restrict to critical points of only even orders. Theorem 3. If ξ ∈ D has a finite number of critical points {ζj }qj=1 of even orders {νj }qj=1 , then the mapping ω(·, ξ) : H r (S 1 , Rn ) → H s (S 1 , Rn ) has a range E(ξ) with the following properties:

320


(i) Any η ∈ E(ξ) satisfies (d/dφ)i η(ζj ) = 0 for 1 ≤ i ≤ νj . (ii) The H s -closure E(ξ) of E(ξ) has finite codimension in H s (S 1 , Rn ). In particular, E(ξ) contains all functions in H s which vanish at ζj of order (s − 1), for j = 1, . . . , q. Proof. Property (i) follows immediately from the chain rule, and, with the argument as in Theorem 5, we see that E(ξ) contains a dense subset in the space of all functions v ∈ H s (S 1 , Rn ) which vanish in some neighbourhood of all critical points. But the H s -closure of this space coincides with the space of all η ∈ H s , such that (d/dφ)j η(ζj ) = 0 if 1 ≤ j ≤ (s − 1). It clearly has codimension at most n · q. Lemma 1. Let ω : A × D → Hn be defined as above. If α ∈ A, and ξ ∈ D has no critical points, then the derivative of ω, Dω(α, ξ) : Tα A × Tξ D → Hn , has dense range in Hn . Proof. If α ∈ A, then α : S 1 → Rn \{0} denotes the tangent vector field along α. Now, if (γ, ρ) ∈ T(α,ξ) N, then the derivative of ω can easily be computed: Dω(α, ξ)[γ, ρ] = γ ◦ ξ + α ◦ ξ · ρ. But T0 A = H r (S 1 , Rn ); thus setting ρ = 0 we see that the range of Dω(α, ξ) contains the range of ω(·, ξ) : H r (S 1 , Rn ) → H s (S 1 , Rn ), and then the result follows from Theorem 3 above. Lemma 2. If α ∈ A, and ξ ∈ D has q critical points {ζj }qj=1 of even orders {νj }qj=1 and νj ≤ (s − 2) for all j, then the range of Dω(α, ξ) : T(α,ξ) N → Hn has the following property: The closure F of (range Dω(α, ξ)) has finite codimension in Hn , and F contains all functions η ∈ Hn such that (∂/∂φ)k η(ζj ) = 0 for 1 ≤ k ≤ s − 1. Proof. This follows again from the formula Dω(α, ξ)[γ, ρ] = γ ◦ ξ + α ◦ ξ · ρ, with ρ = 0, applying the same methods as in the proof of Theorem 3 of Section 5.3. Definition 3. If q ∈ N, s = (s1 , . . . , sq ) is a q-tuple of distinct points of S 1 , ν ∈ Nq , and νj ≤ sj − k − 2 for all j we set Dν (s) := {ξ ∈ D : ξ has a critical point of order νj at sj } and Dν :=

s∈S

Dν (s),

N∗,ν := A × Dν = Pν .


321

Theorem 4. Dν is a C k -submanifold of D of codimension |ν| − q. If ξ ∈ Dν (s) ⊂ Dν , then η ∈ Tξ Dν if and only if (d/dφ)i η(sj ) = 0 for 1 ≤ i ≤ νj − 1, j = 1, . . . , q. Proof. Let Y := {g ∈ H s (R, R) : g(2π) = g(0) + 2π, g˜(t) := g(t) − 2πt being H s -periodic mod 2π}. Then Y can be identified with D, and an argument like that in Lemma 2 in Section 5.3 implies the result. Theorem 5. Let s = (s1 , . . . , sq ) and ν ∈ Nq be as above. Then ω(α, ξ) ∈ n (s) if and only if (α, ξ) ∈ A × Dν (s). H∗,ν Proof. If (α, ξ) ∈ A × Dν (s) then by the chain rule (d/dφ)k (α ◦ ξ)|sj = 0 for n (s) (cf. Definition 6 in Section 5.3). On the 1 ≤ k ≤ νj . So ω(α, ξ) ∈ H∗,ν k other hand, if (d/dφ) (α ◦ ξ)(sj ) = 0 for 1 ≤ k ≤ νj , then using the fact that α never vanishes, we conclude that ξ must have a critical point of order νj exactly at sj . This result is an important step in our present work; that is, we notice that on the bundle level codim(A × Dν , A × D) = |ν| − q, whereas we know n , Hn ) = n|ν| − q. codim(H∗,ν Definition 4. If p, q ≥ 0, λ ∈ Np , ν ∈ Nq , and for all j, νj ≤ s − k − 2, k ≥ 1, if t ∈ T, and s ∈ S, as above, then we define n (t, s)} Nνλ (t, s) := {(α, ξ) ∈ N : ω(α, ξ) ∈ Hλν

and Nνλ :=

Nνλ (t, s).

s∈S,t∈T

Our main goal is: Theorem 6. If p, q ≥ 0, λ ∈ Np , ν ∈ Nq , νj ≤ s − k − 2, k ≥ 2 for all j and all νj ≡ 0 mod 2, then Nνλ is a C k -submanifold of N of codimension n , then 2n|λ| + (n + 1)|ν| − 2p − q. If (α, ξ) ∈ Nνλ and X := α ◦ ξ ∈ Hλν (γ, ρ) ∈ T(α,ξ) N = Tα A × Tξ D is in T(α,ξ) Nνλ if and only if the function n . η := γ ◦ ξ + α ◦ ξ · ρ in Hn satisfies η ∈ TX Hλν Proof. The proof of this result again depends on several preparatory steps. Lemma 3. Let M and N be differentiable Hilbert manifolds and W ⊂ N be a submanifold of finite codimension, say γ. Let ω : M → N be C 1 with m0 ∈ M linearly independent linear and ω(m0 ) = ω0 ∈ W, and let l1 , . . . , lγ be a set of γ functionals lj : Tω0 N → R such that Tω0 W = j=1 ker lj . Suppose that

322


the functionals lj remain linearly independent when restricted to the range of Dω(m0 ) : Tm0 M → Tω0 N. Then ω −1 (W) is a submanifold of M near m0 of codimension γ. If the above holds for all points m0 ∈ ω −1 (W) then ω −1 (W) is a submanifold of M of codimension γ. Proof. This lemma follows at once from the implicit function theorem once we have shown that the above conditions imply that in a neighbourhood of m0 the map ω is “transverse” to W. This means that Tω0 W ⊕ Dω(m0 )(Tm0 M) = Tω0 N. This fact follows from the following sublemma. Lemma 4. Let E be a Banach space and E0 and F0 subspaces with F0 closed and defined as the kernel of a finite set of linearly independent functionals {lj }. Suppose that the {lj } are linearly independent on E0 . Then E = E0 + F0 . Proof. Since the {lj } are independent on E0 we can find elements {vi } on E0 with lj (vi ) = δij (Kronecker’s δ). Let x ∈ E be arbitrary. Then ⎞ ⎛ ⎞ ⎛ lj (x)vj ⎠ + ⎝ lj (x)vj ⎠ . x = ⎝x − j

j

The first term is in F0 and the second in E0 . This ends the proof of both Lemmata 3 and 4. We now repeat a very crucial observation, necessary for the construction in the proof of Theorem 6: If |ν| > 0, n ≥ 2, and if (α, ξ) ∈ A × Dν = N∗,ν = Pν ⊂ N, and if X = ω(α, ξ) ∈ Hn , then we know from Theorem 5 n n , hence ω(α, ξ) ∈ H∗,ν . We also know that the tangent space at that X ∈ H∗,ν n X, TX H∗,ν , can be described by n|ν| − q linearly independent functionals in TX Hn = Hn . But since the range of Dω(α, ξ) : T(α,ξ) N → Hn is not dense, it n ) is not necessary and in fact not true that the codimension of Pν = ω −1 (H∗,ν n n in N equals the codimension of H∗,ν in H . n , and if, in addition, X = Lemma 5. If (α, ξ) ∈ Pν , X = ω(α, ξ) ∈ H∗,ν n n n is described by ω(α, ξ) ∈ Hλν , then the tangent space TX Hλν ⊂ TX H∗,ν a set of 2n|λ| + n|ν| − 2p linearly independent functionals on the range of n Dω(α, ξ) : T(α,ξ) Pν → TX H∗,ν . n of Proof. By Theorems 5 and 6 of Section 5.3, Hνn is a submanifold of H∗,ν n ⊂ Hνn is a submanifold of codimension 2n|λ| − 2p. Let codimension n|ν|; Hλν n (s). By Lemma 2 the range of Dω(α, ξ) : T(α,ξ) Pν → Hn has a X = α◦ξ ∈ H∗,ν closure F containing all η ∈ Hn , such that (∂/∂θ)j η(sk ) = 0 if 1 ≤ j ≤ (s−1). Moreover the space F is finite codimensional and is determined via kernels of point functionals which involve only derivatives of elements h ∈ Hn at the points {sk } with respect to the angular variable θ. Thus it follows that n n in TX H∗,ν (as the linearly independent functionals which determine TX Hλν described in Theorems 3 and 4) are also linearly independent when restricted to F .


323

We are now ready to complete the proof of Theorem 6. Note that W := n n ⊂ H∗,ν =: N is a C k -submanifold of codimension γ = 2n|λ| − 2p + n|ν|. Hλν n can be viewed as a map ω : Let M := Pν = A×Dν ⊂ N. Then ω : N∗,ν → H∗,ν M → N. On account of Lemma 5 the hypotheses of Lemma 3 are fulfilled, and n ) = Nνλ is a C k -submanifold of Pν of codimension 2n|λ|−2p+n|ν|. so ω −1 (Hλν λ Thus Nν is a C k -submanifold of N of codimension 2n|λ| − 2p + (n + 1)|ν| − q because of codim (Dν , D) = |ν| − q. There is only one point left which should be discussed in the present section. In the definition of A one finds a restriction in terms of s for the total curvature of the embeddings α which are allowed in A. Yet in the main Theorem 6 one finds not an assumption on the total curvature, but the assumption (in terms of s) that the orders νi of the boundary branch points should not be larger than s − k − 2. Clearly the assumption on the curvature κ that κ < π(s − 2) will generally not imply a condition like the one used in Theorem 6. But later we will restrict our attention to the minimal surfaces in Nνλ , and for these surfaces, the total curvature of α and their total branching orders are related. This follows from the Gauss–Bonnet theorem (cf. Section 2.11 of Vol. 2, Theorem 2). Theorem 7. Let X be any minimal surface, bounded by a regular α ∈ H 5 (S 1 , Rn ), and as always let λ ∈ Np denote the multiplicities of the interior and ν ∈ Nq the multiplicities of the boundary branch points. If ν is even, if K denotes the Gaussian curvature of X, and kg the geodesic curvature of α, then the following formula holds:

kg ds + α(S 1 )

K dA = 2π + π|ν| + 2π|λ|. X(D)

Corollary 2. If α ∈ A, and ξ ∈ D such that ω(α, ξ) = X is a minimal surface without odd order boundary branch points, then the assumption that the total curvature of α be less than π(s − 2) implies that |ν| ≤ s − 4. Proof. Since for a minimal surface X the Gaussian curvature of X is nonpositive, Theorem 7 implies the inequality |ν| ≤ s−4. On account of Theorem 6 this observation has the following consequence. If α ∈ A, ξ ∈ D, α ◦ ξ is a minimal surface, and λ and ν are such that (α, ξ) ∈ Nνλ ⊂ N as defined above, then Nνλ ⊂ N is a submanifold of N at least of class C 2 , near (α, ξ). We conclude this section with the remark that the restriction that ν ∈ Nq has to be even remains the last problem when applying Theorem 6. But this condition follows from the requirement that the surface be parametrized monotonically by ξ, and exactly this assumption is used in the classical approach to minimal surfaces presented in Section 4 of Vol. 1; therefore we make this assumption as well.

324


5.5 The Index Theorem for Classical Minimal Surfaces In this chapter we show that the minimal surfaces Mλν in the bundle N = A × D of branching type (λ, ν), λ = (λ1 , . . . , λp ) ∈ Np ,

ν = (ν1 , . . . , νq ) ∈ Nq ,

νj ≡ 0 mod (2),

arise locally as subsets of local submanifolds Wλν of N. The bundle N has a projection onto its first factor, π : A × D → A. We show that πνλ := π|Wλν is a Fredholm map whose index is given by the formula index πνλ = 2(2 − n)|λ| + (2 − n)|ν| + 3 + 2p + q. From this we shall conclude that an open and dense subset Aˆ of A contains only curves which admit no minimal surface with boundary branch points. In the next section we shall show that for ν = 0, Mλ0 has the structure of a submanifold of N, and we use this fact in proving generic finiteness. Our proof will involve some of the methods developed in Chapter 6 of Vol. 2. Let α ∈ A, then α in Cjr−1 , and let Γα be the image of such an embedding. Consider the manifold of maps H s (S 1 , Γα ). In Chapter 6 of Vol. 2 we showed that H 2 (S 1 , Γα ) is a smooth submanifold of H 2 (S 1 , Rn ). The same proof generalizes to showing that H s (S 1 , Γα ) is a C r−s−4 -smooth submanifold of H s (S 1 , Rn ). Let N(α) denote the component of H s (S 1 , Γα ) determined by α. Recall that the tangent space to N(α) at a point X can be identified with the H s -maps h : S 1 → Rn such that h(eiθ ) ∈ TX(eiθ ) Γα . By harmonic extension we can identify elements of N(α) with the harmonic surfaces spanning α. We shall always assume this identification. If s = 2, we will show in Chapter 6 that there is a smooth vector field Wα on N(α) whose zeros are minimal surfaces spanning Γα . Under the present assumptions, our proofs in Chapter 6 can be easily generalized to show that Wα is a C r−s -smooth vector field on N(α). We note here that each of the zeros is a minimal surface in a more general sense than classically defined, since a zero X of Wα viewed as a harmonic map X : B → Rn is not required to induce a homeomorphism of S 1 onto Γα . If Wα (X) = 0 then we also know from Chapter 6 of Vol. 2 that the Fréchet derivative of Wα at X maps TX N(α) linearly to itself and is of the form identity plus compact. Moreover Wα is equivariant under the action of the conformal group induced on harmonic surfaces via composition. For the purpose of exposition we state the definition of Wα . Let Tj : S 1 → n R , j = 1, . . . , n, be a smooth framing of Γα , i.e. for each p ∈ Γα the ntuple {Tj (p)}nj=1 forms an orthonormal basis of Rn . We shall assume that T1 is always tangential, so that T1 (p) ∈ Tp Γα . Then the vector field Wα is characterized by the following conditions: For each X ∈ N(α), Wα (X) is a

5.5 The Index Theorem for Classical Minimal Surfaces

325

harmonic map on the disk, i.e. ΔWα (X) = 0, satisfying the mixed NeumannDirichlet boundary conditions; ∂X ∂Wα (X) · T1 (X) = · T1 (X), ∂r ∂r

(1)

Wα (X) · Tj (X) = 0,

j = 2, . . . , n,

where Tj (X) denotes the composition Tj (X(p)), p ∈ S 1 , and ∂/∂r denotes the normal (or radial) derivative along S 1 . We can paraphrase these boundary conditions as follows: Let Ω : Γα → OP (Rn ) (= the orthogonal projections on Rn ) be the C r−2 -map such that Ω(p) is the orthogonal projection of Rn onto Tp Γα . Then (1) can be rewritten as Ω(X)

∂X ∂Wα (X) = Ω(X) , ∂r ∂r

Wα (X)(p) ∈ TWα (X)(p) Γα .

ˇ ˇ := Let N α N(α), and π : N → A be the projection of N(α) onto α, ˇ ˇ has the structure of a C ∞ fibre bundle over A since N π(N(α)) = α. Then N is equivalent to the product A × D via the map ω : (α, ξ) → α ◦ ξ, A is open ˇ with in a Hilbert space, and D is C ∞ -smooth since S 1 is. We shall identify N A × D. ˇ and A × D means to identify (α, ξ) with the function Now to identify N α ◦ ξ : S 1 → Rn or with its harmonic extension to the disk. If (γ, ψ) ∈ T(α,ξ) N, then (γ, ψ) has to be identified with the function h := α ◦ ξ · ψ + γ ◦ ξ of S 1 ˆ : B → Rn such that h| ˆ ∂B = h. into Rn or with the harmonic function h The family of vector fields Wα induces a smooth vector field W on N by the rule W(X) = Wα (X) if X ∈ N(α). W will be vertical in the sense that Wα (X) ∈ TX N(α). If X ∈ N(α) is a zero of W (and hence a zero of Wα ) the Fréchet derivative DW(X) (by verticality) can be viewed as a linear map from TX N to TX N(α). Let Xw be the complex gradient of the harmonic extension of X onto B, and for h ∈ TX N, hw be defined similarly. Then we have: Theorem 1. Let DW(X) denote the Fréchet derivative of W(X) at X; then ∂ DW(X)h = −2 Im(w2 Xw · hw ) · T1 (X) ∂r j j where T1 (p) is the unit tangent to Γα at p, Xw · hw = j Xw hw , and Ωα (p) denotes now the orthogonal projection of Rn onto Tp Γα .

(2)

|Xθ |Ωα (X)

326


Proof. Define Ω(X) := Ωα (X) if X ∈ N(α). Thus Xθ Xθ ,m or |Xθ |2 Ω(X)m = Xθ , mXθ . Ω(X)m = |Xθ | |Xθ | By definition, ∂X ∂W(X) = Ω(X) . ∂r ∂r So at a zero X of W, we have h = α (ξ)ψ + γ(ξ) and Ω(X)

Ω(X)

∂X ∂h ∂ DW(X)h = Ω(X) + DΩ(X)h ∂r ∂r ∂r

and from Chapter 6 in Vol. 2, Section 3 it follows that ∂X ∂h ∂X = , T1 (X) |Xθ |DΩ(X)h ∂r ∂r ∂θ since T1 (X) = Xθ /|Xθ | and |Xθ |Ω(X)

∂h = ∂r

Thus (3)

|Xθ |Ω(X)

∂ DW(X)h = ∂r

∂h ∂X , ∂r ∂θ

∂h ∂X , ∂r ∂θ

T1 (X).

+

∂h ∂X , ∂θ ∂r

T1 (X)

= −2 Im(w2 Xw · hw )T1 (X).

Formula (3) suggests a formula for the corank of DW(X) since the set of harmonic functions on B of prescribed regularity class which are of the form Im(w2 Xw · hw ) for fixed Xw has a codimension which depends on the number and orders of the zeros of Xw , and hence on the number and orders of the branch points of X. Definition 1. Define Mλν := {X ∈ Nνλ : W(X) = 0}. The set Mλν is precisely the set of minimal surfaces in the bundle N of branching type (λ, ν). Each X ∈ Mλν belongs to N(α) for some α. The standing assumption that each νj is even ensures that each minimal surface X ∈ Mλν is monotonic as a map from S 1 to Γα . We now present Theorem 2. If X ∈ Mλν then the corank of DW(X) is 2|λ| + |ν| + 3.

5.5 The Index Theorem for Classical Minimal Surfaces

327

Proof. First note that since DWα (X) is of the form identity plus compact, the range of DWα (X) is closed and of finite codimension although in general we do not know this codimension (the corank). Moreover, from the compact part of DWα (X), it follows that the range of DWα (X) is closed in H 1 (B, Rn ) ∼ = H 1/2 (S 1 , Rn ) and thus the range of DW(X) is similarly closed and of finite codimension. Hence we may compute this codimension or equivalently the corank of DW(X) by computing the H 1 (B)-complement of the range of DW(X). So let k be in the orthogonal complement of the range of DW(X). The form of our previous formulas for DW yields

∂ DW(X)h, k dθ 0 = DW(X)h, kH 1 = S 1 ∂r

∂ Ω(X) DW(X)h, k dθ = ∂r

1 ∂X ∂h ∂X ∂h = , + , T1 (X), k dθ. |Xθ | ∂r ∂θ ∂θ ∂r for all h ∈ TX N. Now, writing k = τ Xθ for some function τ : B → C with real boundary values, we see that

∂X ∂h ∂X ∂h , + , dθ = 0 τ ∂r ∂θ ∂θ ∂r for all h ∈ TX N. Integrating by parts we obtain

∂ ∂ (τ Xθ ), h dθ = 0 − (τ Xr ) + ∂θ ∂r for all h ∈ TX N. By Section 5.4, Lemma 2, the set of all these h is dense in L2 (S 1 , Rn ). We immediately conclude that ∂ ∂ (τ Xθ ) = (τ Xr ) ∂r ∂θ

on S 1 ,

or that K(z) := τ Xθ + iτ Xr is holomorphic on B. But on S 1 we have k(w) = Re K(w) = 2 Re(iτ (w)wXw ),

2wXw = Xr − iXθ .

Thus, on B k(w) = 2 Re(iτ (w)wXw ) = Re K(w), and the orthogonal complement of the range of DW(X) can be identified with the set of all C-valued meromorphic functions τ , real on S 1 , such that τ (w)iwXw |S 1 extends to a continuous function on B. Thus τ (w) may have poles where Xw has zeros, and the order of the pole at each respective zero

328


cannot exceed the order of the zero, except at w = 0. For each interior zero, or branch point of order λ, and for each j, 1 ≤ j ≤ λ, there is a two-(real)dimensional family of such τ ’s with a pole of order j, namely, assuming the pole to be at 0 ib a + awj , − ibwj , a, b ∈ R. wj wj For a boundary branch point of order ν and for each j, 1 ≤ j ≤ ν there is a 1-(real)-dimensional family of such τ ’s. Assuming the branch point is at (1, 0), this would be given by j w+1 a i , w−1

a ∈ R.

Summing over all possible branch points and considering the possibility that τ may be constant, we obtain the dimension 2|λ| + |ν| + 2 + 1 = 2|λ| + |ν| + 3. On the other hand, we may use the symmetric version of the Riemann–Roch theorem (cf. Section 5.10, Theorem 7), which gives the same result. Corollary 1. If X ∈ M00 , i.e. if X : B → Rn is a minimal immersion, then the codimension of the range of DW is three, this number arising from the equivariance of Wα under the action of the conformal group of the disk. The quotient space TX N(α)/DW(X)[TX N] is isomorphic to the orthogonal complement of the range of DW(X). We denote this subspace by J(X) and call its elements the forced Jacobi fields, the word “forced” derived from the fact that they are forced upon one by the existence of the branch points alone. We have already seen in Chapter 6 of Vol. 2 that (as expected) they are in the kernel of the Hessian of Dirichlet’s energy. We summarize this as Definition 2. Assume that (α, ξ) ∈ N such that α ◦ ξ = X is a minimal surface in Hn . Assume that (α, ξ) ∈ Nνλ ; so X has p interior branch points w1 , . . . , wp and q boundary branch points ζ1 , . . . , ζq of orders λ1 , . . . , λp and ν1 , . . . , νq , respectively, with νj even. Set F (w) := Xw (w), and consider the holomorphic maps K : B → Cn such that K(w) := iwF (w)τ (w) where τ : B → C is a meromorphic function on B, such that iwτ (w) is tangent to S 1 = ∂B at w ∈ S 1 , and with poles at the zeros of K such that the order of any pole of τ does not exceed the order of the corresponding zero of wF (w). Then J(X) := {Re K : K(w) = iwF (w)τ (w) for w ∈ B}

5.6 The Forced Jacobi Fields

329

or, equivalently J(X) := {Re K : K(w) = iwF (w)τ (w), where τ (w) is a meromorphic function on B, real on ∂B, with poles of orders ≤ λj at wj = 0 and ≤ νj at ζj . If wj = 0 for any j, τ (w) may have a pole of order ≤ 1 at 0, and if wj = 0 for some j, τ (w) has a pole of order ≤ λj + 1 at 0}. Since the forced Jacobi fields are so central to the question of whether or not minima can be branched we devote a special section to study them a bit further.

5.6 The Forced Jacobi Fields We will use the following definition. Suppose X = α ◦ ξ is a minimal surface with (α, ξ) ∈ Nνλ . Let F (w) := Xw (w). The space J(X) of forced Jacobi fields at X is the space of functions k : B → Rn such that Δk = 0,

k = Re K,

K(w) = iwτ (w)F (w),

τ a meromorphic function on B, Im τ = 0 on ∂B (hence by reflection τ is meromorphic in a neighbourhood of B), τ has poles at w1 , . . . , wp ∈ B and ζ1 , . . . , ζq ∈ ∂B of orders ≤ λj at wj and ≤ νj at ζj . If wj = 0 for any j, τ (w) can have a pole of order ≤ 1 at 0 and if wj = 0 for some j, τ (w) can have a pole of order ≤ λj + 1 at 0}. We now present an alternative (but equivalent) definition. Definition 1. A mapping Y : B → Rn is called a forced Jacobi field for a minimal surface X = α ◦ ξ if and only if it satisfies (i) Y ∈ H s+1/2 (B, Rn ), ΔY = 0. (ii) for all (u, v) ∈ B the function Y satisfies Y (u, v) = γ(u, v)Xu (u, v) + β(u, v)Xv (u, v) with C 1 -real valued γ, β, and is defined everywhere except possibly at the branch points of X. Hence Y is tangent to X at all points except where Xu and Xv vanish. (iii) For all ζ = eiθ ∈ ∂B the function Y satisfies Y (ζ) = ψ(ζ)α (ξ(ζ)) with ψ : S 1 → R. Thus Y is everywhere tangent to α ◦ ξ on S 1 . (iv) For all w = u + iv ∈ B, we have Yw · Xw = 0. This says that Y is “infinitesimally conformal with respect to X”. Let us denote this space by J(X). We will now prove the following result which justifies the use of the same name for both J(X) and J(X). Theorem 1. The space J(X) is identical with the space J(X).

330


We first show Lemma 1. J(X) ⊂ J(X). Proof. Let X = α ◦ ξ be minimal, (α, ξ) ∈ Nνλ . Write F (w) := Xw (w) = RFˆ (w), R ∈ θνλ . Then by regularity Fˆ ∈ H s (B, Cn ). If Y ∈ J(X) then Y = Re(iτ (w)wF (w)), and Y (w) = Re(iτ (w)w) Re F − Im(iτ (w)w) Im F 1 = [Re(iτ (w)w)Xu + Im(iτ (w)w)Xv ] ; 2 so (ii) holds. If ζ = eiθ ∈ ∂B we have Y (ζ) = Re(τ (ζ)) Re(iζF (ζ)) − Im(τ (ζ)) Im(iζF (ζ)) 1 = Re(τ (ζ))Xθ = ψ · α (ξ)(ζ); 2 so (iii) holds. Let τ(w) = iwτ (w), G = τF . Since Y = Re( τ F ) = Re(G), Y is harmonic and we have τ F ) = ( τ F + τF ) Yw = G = ( τ F + τF ) · F . But everywhere where τ is defined. Therefore Xw · Yw = ( F · F = 0 and F · F = 0, and so we obtain (iv). Lemma 2. If Y ∈ J(X) and Y = γXu + βXv as in (ii) of Definition 1 then γ and β are conjugate harmonic away from the branch points of X (zeros of F ). Proof. Computing we have Yv = γv Xu + βv Xv + γXuv + βXvv , Yu = γu Xu + βu Xv + γXuu + βXvu , and Yu · Xu = γu |Xu |2 + γXuu · Xu + βXvu · Xu , Yv · Xv = βv |Xv |2 + γXuv · Xv + βXvv · Xv , Yu · Xv = βu |Xv |2 + γXuu · Xv + βXvu · Xv , Yv · Xu = γv |Xu |2 + γXuv · Xu + βXvv · Xu . Using the relations Yu · Xu = Yv · Xv and |Xv |2 = |Xu |2 , we get that by (iv) of Definition 1 γu |Xu |2 + γXuu · Xu + βXvu · Xu = βv |Xv |2 + γXuv · Xv + βXvv · Xv = βv |Xv |2 + γXuu · Xu + βXvu · Xu . Therefore γu = βv when |Xu |2 = 0. Using that Xu · Yv = −Yu · Xv again by (iv) of Definition 1 we obtain that γv = −βu .


331

Lemma 3. Define ϕ := −2i(γ + iβ). Then by Lemma 2, ϕ is holomorphic except possibly at the branch points of X, and Y = Re(iϕF ) = Re{(γ + iβ)(Xu − iXv )}. (a) If Y = Re(iϕF ) ∈ H s+1/2 (B, Rn ), then iϕF ∈ H s+1/2 (B, Cn ). (b) If F = RFˆ , then Fˆ ∈ H s+1/2 (B, Cn ) and ϕ(w) · R(w) ∈ H s+1/2 (B, C). (c) If Y = Re(iϕF ) satisfies condition (iii) of Definition 1, then the imaginary part of τ (w) = ϕ(w)/w satisfies Im(τ (w)) = 0 on S 1 at those points where Xu = 0. Proof. Assertion (a) follows from the regularity theory for the Cauchy– Riemann equations, (b) from the ring properties of H s+1/2 (B, C), and (c) from a direct computation. To see this, we notice that on S 1 Y = Re(iτ (w)wF (w)) = Re(τ (w)) Re(iwF (w)) − Im(τ (w)) Im(iwF (w)) 1 = [Re(τ (w))Xθ − Im(τ (w))Xr ]. 2 Since Y on S 1 is equal to ψ(θ) · Xθ and Xθ and Xr are orthogonal to each other we must have Im(τ (w)) = 0 at those points w ∈ S 1 where F = 0. This finishes the proof of (c). Lemma 4. J(X) ⊂ J(X). Proof. If Y ∈ J(X), Y = Re(iτ (w)wF (w)) where, by Lemma 3, τ is meromorphic. It will be convenient to have a complex analytic description of the kernel of DW(X), which we denote by ΣX ; so we set ΣX := ker DW(X) ⊂ TX N. Recall that N = A × D and TX N = {(γ, ψ) : γ ∈ H r (S 1 , Rn ), ψ ∈ H s (S 1 , R), where ψ(eiθ ) = λ(θ)ieiθ , λ(θ) is real for all θ, or equivalently ψ(eiθ ) ∈ Teiθ S 1 }. We also have the map ω : N → Hn and its differential Dω(α, ξ)[γ, ψ] = γ ◦ ξ + α (ξ)ψ = h. ˜ ˜ is For such h, let H : B → Cn be defined by H(w) := h(w) + ih(w), where h ˜ the harmonic conjugate to h, h(0) = 0. Then we have: Theorem 2. If X = α◦ξ ∈ Mλν , the kernel ΣX of DW(X) : TX N → TX N(α) can be characterized n as the set of all h ∈ TX N such that for F := Xw : B → Cn the equation j=1 Fj (w)Hj (w) = 0 is satisfied (a formula which will be abbreviated in the sequel as F (w) · H (w) = 0).

332


Proof. This follows from formula (3) of Theorem 1 in Section 5.5 and the fact ∂ is an isomorphism from H s - to H s−1 -vector fields over that LX := Ωα (X) ∂r X, which will be proved in Chapter 6. Corollary 1. As before, let J(X) denote the subspace of forced Jacobi fields in TX N. Then J(X) ⊂ ΣX . Proof. By Definition 2 in Section 5.5, any k ∈ J(X) is of the form k = Re K with K(w) = F (w)ϕ(w). Since X is minimal, F (w) · F (w) ≡ 0. So we have F (w) · F (w) ≡ 0, and consequently F (w) · K(w) ≡ 0. Differentiating this last expression we obtain F (w) · K(w) + F (w) · K (w) ≡ 0, but since F (w) · F (w) ≡ 0, we get F (w) · K (w) ≡ 0; hence Re K ∈ ΣX by Theorem 2, and the lemma follows. Recall that in Definition 3 of Section 5.4 we introduced the submanifold Dν of D, which is a submanifold of codimension |ν| − q (Theorem 4). Again denote by Pν the subbundle A × Dν of N = A × D. Thus, as a submanifold of N, Pν also has codimension |ν| − q. We know that the Jacobi fields at a zero X of W form a subspace of ΣX . Let Jν (X) denote the set of those Jacobi fields which arise only from the boundary branch points or more precisely from those ϕ’s which have poles on S 1 . Again, every element of Jν (X) is of the form Re K(w), K(w) = F (w)ϕ(w), ϕ a meromorphic vector field on B, with poles only at the boundary branch points of X, again say ζ1 , . . . , ζq . The poles are required to have orders less than or equal to νj at ζj . The next result is a consequence of the characterization of Jν (X) and the characterization of the tangent space to Dν . Theorem 3. We have (1)

TX Pν + Jν (X) = TX N.

Proof. At this point encounter some confusion between the bundles we may ˇ The Jacobi fields are the real parts of functions N = A × D and N(α) = N. of the form K(w) = F (w)ϕ(w), ϕ meromorphic and tangent to S 1 . Thus as a map of the disk, Re K ∈ TX N(α) since for each w ∈ S 1 , Re K(w) ∈ ˇ However A × Dν = Pν TX(w) Γα , Γα = α(S 1 ). These K’s sit in the bundle N. sits in N. Thus to show (1) we have to “pull back” the K’s or “push forward” ˇ We choose to push Pν = A × Dν via the identification of A × D with N. forward the Pν . The codimension of Pν in N is |ν| − q, the dimension of the space of Jacobi fields is 2|λ| + |ν| + 3, and the dimension of Jν (X) is |ν|. Thus the space Jν (X) has a dimension large enough to be a possible complement of TX Pν . To prove explicitly that Jν (X) spans the complement we recall that the identification


333

ω ˇ is given by (α, ξ) → ˇ and N as of A × D with N α ◦ ξ, or we can identify N r 1 n s 1 submanifolds of the product Hilbert spaces H (S , R ) × H (S , Rn ) via the ω map (α, ξ) → (α, α ◦ ξ). If we use this last identification the image of T Pν under the differential of ω contains as subspaces all pairs (β, h) such that β ∈ H r (S 1 , Rn ) is arbitrary and (∂ k h/∂θk )(ζj ) vanishes for 1 ≤ k ≤ νj − 1. Now let ζj be a fixed boundary branch point. By a conformal map we can identify the unit disk with the upper half plane, and S 1 with R, and ζj with the origin. Then, about 0, the function (F1 , . . . , Fn ) has a Taylor expansion

Fk (w) =

L

Al wl + o(|w|L )

l=lk

with lk ≥ νj and Fk the k th component of F . In some k, say k = 1, this sum starts at l1 = νj . There exist admissible vector fields ∞τ in Definition 2 of Section 5.5 which have an expansion about 0, τ (w) = l>−l0 bl wl +bl0 w−l0 , bl0 = 0 with 0 ≤ l0 ≤ νj and no poles other than ζj = 0. This implies that for any ˜ where l < νj the Jacobi fields as elements of TX N(α) contain pairs (0, h) s˜ s l+1 ˜ l+1 s˜ s (∂ h/∂θ )(ζj ) = 0, 1 ≤ s ≤ l, (∂ h/∂θ )(ζj ) = 0 and (∂ h/∂θ )(ζm ) = 0, 1 ≤ s ≤ νj , for all other branch points ζm , ζm = ζj . This observation clearly yields the proof of Theorem 3. Our main goal now is to compute the codimension of the range of DW(X) restricted to TX Nνλ ; i.e., what is dim TX N(α)/DW(X)[TX Nνλ ]? Theorem 4. dim TX N(α)/DW(X)[TX Nνλ ] = 4|λ| + 2|ν| + 3 if X is a minimal surface in Nνλ . Therefore the additional loss σ of rank of DW(X) coming from restricting DW to the subvariety Nνλ is σ = 2|λ| + 2|ν|. We shall prove this main result in stages. We shall first consider the (easier) case when ν = 0. We actually first show that in this case the additional loss of rank, σ, is 2|λ|. As before let ΣX denote ker DW(X) in TX N. We know that TX Nνλ is a subspace of TX N of finite codimension 2n|λ| + (n + 1)|ν| − 2p − q (cf. Theorems 5 of Section 5.3 and 4 of Section 5.4). Thus the loss of rank of DW(X) is clearly equal to the dimension of the quotient space DW(X)[TX N]/DW(X)[TX Nνλ ]. However it is not hard to see that DW(X)[TX N]/DW(X)[TX Nνλ ] ∼ = F, where TX N = (ΣX + TX Nνλ ) ⊕ F. Thus in order to compute the loss of rank coming from restriction of DW(X) to TX Nνλ , it is only necessary to compute the dimension of the space

334


TX N/(ΣX + TX Nνλ ). The dimension of this quotient space seems difficult to compute directly. However, by a little algebra we can reduce this computation to an easier one; that is, by the Noether isomorphism theorem we have (TX N/TX Nνλ )/(ΣX /ΣX ∩ TX Nνλ ) ∼ = TX N/(ΣX + TX Nνλ ). In the case ν = 0 we shall now compute the dimension of ΣX /ΣX ∩ TX N0λ . Lemma 5. dim ΣX /ΣX ∩ TX N0λ = 2|λ|(n − 1) − 2p. Corollary 2. Since dim TX N/TX N0λ = 2n|λ| − 2p, it follows by the Noether theorem that on the set Mλ0 the additional loss of rank σ is exactly 2|λ|. Proof of Lemma 5. Let X = (α, ξ) ∈ Mλ0 with F (w) := Xw . Since X is minimal we have the relation F (w) · F (w) ≡ 0.

(2)

By Theorem 2 we can identify the kernel ΣX of DW(X) with the set of all holomorphic H : B → Cn , where Re H = γ ◦ ξ + α (ξ)ψ = h, such that F (z) · H (z) ≡ 0.

(3)

Suppose that (γ, ψ) is tangent to N0λ at X and let h = Dω(α, ξ)[γ, ψ] ∈ TX Hλn . Then if F (w) = P (w)G(w), G(w) = 0 for all w, P ∈ θ0λ , we have by Theorem 3 of Section 5.3 the relations ∂ k h = 0, ∂wk wj and

∂ λj h = βj · G(wj ) ∂wλj wj

1 ≤ k < λj ,

for some βj ∈ C,

where wj denote the branch points of X in B. In terms of H this becomes (a) (4) (b)

∂ k H = 0, 1 ≤ k < λj , ∂wk wj ∂ λj H = βj · G(wj ) for some βj ∈ C. ∂wλj wj

It now suffices to compute first the codimension in An (cf. Definition 1, Section 5.3) of all functions H satisfying (4) and then to compute the codimension of the space of all functions satisfying (2) and (4) in the space of all functions satisfying (3), the latter being the dimension we are looking for. Now F 2 ≡ 0, implies G2 ≡ 0, and in particular for some fixed wj we have

5.6 The Forced Jacobi Fields n

335

Gk (wj ) · Gk (wj ) = 0.

k=1

Since G(wj ) = 0 by definition of G (as X ∈ Mλ0 and F = P ·G, where P ∈ θ0λ ), this implies that at least two of the coordinates of G at wj , {Gi (wj )}ni=1 , are non-zero, say G1 (wj ) and G2 (wj ). Subject to no other relations, the real codimension of the set of H ∈ An satisfying (4) for one wj is equal to 2n(λj − 1) + 2(n − 1). To see this it suffices to show that the complex codimension is n(λj − 1) + (n − 1). Clearly the codimension of the space of H ∈ An satisfying (a) is n(λj − 1). For (b) we can define (n − 1) linearly independent functionals whose kernels yield relation (b), as follows. For simplicity of exposition let us λ denote by H λj the vector (∂ λj H/∂wλj )|wj , and by Hk j its k th component. Define (n − 1) complex linear functionals {φk }nk=2 on An by λ

φk (H) = Hk j −

Gk (wj ) λ · H1 j , G1 (wj )

k = 2, . . . , n.

λ

Clearly H1 j = βj G1 (wj ) for some βj ∈ C, and so if and only if H ∈ n k=2 ker φk , then H satisfies (4)(b). Now we must compute the codimension of the set of all H satisfying (4) subject to relation (3). The claim is that if there is only one branch point wj of multiplicity λj then the real codimension is 2(n − 1)(λj − 1) + 2(n − 2). To see this we first note that if (4)(a) is satisfied for the last components Hk , k = 2, . . . , n, then it must be satisfied by the first H1 , too. Because of F (w) · H (w) ≡ 0 we have that (5)

G(w) · H (w) ≡ 0.

Therefore Hk (wj ) = 0, k = 2, . . . , n, together with G1 (wj ) = 0 imply that H1 (wj ) = 0. Now assume that Hk (wj ) = 0 for all k and differentiate (5) and evaluate at wj to get G(wj ) · H (wj ) = 0, whence Hk (wj ) = 0, k = 2, . . . , n, which again clearly implies that H1 (wj ) = 0 and so on until we reach the (λj − 1) st derivative of Hk evaluated at wj . Next let us assume that H already satisfies (4)(a). Then if (6)

λ

Hk j =

Gk (wj ) λ · H1 j G1 (wj )

for k = 3, . . . , n, then it must hold for k = 1, 2. That it must hold for k = 1 is λ clear. Let βj = H1 j /G1 (wj ). Since H satisfies (4)(a) and G(w) · H (w) ≡ 0 it follows that (7)

n k=1

λ

Gk (wj )Hk j = 0.

336


If we assume (6) we must have λ

0 = βj G1 (wj )2 + G2 (wj )H2 j + βj

n

Gk (wj )2

k=3

= −βj G2 (wj ) + 2

λ G2 (wj )H2 j ,

λ

and thus we have that H2 j = βj G2 (wj ). We therefore need only (n−1)(λj −1) relations or functionals to satisfy (4)(a) and (n − 2) relations to satisfy (4)(b) subject to (3). It is clear that this set of relations is the minimal set. Doing this for each wj we find that the minimal number of real linear functionals which annihilate ΣX ∩ TX N0λ in ΣX is 2(|λ| − p)(n − 1) + 2(n − 2)p = 2|λ|(n − 1) − 2p, which completes the proof of Lemma 5. Thus the loss of rank from restriction of DW to N0λ is exactly 2|λ|. We now wish to consider the case ν = 0, which is more difficult. Let X ∈ Nνλ . Then X ∈ Pν = A × Dν and so by Theorem 3 TX Pν + Jν (X) = TX N, and since Jν (X) ⊂ ΣX , in order to compute the loss of rank of DW(X), it suffices to compute the loss of rank of DW(X)|TX Pν . ν := ΣX ∩ TX Pν . As before we wish to compute the dimension of Let ΣX ν TX N(ΣX + TX Nνλ ). This will give us the additional loss of rank, σ. By the Noether theorem this reduces to computing the dimension of the quotient ν ν /ΣX ∩ TX Nνλ . space ΣX Lemma 6. If X ∈ Mλν , then ν ∩ TX Nνλ = 2(n − 1)|λ| + (n − 2)|ν| − 2p. dim TX Pν /ΣX

Corollary 3. Since by Theorems 4 and 6 of Section 5.4 we have dim TX Pν /TX Nνλ = 2n|λ| + n|ν| − 2p then Lemma 6 implies that ν + TX Nνλ ) = 2|λ| + 2|ν|. dim TX Pν /(ΣX

Thus the additional loss of rank, σ, coming from restriction of DW(X) to Nνλ is exactly 2|λ| + 2|ν|. Proof of Lemma 6. As before let X = (α, ξ) ∈ Mλν , F (w) = ∂X/∂w, and ν be identified with the set of all H ∈ An , where Re H = γ ◦ ξ + α ◦ ξ · ψ, ΣX (γ, ψ) ∈ TX Pν , such that F (w) · H (w) ≡ 0. Since ω maps Pν into H∗,ν (cf.


337

n Theorem 5 of Section 5.4), we have Dω(X) : TX Pν → Tω(X) H∗,ν . From this ν it follows that (γ, ψ) ∈ ΣX if and only if

(a)

(8)

(b)

F (w) · H (w) ≡ 0, n Re H = Dω(X)[γ, ψ] ∈ Tω(X) H∗,ν .

Now suppose that (γ, ψ) is tangent to Nνλ at X. Since G is the complex n gradient of X ∈ Hλν , F (w) = P Qfˆ = P G = Qf = Πj (w − ζj )νj f , P ∈ θ0λ , 0 Q ∈ θν , f as in Theorem 6 of Section 5.3, it follows from the characterization of the tangent space to Nνλ that any h = Re H ∈ TX Nνλ must satisfy (a) (9) (b)

∂ k h ∂ k H = = 0, 1 ≤ k < λj ; k ∂w wj ∂wk wj ∂ λj H ∂ λj h = = βj G(wj ), βj ∈ C, j = 1, . . . , p; ∂wλj wj ∂wλj wj

and for the boundary branch points we have the conditions (c) (10) (d)

∂ k H ∂ k h = = 0, 1 ≤ k < νj ; ∂wk ζj ∂wk ζj ∂ νj H ∂ νj h = = iμj ζj f (ζj ), μj ∈ R, j = 1, . . . , q. ν ∂w j ζj ∂wνj ζj

Remark 1. From Theorem 5 of Section 5.4 we infer that since h is in the n , it may be annihilated by other higher order range of Dω(X) and in TX H∗,ν point functionals at the ζj , though this is without influence in what follows. ν ν Since our immediate goal is to compute the dimension of ΣX /ΣX ∩ TX Nνλ , we must ask for the minimal set of linearly independent functionals whose ν ∩ TX Nνλ . In Lemma 5 of Section 5.6 we argued with kernels determine ΣX one branch point at a time, and it is evident that the same arguments still apply to the case of interior branch points in our new situation; thus we must have at least 2|λ|(n − 1) − 2p linearly independent functionals for these branch points. We shall now argue for the boundary branch points, one at a time. Again it suffices to compute the codimension of the subspace consisting of all H ∈ An satisfying (9) and (10) in the subspace satisfying (8). Let ζj = eiθj be some branch point of X on S 1 . From Theorem 6 of Section 5.3 it follows n that Re H = h ∈ TX H∗,ν implies that

(a) (11) (b)

∂ k h = 0, 1 ≤ k < νj , ∂θk θj ∂ νj h = μj Re(ζj if (ζj )). ∂θνj θj

338


ν Thus H ∈ ΣX if and only if F (w) · H (w) ≡ 0 and (11) holds (always with the requirement that h ∈ range Dω(X). For fixed ζj we must determine the minν which yield relation imal number of linearly independent functionals on ΣX (10), (c) and (d). If we did not have the relation F (w) · H (w) ≡ 0, then from Theorem 6 of Section 5.3 it would follow that we need only consider the n(νj − 1) additional functionals (∂ k H/∂wk )|ζj = 0, k = 1, . . . , νj −1, together with the n additional relations νj ν j ∂ Hk ν +1 = μj Im(ζj j ifk (ζj )), k = 1, . . . , n, Im ζj ∂wνj ζj

where H = (H1 , . . . , Hn ), and where for given H, the value μj is determined by (11), (b). For the interpretation of this last condition see the discussion at the end of Theorem 6 in Section 5.3. Therefore, without the restriction of the relation F (w) · H (w) ≡ 0, the additional number of relations needed to determine TX Nνλ in TX Pν (or equivalently the number of functionals needed to annihilate its quotient) would be n(νj − 1) + n, and summing over all branch points we would get n(|ν| − q) + nq = n|ν| such functionals. Assuming we have the additional relation (8)(a), F (w) · H (w) ≡ 0, we claim that the minimal number of functionals required is (n − 2)(|ν| − q) + (n−2)q = (n−2)|ν|, and we shall argue as in Lemma 5 of Section 5.4. Without loss of generality we may assume that ν +1

Re(ζj j

if1 (ζj )) = 0

n , we (cf. discussion following Theorem 6 of Section 5.3). Since Re H ∈ TX H0,ν must have ∂H ∂H Re = 0 = Re iζj . ∂θ ζj ∂w ζj ν +1

For notational simplicity let ρjk = iζj j fk (ζj ). We know that f 2 (w) ≡ 0 which implies that {Re ρjk Re ρjk − Im ρjk Im ρjk } = 0 k

and

{Im ρjk Re ρjk } = 0, k

where we are assuming that Re ρj1 = 0. Lemma 7. Let {aj }, {bj }, j = 1, . . . , n, be real numbers satisfying Σj a2j = Σj b2j with a1 = 0. Then for some k = 1,

and

Σ j a j bj = 0


det

a1 b 1 a k bk

339

= a1 bk − ak b1 = 0.

Proof. Otherwise, for all k, we have bk = ak b1 /a1 , which implies that a j b1 = 0, Σj aj a1 or that b1 = 0. This says that all bj = 0, which contradicts that Σa2j = 0. Using this lemma we can conclude that for some k Im ρj1 Im ρjk 0. = det Re ρj1 Re ρjk Without loss of generality we can assume k = 2. Now assume that Im(iζj ∂Hk /∂w|ζj ) = 0 for k = 3, . . . , n. The condition F (w)H (w) ≡ 0 implies that f (w)H (w) ≡ 0

(12) or, at ζj , that ν +1

0 = (ζj j

∂H ∂Hk ν +1 = if (ζj )) · ζj i ζj j ifk (ζj ) · ζj i , ∂w ζj ∂w ζj k

and so 0=

∂Hk ∂Hk + Re ρjk Im ζj i Im ρjk Re ζj i ∂w ∂w

k

and 0=

∂Hk ∂Hk − Im ρjk Im ζj i . Re ρjk Re ζj i ∂w ∂w

k

Using (11), the fact that Re(ζj i(∂Hk /∂w)) = 0 and our assumption that Im(ζj i(∂Hk /∂w)) = 0 for k = 3, . . . , n, we obtain the system of equations ∂H1 ∂H2 j j + Re ρ2 · Im ζj i = 0, Re ρ1 · Im ζj i ∂w ∂w ∂H1 ∂H2 Im ρj1 · Im ζj i + Im ρj2 · Im ζj i = 0. ∂w ∂w Since we know that the determinant of the coefficient matrix is non-zero we can conclude that ∂Hk Im ζj i = 0, k = 1, 2, ∂w ζj

340


and that

∂H = 0. ∂w ζj

n , Now assume that (∂H/∂w)|ζj = 0. Again since Re H ∈ TX H0,ν

Re

2 ∂ 2 H 2∂ H = Re ζ = 0. j ∂θ2 θj ∂w2 ζj

By differentiating (12) and evaluating at ζj we get that f (ζj ) · H (ζj ) = 0, which implies that ν +1

ζj j

f (ζj ) · ζj2 H (ζj ) = 0.

As before we can show that Im(ζj2 (∂Hk2 /∂w2 )|ζj ) = 0 for k = 3, . . . , n implies that both H1 (ζj ) and H2 (ζj ) = 0, and hence that H (ζj ) = 0 and so on through all derivatives up to but not including νj . Therefore to annihilate the ν we need only (n − 2)(νj − 1) first νj − 1 derivatives of H on the subspace ΣX linearly independent functionals, and this set is clearly minimal. ν ν Now let Hk j denote ζj j (∂ νj Hk /∂wνj )|ζj . We still need the relations (13)

ν

Im Hk j = μj Im ρjk ,

k = 1, . . . , n, for μj ∈ R.

Assuming that the first νj − 1 complex derivatives of H at ζj have been ν n , we must have Re Hk j = μj Re ρjk for all annihilated, and since H ∈ TX H0,ν k (again see Theorem 6 of Section 5.3). Differentiating (12) νj − 1 times and evaluating at ζj we can conclude that (14)

ν +1

ζj j

ν

if (ζj ) · ζj j

∂ νj H = 0 or ∂wνj ζj

ν

Hk j ρjk = 0

k

with Re ρj1 = 0. Thus we get that both ν ν {Re ρjk Re Hk j − Im ρjk Im Hk j } = 0, (15)

k

ν ν {Im ρjk Re Hk j + Im Hk j Re ρjk } = 0. k

We also know that

(16)

{Re ρjk Re ρjk − Im ρjk Im ρjk } = 0, k {Re ρjk Im ρjk } = 0 k

5.7 Some Theorems on the Linear Algebra of Fredholm Maps

are satisfied, and that Re ρj1 = 0, Im ρj1 det Re ρj1

Im ρj2 Re ρj2

341

= 0.

ν

Now assume that Im Hk j = μj Im ρjk , k = 3, . . . , n. Then (15), (16) and the ν relations Re Hk j = μj Re ρjk , k = 1, . . . , n, imply that the following set of equations holds: ν

ν

ν

ν

(Im ρj1 )(Im H1 j ) + (Im ρj2 )(Im H2 j ) = μj {(Im ρj1 )2 + (Im ρj2 )2 }, (Re ρj1 )(Im H1 j ) + (Re ρj2 )(Im H2 j ) = μj {Im ρj1 Re ρj1 + Im ρj2 Re ρj2 }. ν

Since Im Hk j = μj Im ρjk , k = 1, 2, is a solution and the solution is unique ν we can conclude that Im Hk j = μj Im ρjk , k = 3, . . . , n, implies that (10), (d) holds. Thus the number of additional relations to infer (13) is n−2 for each ζj . It remains to show that this is actually the minimal number. The proof of this is rather technical and must be carried out also in the higher genus case; see Lemma 3 of Section 5.17. Assuming this, we now know that if there is only one boundary branch point of order νj and no interior branch point, then ν ν dim ΣX /ΣX ∩ TX Nν0 = (n − 2)(νj − 1) + (n − 2) = (n − 2)νj .

This is clearly additive over branch points and so by the previous remarks we can conclude that in the case of both interior and boundary branch points ν ν /ΣX ∩ TX Nνλ = 2(n − 1)|λ| + (n − 2)|ν| − 2p. dim ΣX

This completes the proof of Lemma 6. By Corollary 3 we know that the additional loss of rank, σ, is exactly 2|λ| + 2|ν|.

5.7 Some Theorems on the Linear Algebra of Fredholm Maps Before we prove the stratification theorems of this section there are some transversality results which will be necessary. Theorem 1. Let E and F be Hilbert spaces. Suppose f∗ : E × F → RN is linear, continuous and onto, and let π∗ : E ×F → E be the projection onto the first factor. Assume also that f∗ |(ker π∗ ={0}×F ) has rank μ. Then the corank of π∗ |ker f∗ equals N − μ. Proof. Let M = ker f∗ , f1 = f∗ (·, 0) and f2 = f∗ (0, ·). We then have f∗ (x1 , x2 ) = f1 (x1 ) + f2 (x2 ).

342


Since f∗ is surjective, RN = range f1 + range f2 . ˜ ⊂ E be a subspace such that Let E ˜ ⊕ ker f1 = E, E and let F˜ ⊂ F be a subspace such that F˜ ∩ ker f2 = {0}. Then ˜ 0) + f2 (0, F˜ ). RN = f1 (E, Since dim range f2 = μ, ˜ 0) = N − μ. ˜ = dim f1 (E, dim E ˜ 0) ⊕ (0, F˜ ) is not contained in the kernel of f∗ . Since f∗ is Furthermore (E, surjective, we have the direct sum ˜ 0) ⊕ (0, F˜ ). E × F = M ⊕ (E, Applying π∗ to both sides of this equation we obtain ˜ E = π∗ (M ) ⊕ E which immediately implies the theorem.

As a consequence we have Corollary 1. Suppose N = A × D and f : N → RN is C 1 and its Fréchet derivative f∗ at a point X = (α, ξ) is onto. Let π : N → A denote the bundle projection map and π∗ := Dπ : TX N → Tα A its derivative. Now if f∗ := Df : ker π∗ → RN has rank μ then the corank of π∗ : ker f∗ → Tα A equals N − μ. Next we have Theorem 2. Suppose E, F and G are Hilbert spaces and that k∗ : E → F and π∗ : E → G are linear, continuous and onto. Assume that ker π∗ splits and that k∗ : ker π∗ → F is a Fredholm operator of index m. Then π∗ : ker π∗ → G is also Fredholm of index m. Proof. Write E = E0 ⊕E1 , π∗ : E0 → G an isomorphism, E1 = ker π∗ . Further write F = range(k∗ |E1 ) ⊕ F1 , Σ = ker k∗ , E2 a complement to (range π∗ |Σ); G = range(π∗ |Σ) ⊕ E2 . Finally we may identify G with E0 , so that π∗ : E0 ⊕ E1 → E0 is projection onto the first factor.


343

We are assuming that k∗ |E1 is Fredholm of index m. The claim is that π∗ |Σ is also Fredholm of index m. First ∼ ker(k∗ |E1 ) where = ∼ denotes isomorphism. Suppose (e0 , e1 ) ∈ 1. ker(π∗ |Σ) = Σ and π∗ (e0 , e1 ) = e0 = 0. Then k∗ (0, e1 ) = 0 and e1 ∈ ker(k∗ |E1 ). Thus (e0 , e1 ) ∈ ker π∗ |Σ if and only if e0 = 0 and e1 ∈ ker(k∗ |E1 ). This establishes assertion 1. 2. Coker(π∗ |Σ) ∼ = Coker(k∗ |E1 ). Write E0 = range(π∗ |Σ) ⊕ E2 ,

F = range(k∗ |E1 ) ⊕ F2

with p : E0 → E2 , q : F → F2 the respective projections given by these decompositions. Define a map L : F2 → E2 as follows. For f ∈ F2 , by the surjectivity of k∗ , f = k∗ (e0 , e1 ) = k∗ (0, e1 ) for any e1 . Define L(f ) = p(e0 ). (a) L is well defined, i.e. does not depend on the choice of (e0 , e1 ). To see this, suppose f = k∗ (e0 , e1 ). Then k∗ (e0 − e0 , e1 − e1 ) = 0. Thus e0 − e0 ∈ range(π∗ |Σ) and therefore p(e0 − e0 ) = 0 or p(e0 ) = p(e0 ). (b) L is injective. Suppose L(f ) = 0. Then e0 ∈ range(π∗ |Σ) or e0 = π∗ (e0 , e˜1 ), k∗ (e0 , e˜1 ) = 0. Now f = k∗ (e0 , e1 ) = k∗ (e0 , e1 ) − k∗ (e0 , e˜1 ) = k∗ (0, e1 − e˜1 ). But f = q(f ) = q(k∗ (0, e1 − e˜1 )) = 0. This establishes (b). (c) L is surjective. Let e ∈ E2 , e = 0. Define f = qk∗ (e, e1 ) for any e1 ∈ E1 . Claim. f does not depend on the choice of e1 . To see this, suppose qk∗ (e, e1 ) − qk∗ (e, e1 ) = qk∗ (0, e1 − e1 ) = 0. Now we only need show that L(f ) = e. Since k∗ is surjective, e, e1 ) = qf = qk∗ (˜ e, e1 ) f = k∗ (˜ and L(f ) = p(˜ e), by definition. Since again by definition f = qk∗ (e, e1 ) we obtain qk∗ (e − e˜, 0) = 0, k∗ (e − e˜, 0) = k∗ (0, e1 ) for some e1 ∈ E1 or

k∗ (e − e˜, e1 ) = 0.

This says that (e − e˜) ∈ range(π∗ |Σ) or p(e − e˜) = 0 = e − p(˜ e) = e − L(f ). Thus L(f ) = e and surjectivity of L is established.

344


Theorem 3. Suppose as in Theorem 1 that f∗ : E × F → RN is linear, continuous and onto. Let M = ker f∗ , and π∗ : E × F → E again as in Theorem 1. Suppose that there exists a Hilbert space Y and a continuous linear map k∗ : E × F → Y such that k∗ | ker f∗ is onto and that k∗ : ker π∗ → Y is Fredholm of index m. Then π∗ : ker k∗ ∩ ker f∗ → E is Fredholm of index m − N. Proof. By assumption, the mapping k∗ : ker π∗ → Y has index m. Therefore if the rank of f∗ on ker π∗ is μ, the mapping k∗ : ker π∗ ∩ ker f∗ → Y has index m − μ. By Theorem 1 the mapping π∗ : ker f∗ → E has corank N − μ. Define G := range{π∗ : ker f∗ → E}. Then G is closed and has codimension N − μ in E. But k∗ : ker f∗ → Y is onto and π∗ : ker f∗ → G is onto. By Theorem 1, we have that index π∗ : ker f∗ ∩ ker k∗ → G is m − μ. Then index π∗ : ker f∗ ∩ ker k∗ → E is (m − μ) − (N − μ) = m − N . As an immediate corollary we obtain Corollary 2. Suppose now as in Corollary 1 that the derivative Df =: f∗ : T(α,ξ) N → RN is continuous and onto. Assume also that there exists a Banach space Y and a continuous linear mapping k∗ : T N → Y such that k∗ : ker π∗ → Y is onto and Fredholm of index m. Then π∗ : ker k∗ ∩ ker f∗ → Tα A is Fredholm of index m − N . We now remark that in Theorems 1–3 one could replace “Hilbert space” by “Banach space” and the conclusion of the theorems (with slight alterations) would still hold. We are now ready to prove a stratification result which was a main goal of this section. Consider the vector field W on the bundle N described in the beginning of Section 5.5. We know that if we restrict W to the submanifold Nνλ then the Fréchet derivative of W at a minimal surface X ∈ Nνλ which spans a curve α maps TX Nνλ onto a 4|λ| + 3|ν| + 3-codimensional subspace of TX N(α). Since W is a vertical vector field on N we can locally view W as a mapping of a neighbourhood U (X) of X in Nνλ to TX N(α), but DW(X) maps onto a 4|λ| + 3|ν| + 3-codimensional subspace. If DW(X) were onto we could then say that the zero set in Nνλ were a submanifold of this variety. The failure of this surjectivity is a major difficulty in achieving a complete stratification of the zeros of W or the minimal surfaces. Such a stratification, using weighted Sobolev spaces, has been carried out by Thiel [3]. In the next section we show that λ Mλ0 , with each Mλ0 a manifold, provides a stratification of the minimal surfaces without boundary branch points. In what follows we show that locally Mλν ⊂ Wλν , Wλν a manifold, and if π : A × D → A and πνλ := π|Wλν then πνλ is Fredholm, and index πνλ = 2|λ|(2 − n) + |ν|(2 − n) + 2p + q + 3.


345

To this end we set F := TX N(α)

and

F1 := range DW(X)|TX Nνλ

and let k : Nνλ → F1 be defined by k(X) := P W(X) where P is the H s -Hilbert space projection of F onto F1 . The main observation here is that the zeros of W are certainly zeros of k, but not necessarily vice ˜ is onto F1 for versa. Since Dk(X) := k∗ (X) is onto F1 , it follows that Dk(X) ˜ sufficiently close to X. X ˜ ∈ V (X) implies that Dk(X) ˜ Let V (X) be a neighbourhood of X so that X ˜ ∈ V (X)∩Nλ : k(X) ˜ = 0}. Then, as a consequence is surjective. Let Wλν := {X ν of the implicit function theorem and Corollary 2 we have Theorem 4. The set of zeros of k, i.e. Wλν in Nνλ ∩ V (X), is a submanifold of Nνλ . The bundle projection map πνλ = π|Wλν is Fredholm with index πνλ = 2|λ|(2 − n) + |ν|(2 − n) + 2p + q + 3. Proof. Let π : N → A be the bundle projection, as always, and π∗ be its derivative. That the zeros of k are a submanifold follows from the implicit function theorem. Since k∗ (X) = P DW(X) and since DW(X)| ker π∗ has index zero (this is a consequence of the fact that DWα (X) is of the form identity plus compact, which we prove in Chapter 6) it follows that k∗ (X) (or in fact k) has index m = 4|λ| + 3|ν| + 3 on ker π∗ . Since Nνλ has codimension N := 2n|λ| + (n + 1)|ν| − 2p − q in N it arises (at least locally) as the set of zeros of a map f : N → RN . By Corollary 2, πνλ has index m − N = 2|λ|(2 − n) + |ν|(2 − n) + 2p + q + 3. This concludes the proof of Theorem 4. We wish now to show that the image of |ν|>0 Mλν under the map π is a closed, nowhere dense subset of A. We state this as Theorem 5. π( |ν|>0 Mλν ) is a closed, nowhere dense subset of A. Proof. Let X ∈ Nνλ , and let Q1 , Q2 , Q3 be three points in S 1 not equal to any of the ζj . Then as in Böhme [5] we can consider the subbundle S of N defined by S(α) := {X ∈ N(α) : X(Qj ) = α(Qj )}. S := α∈A

α∈A

From our construction in Chapter 6 of Vol. 2 it follows that each S(α) is a submanifold of N(α) with tangent space TX S(α) consisting of those h ∈ TX N(α) such that h(Qj ) = 0 for all j. By the regularity theorem for minimal surfaces it follows that X is a priori bounded in the H¨ older class C r−1,β , 0 < β < 1, and so the orbit OG (X) of the conformal group G of the disk through X is a three dimensional submanifold of

346


N(α). Moreover Tid OG (X), the tangent space to this orbit at the identity map, is spanned by the forced Jacobi fields Re(iwXw ) and Re((iw −i/w)iwXw ) and Re((w + 1/w)iwXw ). Let us state this formally as Lemma 1. Tid OG (X) is spanned by Re(iwXw ), Re((iw − i/w)iwXw ) and Re((w + 1/w)iwXw ).

Proof. See Vol. 2, Section 6.1.

Thus, since elements of Tid OG (X) vanish only at the branch points, and elements of TX S(α) do not vanish at the branch points we see that Tid OG (X) is complementary to TX S(α), i.e. TX S(α) ⊕ Tid OG (X) = TX N(α), and Xw ∈ TX Nνλ ,

Tid OG (X) ⊂ TX Nνλ .

Thus, it follows that TX S(α) + TX Nνλ = TX N(α). This means that, locally, S and Nνλ intersect transversally. Thus, by elementary transversality theory, locally S ∩ Nνλ is a submanifold with TX (S ∩ Nνλ ) = TX S ∩ TX Nνλ . We denote this submanifold by Sλν . It is clear that the codimension of Sλν in N is 2n|λ| + (n + 1)|ν| − 2p − q + 3. The effect of restricting DW to Sλν is to “factor out” the action of the conformal group, an action which does not produce distinct geometric surfaces. We note that, by restricting to TX S, the corank of this restriction is still 4|λ| + 3|ν| + 3 since Tid OG (X) ⊂ ker DW. Now we are again in a position as in Theorem 4 to conclude that the zero ˜ λ of Sλ , and the index of the set of P W|Nνλ = k|Nνλ is a submanifold, say W ν ν λ λ ˜ is now 2|λ|(2−n)+|ν|(2−n)+2p+q. Moreover, restriction of π, say π ˜ν , to W ν every geometric minimal surface (every equivalence class under the action of the conformal group) in a sufficiently small neighbourhood U (X) of X in Nνλ is a zero of k on Sλν . Also, 2(2 − n)|λ| + (2 − n)|ν| + 2p + q ≤ −1

if n ≥ 3 and |ν| > 0

(remember that each νj is even). Thus by Smale’s version of the Sard theorem ˜ λ ) is nowhere dense in A; but for a sufficiently small U (X) the image π ˜νλ (W ν ˜ λ) πνλ (Mλν ∩ U (X)) ⊂ π ˜νλ (W ν and so πνλ (Mλν ∩U (X)) is nowhere dense in A. We can clearly cover |ν|>0 Nνλ by a countable family of such open sets U (Xi ) which implies that π(Mλν ) =

5.8 Generic Finiteness, Stability, and the Stratification of the Sets Mλ0

347

πνλ (Mλν ) is of first category in A. Now choose any three points Q1 , Q2 , Q3 in S 1 and define S ⊂ N as above. Then Mλν = π Mλν ∩ S . π |ν|>0

|ν|>0

Theorem 5 will now follow from Lemma 2. The map π restricted to

|ν|>0

Mλν ∩ S is proper.

Proof. This is a consequence of the bundle topology and the a priori or regularity estimates for classical minimal surfaces. Suppose π(Xn ) → α, Xn = (αn , ξn ), Xn ∈ Mλν ∩ S. |ν|>0

Then αn → α and the regularity results imply due to restriction to S that {ξn } is uniformly bounded in C r−1,β for any 0 < β < 1, and hence has a subsequence which converges in H s to ξ. Thus (α, ξ) ∈ Mλν , |ν| > 0, and the image is closed in A. But a closed set of first category is nowhere dense, which completes the proof both of the lemma and of the theorem. Corollary 3. There exists an open and dense set Aˆ ⊂ A such that for all α ∈ Aˆ there exists no boundary branched minimal surface which spans Γα .

5.8 Generic Finiteness, Stability, and the Stratification of the Sets Mλ0 In this section we show first that the local sets Wλ0 of Section 5.2 are equal to Mλ0 . Then we can use this information to discuss generic finiteness and stability. We begin by discussing the conformality operator. Definition 1. On the bundle N = A × D we define a map k : N → H s−1 (S 1 , R) by (1)

k(α, ξ) =

∂ ∂ (α ◦ ξ) · (α ◦ ξ), ∂r ∂θ

where r and θ are the polar coordinates of the unit disk B, and functions α ◦ ξ on S 1 are identified with their harmonic extensions onto B. If we use the complex differential ∂/∂w, k can be written as ∂ 1 2 ∂ (α ◦ ξ) · (α ◦ ξ) , (1∗) k(α, ξ) = − Im w 2 ∂w ∂w and has partial derivatives

348

(2)


Dα k(α, ξ)[γ] =

∂ ∂ ∂ ∂ (γ ◦ ξ) · (α ◦ ξ) + (α ◦ ξ) · (γ ◦ ξ) ∂r ∂θ ∂r ∂θ

and (3)

Dξ k(α, ξ)[ρ] =

∂ ∂ ∂ ∂ [α (ξ)ρ] · (α ◦ ξ) + (α ◦ ξ) · [α (ξ)ρ]. ∂r ∂θ ∂r ∂θ

It follows from the classical theory of minimal surfaces that k = 0 if and only if α ◦ ξ = X is a minimal surface. We will shortly see that if X ∈ Mλ0 , then the partial derivative in the direction ξ, Dξ k is a Fredholm map of index zero and consequently has a closed, finite codimensional range. (4)

Dξ k(α, ξ)h =

∂X ∂h ∂h ∂X · + · , ∂r ∂θ ∂r ∂θ

where h := α (ξ)ρ.

Theorem 1. h → Dξ k(α, ξ)h is Fredholm of index zero. Proof. First note that (5)

Dξ k(α, ξ)h = −2 Im(w2 G(w) · H(w))

where G(w) = Xw (w),

H(w) = hw (w).

By formula (3) of Section 5.5 we have |Xθ |Ω(X)

∂ DWα (X)h = −2 Im(w2 G(w) · H(w))T1 (X) ∂r

where |Xθ | > 0, and again, as we shall see in Chapter 6, V → Ω(X)

∂V ∂r

is an isomorphism from the H s -vector fields over X to the H s−1 -vector fields V over X. Since λT1 (X) → λ is an isomorphism of the H s−1 -vector fields over X with the H s−1 -real valued functions and DWα (X)h is of the form identity plus compact (Chapter 6) and hence Fredholm of index zero, the result follows. The fact that |Xθ | > 0 is essential in passing from a vector field operator W and its derivative DW to a nonlinear operator k on functions, ξ → k(α, ξ). Now consider k as a function of both α and ξ. Its derivative Dk(α, ξ)[γ, ρ] := k∗ (α, ξ)[γ, ρ] =

∂X ∂h ∂h ∂X · + · ∂r ∂θ ∂r ∂θ

where h := ω∗ (α, ξ)[γ, ρ] = γ · ξ + α (ξ)ρ. We already know (Lemma 1 of Section 5.4) that if ξ is a diffeomorphism then Dω(α, ξ) has a dense range in H s (S 1 , Rn ). As in the previous sections we set


G(w) := Xw (w),

H(w) := hw (w) =

349

∂ {ω∗ (α, ξ)[γ, ρ]}. ∂w

Then as before Dk(α, ξ)[γ, ρ] =: k∗ (α, ξ)[γ, ρ] = −2 Im(w2 G · H). We can now prove: Theorem 2. Let X = α ◦ ξ be a minimal surface with (α, ξ) ∈ N0λ . Then the corank of k∗ |T(α,ξ) N0λ in H s−1 (S 1 , R) is 4|λ| + 3. Proof. Again by formula (3) of Section 5.5 we have |Xθ |Ω(X)

∂DW(X)h = −2 Im(w2 G · H)T1 (X) = k∗ (α, ξ)[γ, ρ]T1 (X). ∂r

Then, using the reasoning in Theorem 4 of Section 5.3 and that V → Ω(X) ∂V ∂r yields an isomorphism, it follows that the corank of Dk|TX N0λ is equal to the corank of DW(X), i.e. 4|λ| + 3. Now for each P ∈ θ0λ , (cf. Definition 2 in Section 5.3) let B2λ (P ) := {h ∈ H s−1 (S 1 , R) : h = Im(w2 P 2 g), g ∈ H s−1/2 (B, C) holomorphic}. 2λ s−1 For each fixed P , the set B (S 1 , R) is a subspace of codimension (P ) ⊂2λH 2λ 4|λ|+3. The space B = P ∈θ0λ B (P ) is clearly a differentiable fibre bundle over θ0λ . We now have:

Theorem 3. The operator k maps N0λ into B2λ . For each minimal surface ˆ = G, P0 ∈ θλ , we have X = α ◦ ξ0 ∈ N0λ , where Xw = P0 G 0 Dk(X) = Dk(α, ξ0 ) maps TX N0λ

onto B2λ (P0 ).

Proof. We have 1 k(X) = k(α, ξ0 ) = − Im(w2 G2 ) ∈ B2λ (P0 ) and 2

B2λ (P0 ) ⊂ B2λ .

For the second part we know by Theorem 2 that the corank of k∗ (X) equals the corank of DW at X and thus is 4|λ| + 3. From the characterization of the tangent space TX N0λ it follows that k∗ (X) : TX N0λ → B2λ (P0 ). Since the codimension of B2λ (P0 ) is also 4|λ| + 3, k∗ (X) must be surjective. ˆ P ∈ θλ , P depends C ∞ -differentiably on Y . For each Y ∈ N0λ , Yw = P G, 0 Therefore there exists a differentiable map ρ : N0λ → θ0λ such that Yw = ˆ Then we obtain a linear isomorphism LP : B2λ (P ) → B2λ (P0 ), ρ(Y ) · G.

350


mapping h = Im(w2 P 2 g) to Im(w2 P02 g), which (because of finite codimension) extends to a linear isomorphism LP : H s−1 → H s−1 . Define the operator k˜ on N0λ by ˜ ) := Lρ(Y ) k(Y ). k(Y Then the zeros of k˜ on N0λ are precisely the zeros of k (and those are minimal surfaces). By construction k˜ : N0λ → B2λ (P0 ), and thus has fixed range. ˆ Theorem 4. Let X = α ◦ ξ0 be a minimal surface in N0λ , with X = P0 G, λ λ λ ˆ P0 ∈ θ0 , and suppose Y ∈ N0 , Yw = P G, P ∈ θ0 , is a minimal surface too. Then k˜∗ (Y ) : TY N0λ → B2λ (P0 ) is surjective and hence k˜∗ (Y ) : TY N0λ → H s−1 (S 1 , R) has corank 4|λ| + 3. Proof. At Y , we have k˜∗ (Y ) = Lρ(Y ) ◦ k∗ (Y ), k∗ (Y ) maps onto B2λ (P ) and Lρ(Y ) is an isomorphism of B2λ (P ) onto B2λ (P0 ). Theorem 5. The set Mλ0 is a C r−s -submanifold of N, and the restriction map π0λ of the bundle projection map π to Mλ0 has index 2(2 − n)|λ| + 2p + 3. Proof. The proof is essentially the same as that of Theorem 4 in Section 5.7. We apply Corollary 2 of Section 5.7 with k˜ above replacing the k of the ˆ Let F1 = corollary. Let X = α(ξ0 ) be a minimal surface with Xw = P0 G. 2λ s−1 λ , and f be a mapping on N defining N0 . Since k∗ | ker π∗ B (P0 ), F = H is Fredholm of index zero, the mapping k∗ | ker π∗ : ker π∗ → F1 has index 4|λ| + 3. By Theorem 3, if Y = β ◦ ξ ∈ N0λ , Y minimal, k∗ (β, ξ) maps ker f∗ (X) = TX N0λ onto F1 . Thus again by Corollary 2 of Section 5.7 and by codim (N0λ , N) = 2n|λ| − 2p the index of the mapping π∗ (X)| ker kˆ∗ (X) ∩ TX N0λ is just the difference 4|λ| + 3 − {2n|λ| − 2p} = 2(2 − n)|λ| + 2p + 3. Therefore if π0λ denotes the restriction of π to Mλ0 the index of the mapping π0λ is 2(2 − n)|λ| + 2p + 3. Corollary 1. The sets Mλ0 and Wλ0 coincide. Proof. Both are submanifolds of N and by definition Mλ0 ⊂ Wλ0 . Therefore Mλ0 is a submanifold of Wλ0 . But the indices of the restrictions of the projection map π to both Mλ0 and Wλ0 coincide and hence the manifolds must coincide. Theorem 6. If n > 3, π( |λ|+|ν|>0 Mλν ) is a closed, nowhere dense subset of A. Proof. By Theorem 5 of Section 5.7 we know that π( |ν|>0 Mλ ) is a closed, nowhere dense subset of A. As in this theorem define


351

S(α) := {X ∈ N(α) : X(Qj ) = α(Qj )} where Q1 , Q2 , Q3 are three prescribed points on S 1 . The codimension 3subbundle S = α∈A S(α) intersects the manifolds Mλ0 transversally and so S ∩ Mλ0 is a submanifold of N and π(Mλ0 ) = π(Mλ0 ∩ S). For n > 3 and π ˆ0λ has index 2(2 − n)|λ| for |λ| > 0, π|S ∩ Mλ0 := + 2p < 0. From the Sard theorem it follows that π( |λ|>0 S ∩ Mλ0 ) = π( |λ|>0 Mλ0 ) is of first category in A. Let A1 := π( |ν|>0 Mλν ) and := |λ|>0 (S ∩ Mλ0 ) \ π −1 (A1 ). Then π : → A \ A1 is a proper map (as in Theorem 5 of Section 5.7), and so its image is closed in A \ A1 . Thus π( ) is nowhere dense, and so (Mλν ∩ S) = π Mλν π |λ|+|ν|>0

is nowhere dense.

|λ|+|ν|>0

Theorem 7. If n ≥ 3, π( |λ|+|ν|>p Mλν ) is a closed, nowhere dense subset of A. (Recall p is the number of interior branch points.) Proof. If n > 3, this follows from Theorem 5 of Section 5.7. In case n = 3 the set |λ|+|ν|>p Mλν is the set of minimal surfaces with non-simple branch points in the interior (and perhaps branch points on the boundary). The proof is similar to that of Theorem 5 of Section 5.7 if we note that on each Mλ0 , |λ| > p, the index of the projection π restricted to S ∩ Mλ0 is negative. Theorem 8. For an open, dense set Aˆ in A the set of minimal surfaces bounded by α ∈ Aˆ is a finite “non-degenerate” set. These minimal surfaces will be stable under perturbations of α. If n > 3, the minimal surfaces spanning α ∈ Aˆ will be immersed up to the boundary. If n = 3, they will have at most simple branch points in the interior. Proof. Case 1 (n > 3): In this case A2 := π( |λ|+|ν|>0 Mλν ) is a closed, nowhere dense set. Let 2 := (S ∩ M00 ) \ π −1 (A2 ). Then π : 2 → A \ A2 is a proper Fredholm map of index zero. Smale’s Sard theorem states that an open and dense set Aˆ in A \ A2 consists of regular values for π. The set Aˆ is ˆ π −1 (α) ∩ S ∩ M00 is a finite clearly open and dense in A, and for each α ∈ A, −1 set. Moreover each X ∈ π (α) ∩ S ∩ M00 is non-degenerate in the sense that π ˆ00 := π00 |S ∩ M00 is a local diffeomorphism of a neighbourhood of X onto a neighbourhood of α. If α ∈ Aˆ is fixed and X1 , . . . , XN denote the minimal surfaces in S ∩ Mλν spanning α, then for β close to α there are N minimal surfaces X1 (β), . . . , XN (β) spanning β and depending smoothly on β. Case 2 (n = 3): In this case A3 := π( |ν|+|λ|>p Mλν ) is a closed, nowhere dense subset of A. Let 3 := |λ|=p (Mλ0 ∩ S) \ π −1 (A3 ). Then π : 3 → A \ A3 is proper Fredholm of index zero. Applying the Sard theorem as in case 1 finishes the proof.

352


In R3 there exist many stable minimal surfaces with simple branch points in the interior. This does not follow immediately from Theorem 5 since it is a priori possible that π( |λ|=p,p=0 Mλ0 ) is nowhere dense in A. This is ruled out formally by an observation of Sch¨ uffler, who examined the index construction and noted that a simply branched minimal surface X ∈ Mλ0 , λ = (1, . . . , 1) is nondegenerate in the sense that π0λ is a local diffeomorphism about X onto a neighbourhood of α = π(X) if and only if (a) the only elements in the kernel of the Hessian of Dirichlet’s integral and equivalently the only elements in the kernel of DWα : TX N(α) → TX N(α) are the forced Jacobi fields, and (b) that, apart from those that arise from the action of the conformal group (Re iwXw , Re{iwXw (w+ w1 )}, Re{iwXw ( wi −iw)}), none of other Jacobi fields are in TX N0λ . This immediately implies from the index construction that π0λ is a local diffeomorphism. Sch¨ uffler observed that one can give a quite easy description of precisely when the Jacobi fields in the case of simple branch points are transverse to N0λ . To this end, consider the only Jacobi fields that, in the simple branch point case, do not arise from the action of the conformal group. If we assume (for simplicity) that the branch point is the origin, we have the Jacobi fields Re{iwXw ( w12 + w2 )} and Re{iwXw ( wi2 − iw2 )}. Let H1 (w) be the harmonic extension of Re{iwXw ( w12 + w2 )} and H2 (w) the harmonic extension of Re{iwXw ( wi2 − iw2 )}. Then

∂ iXw ∂ Xw ∂H1 ∂H2 (0) = (0) = − , . ∂w ∂w w w=0 ∂w ∂w w w=0 We are assuming n = 3. By a linear change of coordinates in R3 we may write (1)

Xw (w) = (A1 w + A2 w2 + · · · , R2 w2 + R3 w3 + · · · )

where Aj ∈ C2 and Rj ∈ C. Using this form for Xw , we obtain ∂H1 ∂H2 (0) = (iA2 , iR2 ), (0) = (−A2 , −R2 ). ∂w ∂w By Theorems 5 of Section 5.3 and 6 of Section 5.4, H1 and H2 will be tangent to N0λ at X ∈ Mλ0 , λ = (1, . . . , 1), if and only if ∂Hj (0) = βXww (c) = β · (A1 , 0), ∂w 0, this is impossible. Thus, we arrive at β ∈ C. If R2 = Theorem 9 (Sch¨ uffler). A simply branched minimal surface X : B → R3 bounded by a smooth curve Γα will be non-degenerate (and therefore isolated) if the only kernel elements of the Hessian D2 Eα (X) are the forced Jacobi fields and, if one moves (by use of the conformal group) each branch point to the origin and employs (1), then R2 = 0. Or, without using the conformal group to move branch points to the origin: X is non-degenerate if and only if (a) the forced Jacobi fields are the only elements in the kernel of the Hessian and (b) for each branch point wk , Xw (w) = (w − wk )G(w), G(wk ) = 0, then G(wk ) and G (wk ) are linearly independent over C.

5.9 The Index Theorem for Higher Genus Minimal Surfaces Statement

353

This Theorem will be of crucial importance in proving a local normalform theorem for Dirichlet’s integral about a nondegenerate minimal surface in R3 , and it also shows that there are many stable, simply branched minimal surfaces in R3 . Using Weierstrass’s representation formula it is easy to produce simply branched minimal surfaces X in R3 which also satisfy the hypotheses of Theorem 9. ˜ such that Now, by the index theorem there is a perturbation of X, say X, 1 λ ˜ ˜ is a ˜ α := X|S is a regular value for π0 , X ∈ Mλ0 , λ = (1, . . . , 1) and thus X 3 nondegenerate, simply branched minimal surface in R . This concludes the discussion of the first part of this chapter. We now move on to the index theorem for higher genus surfaces.

5.9 The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries Let M be a smooth, compact, oriented surface with boundary ∂M ∼ = S1, genus M > 0, and let us denote by A the set of all embeddings of S 1 into Rn of Sobolev class H r , r 1 and by M the set of all branched minimal immersions X : M → Rn such that X|∂M ∈ A and X : ∂M → X(∂M ) is an orientation preserving homeomorphism. Consider the map Π : M → A which assigns to each X ∈ M its boundary curve in A. For k, l ∈ N and μ = (μ1 , . . . , μk ) ∈ Nk , ν = (ν1 , . . . , νl ) ∈ Nl we denote by Mμν the set of all branched minimal surfaces X ∈ M which have exactly k interior branch points of orders μ1 , . . . , μk and l boundary branch points of orders ν1 , . . . , νl . Let us observe that the νj are even numbers since X|∂M is injective. As the main result of this chapter we prove that (i) Mμ0 is a differentiable manifold and Π|Mμ0 is a differentiable Fredholm k μj ) + 2k. map of Fredholm index 2(2 − n)(Σj=1 ˜ , and (ii) For ν = 0, Mμν locally is contained in differentiable manifolds W k l μν Π|Wμν is Fredholm of index 2(2−n)( j=1 μj )+2k+(2−n)( j=1 νj )+l. It follows that there is a subset A1 of A of first category such that all branched minimal surfaces spanning a curve α ∈ A\A1 are immersed up to the boundary for n ≥ 4 and, if n = 3, have at most simple interior branch points. Moreover, Π is a local diffeomorphism near any surface X with Π(X) ∈ / A1 . As a consequence, for α ∈ A\A1 , all minimal surfaces spanning α are isolated. We will be using μ and Π in place of λ and π to distinguish the genus zero case from the higher genus case.

354


5.10 Review of Some Basic Results in Riemann Surface Theory In this section, for the convenience of the reader, we review the basic results on the existence of harmonic differentials on a Riemann surface, the Riemann period relations and the theorem of Riemann–Roch. Let ω be a smooth closed 1-form or differential on a Riemann surface M . Locally we have ω = df where f is a smooth function defined on a simply connected neighbourhood of some point of M . We say that ω is harmonic if f is a harmonic function on this neighbourhood. If ω = A du + B dv in a local complex coordinate system we define formally the conjugate ∗ω = −B du + A dv. If ω is harmonic then ω + i ∗ ω = (A − iB)(du + i dv) where locally A − iB is holomorphic. Thus ω + i ∗ ω is a holomorphic differential on M . Similarly one can define the notion of a meromorphic differential ϕ(w) dw. If ϕ(w) dw is a meromorphic differential then ϕ(w)1 dw is said to be a negative meromorphic differential. Equivalently, we could speak of meromorphic vector fields. Suppose that (u, v) is a coordinate system around a boundary point p ∈ ∂M such that locally ∂M corresponds to the u-axis. A meromorphic differential ϕ(w) dw is said to be real on ∂M if ϕ(w) dw = (A − iB)(du + i dv) where B ≡ 0 on the u-axis for all such coordinate systems. Let ι : ∂M → M be the inclusion map. Then we may pull back any differential ω on M to a differential ι∗ ω on ∂M . We say that ω is zero on ∂M , or ω vanishes on ∂M , if ι∗ ω ≡ 0 on ∂M . We shall henceforth assume that ∂M has only one component. We may then consider M in terms of its polygonal model M ∗ with peripheral sequence of 4· (genus M ) sides ±Qj , ±Qj , 1 ≤ j ≤ genus M , which are identified in pairs subject to their orientations, indicated by the + and −-signs. It is well known (cf. Massey [1], Cohn [1]) that the “cross cuts” Qj , Qj generate the homology and the fundamental group of M . The integral Qj ω of a harmonic differential ω over Qj is said to be the period of ω about Qj and is denoted by the brackets (Qj , ω). We now have the two basic existence theorems for harmonic differentials. Theorem 1. Given arbitrary constants cj , cj ∈ R, j = 1, . . . , genus M , there exists exactly one harmonic differential ω on M such that ∗ω = 0 on ∂M and

5.10 Review of Some Basic Results in Riemann Surface Theory

(Qj , ω) = cj ,

355

(Qj , ω) = cj

for j = 1, . . . , genus M . Let us remark that, as for any closed 1-form, the period of a harmonic differential ω on ∂M is zero. Therefore, ω|∂M = dϕ for some C ∞ -function ϕ : ∂M → R. We may then formulate the second existence theorem which corresponds to Dirichlet data, as opposed to Neumann data in Theorem 1. Theorem 2. Let the constants cj , cj ∈ R and, moreover, a smooth function ϕ : ∂M → R be given. Then there exists exactly one harmonic differential ω satisfying the conditions ι∗ ω = dϕ on ∂M and (Qj , ω) = cj ,

(Qj , ω) = cj ,

j = 1, . . . , genus M,

where ι : ∂M → M is the inclusion. Proofs of the Theorems 1 and 2 can be found in H. Cohn [1], as well as the proof of Theorem 3 (Riemann’s period relations). Let dF and dG be closed differentials on M . Since ∂M dF and ∂M dG are both zero we may define either F or G as single valued functions on M ∗ , the polygonal model of M . Then

G dF =

(1)

genus M

∂M ∗

j=1

det

(Qj , dG) (Qj , dG)

(Qj , dF ) (Qj , dF )

+

G dF. ∂M

Applying Stokes’s theorem to the left hand side of (1) we obtain Corollary 1.

G dF = ∂M

M∗

dG ∧ dF +

genus M j=1

det

(Qj , dF ) (Qj , dF )

(Qj , dG) (Qj , dG)

.

We now briefly give the background material that enables one to state the Riemann–Roch theorem and to compute the dimension of the space of holomorphic quadratic differentials on a Riemann surface. Let (M, c) be a surface of genus greater than one with an associated complex structure c. To start we shall assume that ∂M = ∅. For ease of exposition we shall further suppress the c from the notation (M, c). Recall that a 1-form ω is a holomorphic differential if “locally” ω can be written as ω = f (w) dw, dw = du + i dv, where f is a holomorphic function. It is well known (see H. Cohn [1]) that the complex dimension of the vector space of holomorphic differentials is equal to the genus of M . One also has the obvious notion of a meromorphic differential by requiring f to be meromorphic. A complex meromorphic 1-form on M is called an Abelian differential.

356


It is easy to check that the order of a zero or a pole of either a meromorphic function or an Abelian differential is well defined. We are interested in specifying to some extent the location and orders of poles of both meromorphic functions and Abelian differentials on M . Let P1 , P2 , . . . , Pn be points on M and α1 , α2 , . . . , αn be integers. The symbol a = P1α1 P2α2 · · · Pnαn is called a divisor. The integer αk is called the order of a at Pk . By the degree d[a] of a divisor a we mean the sum d[a] := Σαk . If f is a meromorphic function, not identically zero on M we define the divisor (f ) of f by 1 · · · Q−β (f ) = P1α1 · · · Pkαk Q−β 1 where the zeros of f are P1 · · · Pk with orders α1 · · · αk , all αi ≥ 0, and the poles of f are Q1 , . . . , Q with orders β1 , . . . , β , all βi ≥ 0. The divisor of an Abelian differential is defined in the same way. Since for a meromorphic function the sum of the orders of its zeros is equal to the sum of the orders of its poles, it follows that d[f ] = 0 for any f . The following is a basic result in Riemann surface theory (cf. Cohn [1]). Theorem 4. If ω is an Abelian differential then d[ω] = 2·(genus M )−2. Note that d[ω] > 0 in our case where genus M > 1. Consequently every Abelian differential must have a zero. A divisor a = P1α1 · · · Pkαk is called integral if αj ≥ 0 for all j. If b = Qβ1 1 · · · Qβ then the quotient divisor a/b is defined by 1 · · · Q−β . a/b = P1α1 · · · Pkαk Q−β 1

By a1 we mean the divisor P1−α1 · · · Pk−αk . If a/b is integral we say that b divides a or that a is a multiple of b. Define by L(a) the vector space of meromorphic functions on M whose divisors are an integral multiple of a, and by Ω(a) the vector space of Abelian differentials whose divisors are integral multiples of a. A beautiful relationship between the dimensions of these vector spaces over C is given by Theorem 5 (Riemann–Roch). 1 = dim Ω(a) + d[a] − (genus M ) + 1. dim L a For a proof the reader may consult a text on Riemann surfaces; see for example Ahlfors and Sario [1], Cohn [1], Forster [1]. In addition to permitting us to speak of meromorphic functions and Abelian differentials, a complex structure on M allows us to speak about meromorphic quadratic differentials.

5.10 Review of Some Basic Results in Riemann Surface Theory

357

A complex-valued quadratic differential on M is a complex-valued symmetric (0, 2)-tensor Q. Thus, for each q ∈ M , Q(q) : Tq M × Tq M → C is bilinear and symmetric. Locally Q can be expressed as ˜ du2 + F˜ du dv + G ˜ dv 2 ). Q = E du2 + F du dv + G dv 2 + i(E Q is said to be holomorphic if it can be expressed in a local coordinate system as Q(w) = ϕ(w) dw2 , with ϕ being holomorphic. Let QD(M ) denote the complex linear space of holomorphic quadratic differentials on M . The following theorem on the dimension of QD(M ) is the principal result we will need from elementary Riemann surface theory. Theorem 6. dimC QD(M ) = 3 · genus M − 3. Proof. Let ω0 (w) dw2 be a holomorphic quadratic differential, say, for example the square of a holomorphic differential. Then it follows from Theorem 4 that if a0 is the divisor of ω0 then d[a0 ] = 4 · genus M − 4. If ω(w) dw2 is any other holomorphic quadratic differential then ω(w) dw2 /ω0 (w) dw2 is a is the meromorphic function. If a denotes the divisor of ω and since a a−1 0 divisor of a meromorphic function, it follows that d[a] = d[a0 ] = 4·genus M −4. For arbitrary ω let fω (w) = ω(w) dw2 /ω0 (w) dw2 . Then ω = fω · ω0 where (fω ) is an integral multiple of a−1 0 . It therefore follows that the elements of QD(M ) are in one-to-one correspondence with L(a−1 0 ). By the Riemann–Roch theorem dim L(a−1 0 ) = dim Ω(a0 ) + d[a0 ] − (genus M ) + 1. If τ = ϕ(w)DW ∈ Ω(a0 ) is non-zero, d[τ ] ≥ d[a0 ] = 4 · genus M − 4. But, on the other hand, d[τ ] = 2 · genus M − 2 which is impossible since genus M > 1. Thus dim Ω(a0 ) = 0. Hence dimC QD(M ) = dim L(a−1 0 ) = 4 · genus M − 4 − genus M + 1 = 3 · genus M − 3.

We now consider the case when ∂M = ∅. What we are about to explain also applies to the situation where ∂M has more than one component. By an Abelian differential on M we mean a holomorphic or meromorphic differential which is real on ∂M . Such a differential can be reflected across ∂M to an Abelian differential on the double 2M . Let P1 , P2 , . . . , Pn be points on M , let Pˆ1 , . . . , Pˆn their conjugates on the double 2M , and ξ1 , ξ2 , . . . , ξm be points on ∂M . Furthermore let α1 , . . . , αn , as well as β1 , . . . , βm be integers. We consider the divisor on M

358

5 The Index Theorems for Minimal Surfaces of Zero and Higher Genus βm a = P1α1 P2α2 · · · Pnαn ξ1β1 · · · ξm

and the corresponding divisor βm 2a = P1α1 P2α2 · · · Pnαn Pˆ1α1 · · · Pˆnαn ξ1β1 ξ2β2 · · · ξm

on 2M . The degree of 2a is d[2a] = 2Σj αj + Σk βk . The Abelian differentials on M can be identified with the symmetric Abelian differentials on 2M . If S : 2M → 2M denotes the natural antiholomorphic involution then ω is symmetric on 2M if ω = S ∗ ω. Similarly the meromorphic functions on M that are real on ∂M can be identified with the symmetric meromorphic functions on 2M . Define by L(a) the vector space of meromorphic functions on M that are real on ∂M and whose divisors are integral multiples of a and by Ω(a) the vector space of Abelian differentials on M whose divisors are integral multiples of a. Then the symmetric version of the Riemann–Roch theorem gives a relation between the dimensions of these vector spaces which can easily be deduced from Theorem 5 by splitting the spaces L(1/2a) and Ω(2a) into their symmetric and antisymmetric parts. Theorem 7 (Symmetric Riemann–Roch Theorem). 1 = dim Ω(a) + d[2a] − genus(2M ) + 1. dimR L a Now let QR D(M ) denote the real linear space of holomorphic quadratic differentials on M which are real on ∂M . As an immediate corollary of Theorem 7 we have Theorem 8. If ∂M has only one component then dimR QR D(M ) = 6 · genus M − 3. The Riemann–Roch theorem is in some sense an outgrowth of the fact that on a compact Riemann surface M , ∂M = ∅, one cannot arbitrarily assign the zeros and poles of either a meromorphic function or differential. For example a meromorphic function with one zero and no poles must be identically zero. This is not the case for open Riemann surfaces. For the purpose of this chapter we shall define an open Riemann surface as one which is the interior of a Riemann surface with nonempty boundary. We then have Theorem 9. On an open Riemann surface one can arbitrarily prescribe the zeros and poles along with their orders of a meromorphic function or differential. For a proof see Ahlfors and Sario [1] or Forster [1].

5.11 Vector Bundles over Teichm¨ uller Space

359

5.11 Vector Bundles over Teichm¨ uller Space In this section we study holomorphic and harmonic functions and forms on an open Riemann surface with varying conformal structure, and we shall show how, in a natural way, these objects can be given a smooth vector bundle structure over Teichm¨ uller space. Let us start with a technical lemma on differentiable Hilbert space bundles π : Y → B and π : X → B over a base manifold B with fibres Xτ , Yτ (τ ∈ B). We define a bundle map ϕ : X → Y as a differentiable, fibre-preserving map such that on each fibre the mapping ϕτ : Xτ → Yτ is linear and, additionally, in a local trivialization around any τ0 ∈ B, the family (ϕτ ) depends continuously on τ with respect to the norm topology of the space of linear operators from Xτ0 to Yτ0 . Let us remark that the norm continuity of ϕτ follows automatically from the other conditions if the fibres of X are finite dimensional or if the bundles are holomorphic, cf. Kato [1]. Lemma 1. (i) Let P : X → Y be a bundle map such that on each fibre Pτ : Xτ → Yτ is a projection. Then τ range Pτ is a differentiable subbundle of X. (ii) Let C : X → Y be a bundle map such that on each fibre Cτ : Xτ → Yτ is injective with closed range. Then τ range Cτ is a differentiable subbundle of Y. (iii) Let D : X ¸ → Y be a bundle map such that on each fibre Dτ : Xτ → Yτ is linear and surjective. Then τ ker Dτ is a differentiable subbundle of X. Proof. In all cases we may clearly assume that X = B × X0 , Y = B × Y0 are trivial bundles. (i) Let τ ∈ B be a fixed point and set Qτ = Id −Pτ ,

Sτ := Pτ Pτ0 + Qτ Qτ0 .

Clearly, Sτ0 = id and hence, by the norm continuity of Pτ and Qτ , the map Sτ is a bijection of X0 for τ in some neighbourhood of τ0 . Thus Sτ defines a local bundle diffeomorphism S of X. It suffices therefore to show that range Pτ = Sτ (range Pτ0 ). − + − Writing X0 = X+ 0 ⊕ X0 with X0 := range Pτ0 , X0 := range Qτ0 we have + Sτ (X+ 0 ) = Pτ (X0 ) ⊂ range Pτ , − Sτ (X− 0 ) = Qτ (X0 ) ⊂ range Qτ . − Since, however, Sτ is a bijection we must have Pτ (X+ 0 ) ⊕ Qτ (X0 ) = X0 . Hence + − it follows that Pτ (X0 ) = range Pτ and Qτ (X0 ) = range Qτ , proving (i). − + − (ii) Let Y0 = Y+ 0 ⊕Y0 with Y0 := range Cτ0 and Y0 a closed linear subspace − of Y0 , and define a linear map Sτ : X0 ×Y0 → Y0 by Sτ (x0 , y0− ) := Cτ (x0 )+y0− . Clearly, Sτ0 is a bijection from X0 × Y− 0 onto Y0 . Hence Sτ defines a bundle

360


diffeomorphism S : U × (X0 × Y− 0 ) → U × Y0 for some neighbourhood U of τ0 . Obviously we have Sτ (X0 × {0}) = Cτ (X0 ), proving (ii), since B × X0 is a differentiable subbundle of B × (X0 × Y− 0 ). − + (iii) For fixed τ0 ∈ B choose a splitting X0 = X+ 0 ⊕ X0 with X0 := ker Dτ0 − and X0 a closed linear subspace of X0 and define a linear map Sτ : X0 → − + − + + Y0 × X− 0 by Sτ (x0 ) := (Dτ (x0 ), x0 ) where x0 = x0 + x0 with x0 ∈ X0 and − − − −1 x0 ∈ X0 . Clearly, ker Dτ = Sτ ({0}×X0 ). However Sτ defines a local bundle diffeomorphism S : B × X0 → B × (Y0 × X− 0 ) around τ0 ∈ B. The foregoing simple lemma enables us to deduce the smooth vector bundle structure of various holomorphic objects on a compact Riemann surface M with boundary over Teichm¨ uller space as base space. m For nonnegative integers p, q and a number m ≥ 1 we denote by Tp,q the linear space of complex valued (p, q)-tensor fields on M which are of Sobolev class H m . This space can be given the structure of a Hilbert space. For each τ ∈ T(M ), the Teichm¨ uller space of M , we introduce the space m Em : f holomorphic on (M, τ )}, τ := {f ∈ H

m Dτm := {ω ∈ T1,0 : ω = f (w) dw for each local complex coordinate w

on (M, τ )},

m Dτ m := {ω ∈ T1,0 : ω = f (w) dw for each local complex coordinate w

on (M, τ )},

m Dm τ := {ω ∈ Dτ : ω holomorphic on (M, τ )}.

Finally let QR Dτ denote the R-linear space of holomorphic quadratic difm for every m, in fact, any δ ∈ QR Dτ ferentials real on ∂M . Clearly QR Dτ ⊂ T2,0 extends analytically on the double 2(M, τ ) of (M, τ ). As follows readily from Theorem 8 of Section 5.10 one has dimR QR Dτ = 3 · genus 2M − 3 = −3χ(M ). The basic result is Theorem 1. (i) τ Dτm , τ Dm τ are smooth subbundles of the product bundle m . T(M ) × T1,0 (ii) τ Em is a smooth subbundle of T(M ) × H m . τ m (iii) τ QR Dτ is a smooth subbundle of T(M ) × T2,0 for all m. Proof. Remember that T(M ) is diffeomorphic to the image Σ of a section of the bundle MS−1 → MS−1 /DS0 and that to each g ∈ MS−1 there exists a unique ∞ . The dependence g → Jg is smooth with almost complex structure Jg ∈ T1,1 m respect to any H -norm. In fact, topology does not matter here since Σ is finite-dimensional. We may therefore think of T(M ) as a finite-dimensional submanifold of the space of almost complex structures of class H , ≥ 2, . Let us note that the traditional ∗ operator (or which itself is a subset of T1,1 Hodge dual) depends on the complex structure τ . Therefore, to be precise,


361

one should write ∗(τ ) to denote this dependence. However, for convenience we m we consider mappings ∗, PJ , PJ defined by shall simply write ∗. In T1,0 ∗ω(w) = ω(w) ◦ J(w) 1 (ω + i ∗ ω), 2

PJ ω =

(w ∈ M ), PJ ω =

1 (ω − i ∗ ω). 2

It can easily be seen that these maps define bundle maps of the trivial bundle m and that PJ and PJ are complementary projections with T(M ) × T1,0

m PJ (T1,0 ) = Dτm ,

m PJ (T1,0 ) = Dτ m .

The assertion about τ Dτm and τ Dτ m therefore follows directly from Lemma 1, (i). Next we introduce the differential operators dJ , dJ acting on functions by dJ f =

1 (df + i ∗ df ), 2

1 (df − i ∗ df ). 2

dJ f =

Clearly dJ and dJ are nothing but ∂ = ∂/∂w and ∂ = ∂/∂w for any complex coordinate w of the complex structure associated with J. The operators dJ and dJ act on differentials ω = f du + g dv by means of dJ ω = dJ f ∧ du + dJ g ∧ dv, and similarly for dJ . It is well known that on an open Riemann surface

dJ : H m → Dτ m−1 ,

a,m−1 dJ : Dτm → T2,0

a,m are surjective, where T2,0 are the alternating 2-forms of Sobolev class H m (see Forster [1] where it is proved within C ∞ , but with elliptic theory of partial differential equations the proof can easily be extended to H m ). Moreover, one easily checks that the family of operators (dJ ) defines bundle maps Dτ m−1 , d : T(M ) × H m →

d :

τ

Dτm−1

a,m−1 → T(M ) × T2,0 ,

τ

and that

m Em τ = ker(dJ |H ),

m Dm τ = ker(dJ |Dτ ).

In view of Lemma 1, (ii), this proves (ii) and the remainder of (i). For the proof of the last part of the theorem, which is a bit more involved, we find it convenient to identify T(M ) with Σ ⊂ MS−1 ⊂ MS,m −1 , where S,m m M−1 ⊂ T2,0 denotes the set of S-symmetric Riemannian metrics on 2M of curvature −1 and of Sobolev class H m . Let us first recall the L2 (g)-orthogonal decomposition

362

5 The Index Theorems for Minimal Surfaces of Zero and Higher Genus S,m+1 Tg MS,m ) −1 = Re(QR Dg ) ⊕ αg (T0,1

S,m are the H m -vector fields which are tangent along ∂M and exwhere T0,1 tend as S-symmetric H m -vector fields V on 2M , and αg is the Lie-derivative operator

αg (V ) = LV g. Furthermore Re(QR Dg ) is the space of all real parts of elements in QR Dg . Since m T MS,m is a subbundle of MS,m −1 × T2,0 by definition it suffices to show that −1 S,m+1 S,m m ) is a subbundle of M−1 ×T2,0 . We want to apply Lemma 1, (ii). g αg (T0,1 S,m+1 In fact, αg (T0,1 ) is a closed subspace of Tg MS,m −1 since QR Dg is finite dimenS,m+1 sional, and αg is also injective because any Killing vector field V ∈ T0,1 (i.e. LV g = 0) would extend as a vector field on 2M , being holomorphic with respect to the complex structure induced by g. Since 2M has negative Euler characteristic it follows that V = 0, proving the smooth vector bundle structure of Re(QR Dg ). From this the statement (iii) readily follows since there is a vector bundle isomorphism between Re(QR Dg ) and QR Dg which, in terms of the almost complex structure J associated with g, is given by Re(QR Dg ) β → β − iβ˜ ∈ QR Dg where ˜ W ) = β(JV, W ). β(V,

In our approach to the index theorem it is necessary to use multivalued harmonic maps. By definition, a multivalued harmonic function is a pair consisting of a harmonic differential and a constant (the integration constant). By Gauss’s theorem we have

ω=0 ∂M

for any closed differential ω, and since we consider only surfaces with one boundary component, it follows that any multivalued harmonic function has single-valued boundary values. In fact, a multivalued harmonic function must be single-valued in a tubular neighbourhood of ∂M . For m ∈ 12 N let Hm (τ ) denote the linear space of Rn -valued harmonic ˆ m (τ ) we denote functions on (M, τ ) of Sobolev class H m , τ ∈ T(M ). By H the corresponding space of multivalued harmonic functions which is of course identified with the direct sum of Rn and the space of τ -harmonic differentials of class H m−1 . By a reasoning completely analogous to that used in the proof ˆ m (τ ) are smooth subbundles of of Theorem 1 one can show that Hm (τ ) and H m−1 m ⊕ Rn ). Let now r, s ∈ N, the product bundles T(M ) × H and T(M ) × (T1,0 r > s, say, r > s(s + 3). Then we define


363

Ar := {α : ∂M → Rn : α is an embedding of class H r }, Hαs := the component of α ∈ Ar in H s (∂M ; α(∂M )), 1

N(α)(τ ) := {X ∈ Hs+ 2 (τ ) : X|∂M ∈ Hαs }, ˆ ˆ s+ 12 (τ ) : X|∂M ∈ H s }. N(α)(τ ) := {X ∈ H α Remark 1. Different from Chapter 4 we shall not write X : (M, τ ) → Rn for mappings from (M, τ ) into Rn , but simply X : (M, τ ) → Rn , and we do not distinguish between X and its local representation X = X ◦ ϕ−1 by a coordinate chart (G, ϕ), because in the past section we wrote f (w) for the “local representation” of any mapping f of M in terms of local coordinates w = u + iv on M . ˆ ) can be described as the set of Remark 2. The tangent space TX N(α)(τ ˆ s+ 12 (τ ) such that for u ∈ ∂M , h(u) ∈ TX(u) α(∂M ). Note that maps h ∈ H ˆ ). TX N(α)(τ ) is the set of single-valued elements of TX N(τ We now prove the final result of this section ˆ Theorem 2. For fixed α ∈ Ar the sets τ N(α)(τ ) and τ N(α)(τ ) are trivial 1 fibre bundles. In fact, they are (r − s − 3)-subbundles of τ (Hs+ 2 (τ ))n and ˆ s+ 1 2 (τ ))n , respectively. τ (H ˆ ) since the assertion on Proof. We only prove the statement for τ N(α)(τ N(α)(τ ) can be obtained by specializing the argument. Let us denote by τ ˆ m is the direct ˆ m the multivalued functions of Sobolev class H m on M , i.e. H H m−1 . We shall write sum of R and the closed differentials of Sobolev class H ˆ m in the form (dX, c) with dX ∈ T m−1 and c ∈ Rn . It follows each X ∈ H 1,0 from Theorem 1 of Section 5.10 and the standard theory of elliptic operators that the map

dX, dX (1) (τ, X) → τ, d ∗τ dX, X|∂M, Qj

Qj

1

α,s− 12

ˆ s+ 2 and T(M )×T is a diffeomorphism of the product bundles T(M )×H 2,0 H s (∂M ) × R2·genus M .

×

ˆ s+ 12 such that X|∂M = Denote by Λτ (η,p) the unique element X ∈ H 2·genus M . Since Λ is a restriction η, d∗τ dX = 0, and ( Qj , dX, Q dX) = p ∈ R j

of the inverse of (1) we see that Λ is a smooth embedding, and clearly 1

ˆ s+ 2 (τ ) Λτ (H s (∂M ) × R2·genus M ) = H ˆ s+ 1 2 (τ ). Since H s (∂M, α(∂M )) and i.e. Λ is a trivialization of the bundle τ H s r−s s -submanifolds of H (∂M )n , the theorem follows from the hence Hα are C fact that ˆ N(α)(τ ) = Λ(T(M ) × Hαs × (R2·genus M )n ). τ

364


5.12 Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions In this section we want to prove a result on algebras of holomorphic functions which will be used in an essential way at several places. This result is an adaption of Kap. V, §7 in Grauert and Remmert [1] to the context of Sobolev spaces. In the following we fix some complex structure on 2M . Lemma 1. Let E(M ) denote the space of functions that are holomorphic in an open neighbourhood of M . Then any algebra homomorphism σ : E(M ) → C is a Dirac measure supported by some point p ∈ M , i.e. σ(f ) = f (p) for all f ∈ E(M ). Proof. By definition E(M ) = Ω E(Ω) where Ω runs through all open sets containing M and E(Ω) is the space of holomorphic functions in Ω. The result for E(Ω) is well known (see Grauert and Remmert [1], Kap. V, §7, Satz 2). Hence, to each Ω ⊃ M there exists a point pΩ ∈ Ω such that σ(f ) = f (pΩ ) for f ∈ E(Ω). We claim that pΩ = pΩ for Ω ⊂ Ω. Otherwise we would have f (pΩ ) = f (pΩ ) for two different points pΩ , pΩ ∈ Ω and all f ∈ E(Ω), contradicting the theorem of Weierstrass. It follows that pΩ = p ∈ M for all Ω ⊃ M. ˚ ) ∩ H s (M ). Then E(M ) is dense in Lemma 2. For s ≥ 1 let Es (M ) := E(M s s E (M ) with respect to the H -norm. Proof. Let ϕt : M → 2M , |t| < ε, be some C ∞ -family of smooth embeddings ˚ ) ⊃ M for t > 0. Furthermore, let J denote the such that ϕ0 = id and ϕt (M almost complex structure determined by the given complex structure on 2M and let Jt := ϕ∗t J. As in Section 5.11 we define dt := d − i ∗t d and consider the vector bundle morphism D (s−1) (M, Jt ). d : (−ε, ε) × H s (M ) → |t| 1, is generated by some point p ∈ M , i.e. I = {f ∈ Es (M ) : f (p) = 0}.

5.13 Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators

365

Proof. By the theorem of Gelfand–Mazur, I is the kernel of some continuous algebra homomorphism σ : Es (M ) → C. From Lemma 1 we know that σ(f ) = f (p) for some p ∈ M and all f ∈ E(M ) ⊂ Es (M ). It follows from the density Lemma 2 that σ(f ) = f (p) for all f ∈ Es (M ). Corollary 1. Let F ∈ Es (M, Cn ) be a vector function which is nowhere zero on M . Then the equation F · G = h has a solution G ∈ Es (M, Cn ) for any given h ∈ Es (M ). Proof. If not then the set of all functions F ·G, G ∈ Es (M, Cn ), forms a proper ideal in Es (M ) which is contained in some maximal ideal I. By Theorem 1 there exists p ∈ M with F (p) · G(p) = 0 for all G ∈ Es (M, Cn ). Setting G := F (p) it follows that F (p) = 0, contradicting the hypothesis. Corollary 2. Let Ψ be a Cn -valued holomorphic differential of class H s (M ) which is nowhere zero on M . Then, given any quadratic holomorphic differential ω of class H s (M ) there exists a holomorphic Cn -valued differential ϕ such that Ψ · ϕ = ω. Proof. Choose some holomorphic differential γ, defined and without zeros in an open neighbourhood of M . Then γ −1 Ψ is a function without zeros on M and by Corollary 1 there exists G ∈ Es (M, Cn ) such that (γ −1 Ψ ) · G = γ −2 ω.

Multiplication with γ 2 gives the result.

5.13 Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators ˆ Let N(α)(τ ) and N(α)(τ ) be the smooth vector-valued harmonic maps and multivalued maps spanning the curve α in Rn introduced in the last section, ˆ where τ ∈ T(M ) is fixed for the moment. For X ∈ N(α)(τ ) or N(α) we define ˆ the Riemannian structures on both N(α)(τ ) and N(α)(τ ) by 1 = 2 j=1 n

(1)

h, kX

dhj ∧ ∗dk j . M

If h and k are single valued then (1) is clearly equivalent to 1 = 2 j=1 n

(2)

h, kX

g(w)(∇g hj , ∇g k j ) dμg M

where the metric g ∈ M−1 is chosen so that its conformal class equals τ .

366


ˆ We now define a vector field Wα (τ ) on N(α)(τ ) which on N(α)(τ ) equals the gradient with respect to ·, · of Dirichlet’s integral Eα (X, τ ) =

n j=1

dX j ∧ ∗dX j .

M

We extend the definition of Wα in such a way that Wα stays single-valued ˆ on the larger space N(α)(τ ). For this purpose we introduce the finite codiˆ ˆ mensional subspace (TX N(α)) sv of TX N(α) consisting of the single-valued ˆ tangent vectors and then define a vector Wα (τ )X ∈ (TX N(α)) sv uniquely by the relation

ˆ dY ∧ ∗dX for all Y ∈ (TX N(α)) (3) Y, Wα (τ )Xg = sv . M

Often, when it is clear from the context, we write Wα [X] in place of Wα (τ )X. Alternatively, the vector Wα [X] can be characterized by the following system of differential equations and boundary conditions:

(4)

(i)

d ∗ dWα [X] = 0

(ii)

(id − Pr(X))Wα [X] = 0

on M, on ∂M,

∂X ∂ Wα [X] = Pr(X) (iii) Pr(X) ∂N ∂N

on ∂M,

where Pr(X) : Rn → TX α(∂M ) is the orthogonal projection and N a normal vector (with respect to τ ) along ∂M . Equations (3) and (4) define the vector field Wα [X] in such a way that Wα [X] is an element of H 1 , and it follows ˆ that Wα [X] ∈ TX N(α) for each X. Alternatively this is shown directly in Chapter 6 if M is a disk. ˆ ))sv equals Also we should remark that if X is single-valued, (TX N(α)(τ TX N(α)(τ ). We are now ready to define a “partial” vector field conformality operator. ˆ ˆ ) → T N(α)(τ ) × R2pn , V Cα (τ ) : N(α)(τ

p = genus M,

namely, V Cα (τ )[X] = (Wα (τ )[X], (dX, Qj ), (dX, Qj )), where Qj , Qj (j = 1, . . . , genus M ) discussed earlier are the generators of the fundamental group of M given by a polygonal model for M . If V Cα (τ ) vanˆ ishes at some X ∈ N(α)(τ ) then the periods of dX vanish, i.e. X ∈ N(α)(τ ), and from (3) we see immediately that u is critical for Dirichlet’s energy on N(α)(τ ) for a fixed τ . We define a second partial conformality operator which is essentially function-valued and denoted by Cα (τ ), setting

5.13 Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators

(5)

Cα (τ )(X) :=

367

∂X ∂X · , (dX, Qj ), (dX, Qj ) , ∂N ∂T

ˆ Cα (τ ) : N(α)(τ ) → H s−1 (∂M, R) × R2pn , where N and T are unit-length normal and tangent vectors to ∂M with respect to the metric associated with τ ∈ T(M ). Obviously, Cα and V Cα have the same zeros. We shall now prove Theorem 1. At a minimal surface X ∈ N(α)(τ ) (which is a fortiori a zero of W(α)(τ )) the derivative map DWα (τ ) from TX N(α)(τ ) into itself is Fredholm of index zero, in fact it is of the form identity plus compact. Consequently the linear map ˆ ) → TX N(α)(τ ) × R2pn D(V Cα (τ )) : TX N(α)(τ is Fredholm of index zero. Moreover DWα (τ ) and D(V Cα (τ )) extend to Fredˆ holm maps on the closures of TX N(α)(τ ) and TX N(α)(τ ) in the H 1 (M )(1) (1) ˆ topology denoted by TX N(α) (τ ) and TX N(α) (τ ), and we have (6)

DWα (τ )[X](TX N(α)(τ )) 1

= {DWα (τ )[X](TX N(α)(1) (τ ))} ∩ H s+ 2 (M ). Proof. It follows from (4) that the vector k := DWα (τ )[X]h, h ∈ TX N(α)(τ ), satisfies (i) d ∗ d(k − h) = 0 on M, (ii)

(id − Pr(X))(k − h) = 0 on ∂M,

(iii) Pr(X)

∂X ∂ (k − h) = (D Pr(X)h) ∂N ∂N

on ∂M.

It is important to notice that (i)–(iii) can be brought into an equivalent variational form which has an obvious extension to H 1 (M ). Namely, using an integration by parts we see that for h, k ∈ TX N(α)(τ ) the relation (iii) is equivalent to

∂X · ds for all ∈ TX N(α)(τ ). D Pr(X)h d ∧ ∗d(k − h) = (7) ∂N M ∂M We remark that M dk ∧ ∗dk = 0 implies k = 0 if k ∈ TX N(α)(τ ). Hence, k ∈ TX N(α)(τ ) is uniquely determined by either (iii) or (7). Since obviously both sides of (7) extend continuously onto TX N(α)(1) (τ ) as functions of h, k and , by the Riesz representation theorem the extension DWα (τ )[X] map (1) ping TX Nα (τ ) into itself is well defined. Since ( dh ∧ ∗dh)1/2 is a complete norm on TX N(α)(1) (τ ) and the embedding H 1 (M ) "→ L2 (∂M ) is compact, we deduce by a standard contradiction argument the inequality

368


|h| ds ≤ C

dh ∧ ∗dk

2

∂M

for all k ∈ TX N(α)(1) (τ ).

M

Hence, setting = k − h in (7), we obtain the estimate

1/2

d(k − h) ∧ ∗d(k − h) ≤ C1 (X)|h|L2 (∂M ) M

d(k − h) ∧ ∗d(k − h) M

which again by the compact embedding H 1 (M ) "→ L2 (∂M ) proves the com(1) ˆ pactness of (DWα (τ )[X] − id)|TX N(α) (τ ). The corresponding statement s for the H -topology follows from the elliptic estimates and the fact that the right-hand side of (iii) is of differentiation order zero with respect to h, whereas the left-hand side is of order 1 in k − h. It follows immediately that DWα (τ )(X)|TX N(α)(τ ) is Fredholm of index 0. The same is true for the ˆ mapping DV Cα (τ )(X) : TX N(α)(τ ) → TX N(α)(τ ) × R2pn since the incluˆ sion map TX N(α)(τ ) "→ TX N(α)(τ ) and the projection TX N(α)(τ ) × R2pn → TX N(α)(τ ) are of index −2np and 2np respectively. Statement (6) follows from elliptic regularity theory if we consider (i)–(iii) as a system for h with k given. Let us remark that the coefficients of this system are highly differentiable since X is as often differentiable as its boundary curve α. Corollary 1. Suppose that X is a minimal surface and X|∂M an immersion. ˆ ) → H s−1 (∂M ) × R2pn is Fredholm Then the derivative DCα (τ ) : TX N(α)(τ of index 0. Proof. From (4)(iii) it is immediately seen that ∂X ∂X ∂X ∂ · Wα (τ )(X) = · =: Cˆα (X) ∂T ∂N ∂T ∂N and hence ∂X ∂ · DWα (τ )(X)h = DCˆα (X)h, ∂T ∂N

(8)

what can be rewritten as DCˆα (X) = S(X) ◦ DWα (τ )(X) with an operator S(X) : TX N(α)(τ ) → H s−1 (∂M ), S(X)h :=

∂X ∂h · . ∂T ∂N

One verifies that solving the equation S(X)h = f is equivalent to

d ∧ ∗dh = M

· ∂M

∂X ∂T

∂X −2 ds f ∂T

for all ∈ TX N(α)(τ ).

5.14 The Corank of the Partial Conformality Operators

369

Since the left-hand side induces a complete norm on TX N(α)(τ ), the Riesz representation theorem guarantees that S(X) is an isomorphism in the H 1/2 topology on ∂M to the H −1/2 -topology. Elliptic regularity theory implies that S(X) is an isomorphism in the stronger topologies and hence index(DCˆα (X)) = index(DWα (τ )(X)) = 0.

For another proof see Chapter 6 where the result is proved in the disk case.

5.14 The Corank of the Partial Conformality Operators Again let us denote by Ar the set of immersions α : ∂M → Rn of class H r . We introduce the trivial fibre bundles ˆ ˆ := ˆ ), ˆ ) := N(α)(τ ), N N(τ N(τ α∈Ar

τ ∈T(M )

ˆ ) and and extend the conformality operators introduced in Section 5.13 on N(τ ˆ N by ˆ ˆ V C(τ )|N(α)(τ ) := V Cα (τ ), C(τ )|N(α)(τ ) := Cα (τ ), and ˆ ) := V C(τ ), C|N(τ ˆ ) := C(τ ). V C|N(τ ˆ is a fibre bundle over T(M ). Let us denote by It should be pointed out that N s (∂M, ∂M ) the component of the identity in H s (∂M, ∂M ). Then the map Hid s ˆ ), → N(τ Ar × Hid (α, ξ) → τ -harmonic extension of α ◦ ξ ˆ ). In fact, it is by this trivialization that we is a global trivialization of N(τ ˆ ). It follows that the tangent wish to define the differentiable structure of N(τ ˆ ˆ space of N(τ ) at some X ∈ N(α)(τ ), X = α ◦ ξ, can be identified with ˆ TX N(α)(τ ) + LX

(1) 1

where LX := {h ∈ H s+ 2 (M, Rn ) : h is τ -harmonic, h|∂M = β ◦ ξ with β ∈ H r (∂M, Rn )}. ˆ Theorem 1. Let X ∈ N(α)(τ ) be a minimal surface, X|∂M = α ◦ ξ with s ξ ∈ Hid (∂M, ∂M ) injective. Then the derivative ˆ ) → TX N(α)(τ ˆ DW(τ )(X) : TX N(τ ) has closed range of finite codimension whose weak orthogonal complement in the metric of Section 5.13 is isomorphic to the space of negative meromorphic differentials ω = λ(w)/dw on M , real on ∂M , such that ω(dX + i ∗ dX) is holomorphic.

370


Proof. We may as in the proof of Theorem 1 in Section 5.13 pass to the natural ˆ ) + LX which, as was shown in the course of the H 1 -extension on TX N(α)(τ (1) ˆ proof of this theorem, has finite codimensional range in TX N(α) (τ ). Let us 1 now assume that we can characterize the range of the H -extension of DW(τ ) as stated in the theorem. For the proof of the theorem it is necessary to show that (corresponding to relation (6) of Section 5.13) ˆ DW(τ )(X)(TX N(α)(τ ) + LX )

(2)

(1) ˆ (τ ) + LX ) ∩ H s (∂M ). = DW(τ )(X)(TX N(α)

In order to prove this let k belong to the right-hand side of (2), which means (1) ˆ that k = DW(τ )(X)(γ + h) with γ ∈ TX N(α) (τ ) and h ∈ LX . Since, by the boundary regularity of minimal surfaces, LX ⊂ H r ⊂ H s and since DW(τ )(X) maps H s into H s , as can be seen from Theorem 1 of Section 5.13, it follows that DW(τ )(X)γ ∈ H s and hence γ ∈ H s by (6) in Section 5.13. This proves (2) and we may go on with the computation of the H 1 (M )-complement ˆ of the range of DW(τ )(X). Let k ∈ TX N(α)(τ ) belong to this complement. In local conformal coordinates (u, v) such that M locally corresponds to {v ≥ 0}, we may write k(x) = λ(x)Xu . We remark that λ transforms as a negative differential ω on ∂M . Using the relation ∂X ∂h ∂h ∂X ∂X ∂ · DW(τ )(X)h = · + · , ∂u ∂v ∂u ∂v ∂u ∂v

(3)

which is analogous to (8) of Section 5.13 we obtain

0= dk ∧ ∗DW(τ )(X)h = k ∗ dDW(τ )(X)h M

∂M

∂X ∂h ∂h ∂X · + · = − λ ∂u ∂v ∂u ∂v ∂M

du.

Consider now the Cn -valued function H := H1 − iH2 , where H1 and H2 are τ -harmonic and H1 |∂M := λXu , H2 |∂M = λXv . We may then continue

H1 · hv du + 0=

∂M

H2 · hu du ∂M

dH1 ∧ ∗dh −

= M

h · H2u du ∂M

dh ∧ ∗dH1 −

= − M

h · H2u du ∂M

h · (− ∗ dH1 − dH2 ) −

= ∂M

p n j=1 =1

det

()

(Qj , dh() )

(Qj , ∗dH1 )

(Qj , dh() )

(Qj , ∗dH1 ) ()


371

where we have used the Corollary of Section 5.10. We may now specialize h to have zero boundary values but arbitrary periods. Then it follows that (Qj , ∗dH1 ) = (Qj , ∗dH1 ) = 0.

(4)

Using the fact that X is immersed with the possible exception of finitely many branch points it is not difficult to see that the space LX is dense in L2 (∂M ) (even in H 1 (∂M ) as was shown by U. Thiel). From the above integral relation we may deduce ∗dH1 = −dH2

(5)

along ∂M,

i.e. the antiholomorphic differential d H = dH − i ∗ dH is real on ∂M . Since ∗dH1 has no periods by (4) there is a harmonic function F with ∗dH1 = dF . It follows then from (5) that F = −H2 + const along ∂M and hence on all of M . Consequently we have ∗dH1 = −dH2 and dH1 = ∗dH2 . This means that d H = 0, and so H is holomorphic. On ∂M we have H = ω dX where both H and dX are holomorphic on M . As in the genus zero case, this implies that ω extends as a meromorphic negative differential on M which is real on ∂M , proving the theorem. It is our next goal to extend Theorem 1 to the partial conformality operator V C(τ ), i.e. to take the period relations into account. For this purpose we need to show that the period map h → ((Qj , dh), (Qj , dh))j=1,...,p is onto R2np if h is restricted to the kernel of DW(τ ). Let us note that, as one sees from (3), ˆ ) + LX ) : Im(d h · d X) = 0 on ∂M }, (6) ker DW(τ )(X) = {h ∈ (TX N(α)(τ where d was defined in the proof of Theorem 1 of Section 5.11. For the proof of the surjectivity of the period map we need a sequence of lemmata. Lemma 1. For 1 ≤ j ≤ p let Ψj be a multivalued harmonic function with (7)

Ψj = 1

on ∂M,

(Qk , dΨj ) = δjk ,

(Qk , dΨj ) = 0.

Then for each harmonic ϕ ∈ C 1 (M , R) we have

(8) (Qj , ∗dϕ) = − ϕ ∗ dΨj . ∂M

Proof. The existence of Ψj satisfying (7) follows from Theorem 2 of Section 5.10. We first note that

Ψj ∗ dϕ = ∗dϕ = 0 ∂M

∂M

372


since ∗dϕ is closed. We therefore obtain from Riemann’s period relations that

p (Q , ∗dϕ) (Q , dΨ ) k k j dΨj ∧ ∗dϕ + det 0= (Qk , ∗dϕ) (Qk , dΨj ) M k=1

= ϕ ∗ dΨj + (Qj , ∗dϕ). ∂M

Similarly one shows that (Qj , ∗dϕ) =

(9)

ϕ ∗ dΨ j ∂M

where Ψj = 1 on ∂M , (Qk , dΨj ) = δjk and (Qk , dΨj ) = 0. Lemma 2. For harmonic functions ϕ ∈ C ω (M ) the periods (Qj , ∗dϕ), (Qj , ∗dϕ), j = 1, . . . , p, may arbitrarily be prescribed. Proof. Suppose that the map ϕ → ((Qj , ∗dϕ), (Qj , ∗dϕ))j=1,...,p were not onto R2p , where ϕ runs through all harmonic functions of class C ω (M ). Then we can find a nontrivial linear relation p (cj (Qj , ∗dϕ) + cj (Qj , ∗dϕ)) = 0 j=1

holding for all such ϕ. Thus if ω := j (cj dΨj − cj dΨj ) it follows from (8) and (9) that

ϕ ∗ ω = 0 for all ϕ ∈ C 1 (∂M ), ∂M

and hence ∗ω = 0 along ∂M . But ω = 0 along ∂M by the construction of Ψj , Ψj , and it follows therefore that ω ≡ 0 since ω is harmonic. The dΨj , dΨj being linearly independent (see (7)), this implies cj = cj = 0 contradicting our assumption on the cj , cj . Lemma 3. For holomorphic differentials of class C ω (M ) the periods may be prescribed arbitrarily. Proof. Let γ ∈ C ω (M ) be a real harmonic differential with prescribed periods (Theorem 1 of Section 5.10) and ϕ a real harmonic function such that the periods of ∗dϕ are arbitrary (Lemma 2). This means that the periods of (γ + i ∗ γ) + (dϕ + i ∗ dϕ) may be arbitrarily chosen. Lemma 4. There exist functions Fj , Fj , j = 1, . . . , p, which are holomorphic on a neighbourhood of ∂M and satisfy


(10)

Fj ϕ,

(Qj , ϕ) =

(Qj , ϕ) = −

∂M

373

Fj ϕ ∂M

◦

for all holomorphic differentials ϕ on M continuous up to ∂M . Moreover the Fj , Fj (j = 1, . . . , p) form a linearly independent set. Proof. By Lemma 3 we can select holomorphic differentials ωj , ωj defined in an open neighbourhood of M such that (11)

(Qk , ωj ) = 0,

(Qk , ωj ) = δjk ,

(Qk , ωj ) = 0,

(Qk , ωj ) = δjk .

In a neighbourhood of ∂M we have ωj = dFj , ωj = dFj with holomorphic functions Fj , Fj . From Riemann’s period relations we obtain

Fj ϕ = ωj ∧ ϕ + (Qj , ϕ). ∂M

M

However, ωj ∧ ϕ = 0, since both ωj and ϕ are holomorphic. This proves the first formula in (10). The second one follows similarly. Lemma 5. (i) Let g : ∂M → C such that g extends as a H 2 (M )-function and suppose that

gω = 0 ∂M

for all differentials ω holomorphic in a neighbourhood of M . Then g extends as a holomorphic function on M . (ii) Let ϕ be a differential on ∂M which has an H 2 (M )-extension and suppose that

fϕ = 0 ∂M

for all holomorphic functions f defined in a neighbourhood of M . Then ϕ extends as a holomorphic differential on M . Proof. (i) Let G be the harmonic extension of g, G ∈ H 2 (M ). Then by Gauss’s theorem

(12) gω = d(Gω) 0=

∂M

M

(d G + d G) ∧ ω =

= M

d G ∧ ω

M

for all holomorphic differentials ω in a neighbourhood of M and with dG = d G + d G. It follows from an earlier discussion that the differentials holomorphic in a neighbourhood of M are H 1 -dense in the set of holomorphic differentials of class H 1 (M ), and hence we may insert ω = d G in (12). In terms of a local complex coordinate w this leads to

374


0= M

∂G ∂G dw ∧ dw = ∂w ∂w

∂G 2 dw ∧ dw M ∂w

and hence Gw = 0, proving the first part of the lemma. (ii) Let ϕ0 be a non-vanishing holomorphic differential defined in an open neighbourhood of M . Then g := ϕ/ϕ0 is a function on ∂M , and clearly g satisfies the assumptions of part (i). It follows that g has a holomorphic extension G to M and hence ϕ is holomorphically extended as Gϕ0 . We are now ready to prove Theorem 2. Let η = (η 1 , . . . , η n ) be a vector-valued holomorphic differential on M of class H 2 (M ) and such that η 1 , . . . , η n are linearly independent over R. Then f → Re((Qj , f η), (Qj , f η))j=1,...,p

(13)

is a surjective map from the space of holomorphic functions defined in a neighbourhood of M onto R2pn . Proof. Assuming that (13) is not onto, there are real numbers cjk , cjk , j = 1, . . . , p, k = 1, . . . , n, not all of them zero, such that (cjk (Qj , f η k ) − cjk (Qj , f η k )) = 0 (14) Re j,k

for all f holomorphic in a neighbourhood of M . Since f in (14) is arbitrary we may omit Re in front of the sum in (14). Using (10) to define functions Fj , Fj , this implies

f ω = 0, ω := (cjk Fj η k + cjk Fj η k ) ∂M

j,k

for all f as above. From part (ii) of Lemma 5 we may then conclude that there exists a holomorphic differential σ on M such that ω = σ along ∂M and hence ω = σ in a neighbourhood of ∂M since ω is also holomorphic (where defined). Remembering that dFj = ωj , dFj = ωj are globally defined holomorphic differentials, we may consider the equation ω = σ as well on the ∗ k . By assumption, not all of the differentials polygonal model M k cjk η , k c η vanish identically. Without loss of generality let us assume that k jk k k k c1k η ≡ 0. Dividing the equation ω = σ by k c1k η then we obtain (15)

F1 = g −

p j=2

Fj g j −

p

Fj gj

j=1

where g, gj and gj are meromorphic functions on M and (15) may also be considered as a relation on M ∗ . Denoting the endpoints of the crosscut Q1 ⊂ ∂M ∗ with q0 and q1 , and observing that g, gj , and gj are periodic, i.e.


g(q1 ) = g(q0 ),

gj (q1 ) = gj (q0 ),

375

gj (q1 ) = gj (q0 ),

we conclude from (11) and (15) that 1 = (Q1 , ω1 ) = F1 (q1 ) − F1 (q0 ) p p Fj (q1 ) − Fj (q0 ) gj (q0 ) [Fj (q1 ) − Fj (q0 )] gj (q0 ) − = − j=2

= −

p j=2

j=1

(Q1 , ωj )gj (q0 ) −

p

(Q1 , ωj )gj (q0 ) = 0,

j=1

an obvious contradiction. The lemma is proved.

We are now in a position to prove ˆ Theorem 3. Let X ∈ N(α)(τ ) be a minimal surface, X|∂M = α ◦ ξ with ξ injective. Then (16)

range(DV C(τ )(X)) = range(DW(τ )(X)) × R2pn .

Proof. As in Theorem 1 we first consider the corresponding statement for the H 1 (M )-extension of DW(τ )(X). We know from the genus zero case that the functions of the form β ◦ ξ, β ∈ C ∞ (∂M, Rn ) are dense in H 1 (∂M, Rn ) and (1) ˆ therefore, a fortiori, the domain of the H 1 (M )-extension, TX N(α) (τ ) + LX , 1 is dense in the space of multivalued τ -harmonic maps of class H (M ) which ˆ 1 (τ ). We also know from Theorem 1 of Section 5.13 that is denoted by H 1 the H (M )-extension of DW(τ )(X) and hence of DV C(τ )(X) have closed ˆ 1 (τ ) without ranges. We may therefore extend the latter operator onto all of H 1 changing its range. In order to prove the H (M )-analog of (16) it is therefore sufficient to show that the period map → Re((Qj , d ), (Qj , d )) is surjective ˆ 1 (τ ), i.e. from the space of all ∈ H ˆ 1 (τ ) from the kernel of DW(τ )(X) on H such that (17)

Im(d · d X) = 0

on ∂M.

Let m, 2 ≤ m ≤ n, be the minimal dimension of an affine subspace of Rn which contains the image of X. After applying an isometry of Rn we may assume that X(M ) ⊂ Rm × {0} ⊂ Rn . Then the differentials ηj = d X j , j = 1, . . . , m are linearly independent over R. Let us choose further holomorphic differentials ηj ∈ C ∞ (M ), j = m + 1, . . . , n, such that η1 , . . . , ηn are linearly independent. Then with η = (η1 , . . . , ηn ) we have η · d X = 0. Hence any multivalued harmonic function of the form = Re f η with f holomorphic in a neighbourhood of M satisfies (17). Therefore the surjectivity of the period map from the space defined by (17) follows from Theorem 2, and the H 1 analog of (16) is proved. Let us now turn to the statement (16) as it stands.

376


Given y ∈ R2pn , there are, corresponding to what we have just shown, ω ∈ (1) ˆ (τ ) and h ∈ LX such that DW(τ )(X)h + ω = 0 and TX N(α) Re((Qj , d h + d ω), (Qj , d h + d ω)) = y. Since h ∈ H s (∂M ), we have DW(τ )(X)ω = −DW(τ )(X)h ∈ H s (∂M ) and hence ω ∈ H s (∂M ) by (6) of Section 5.13 proving the theorem.

Corollary 1. If in addition to the hypotheses in Theorem 3, X|∂M is also immersed then the range of DC(τ )(X) is closed in H s−1 (∂M, R) × R2pn and has the same codimension as the range of DV C(τ )(X). The proof is completely similar to that of Corollary 1 of Section 5.13 and may therefore be omitted. Now we wish to compute the corank of DV C(τ ) at a minimal surface X in terms of the branch points of X. By definition these are those points in M where X fails to be immersed, i.e. where the holomorphic differential d X = dX + i ∗ dX vanishes. The order of a branch point is the order of the corresponding zero of d X which is also well defined at the boundary of M (cf. Section 2.10 of Vol. 2). By Theorems 1 and 3 the corank of DV C(τ )(X) equals the dimension of the linear space of negative meromorphic differentials ω on M which are real on ∂M and such that ω d X is holomorphic in the interior of M and continuous up to the boundary. We denote this space by ND(X). ˆ Theorem 4. Let X ∈ N(α)(τ ) be a minimal surface with k interior branch and boundary points w1 , . . . , wk of orders λ1 , . . . , λk branch points ξ1 , . . . , ξ of orders ν1 , . . . , ν . Then with |λ| = λj , |ν| = νj , dim ND(X) = 2|λ| + |ν| − 6p + 3 + ind(a) where p is the genus of M and ind(a) is the dimension of the space of symmetric meromorphic differentials on 2M which are multiples of the divisor class (18)

ˆ1λ1 · · · w ˆkλk ξ1ν1 · · · ξν δ −1 a = w1λ1 · · · wkλk w

where w ˆj denotes the symmetric image of wj , and δ is the divisor class of a symmetric holomorphic differential on 2M . One should note that 6p − 3 is the dimension of the Teichm¨ uller moduli space. Corollary 2. We have dim ND(X) =

0

if 2|λ| + |ν| < 4p − 2,

2|λ| + |ν| − 6p + 3

if 2|λ| + |ν| > 8p − 4.

5.15 The Corank of the Complete Conformality Operators

377

Proof. For any ω ∈ ND(X) the total order of poles (after reflection across ∂M ) is always less or equal to 2|λ| + |ν|. Hence if ω = 0 then ω −1 would be a meromorphic differential on 2M whose total order of zeros on 2M would be strictly less than 4p − 2 in the first case. Since, however, the degree of any meromorphic differential is always 2 · genus(2M ) − 2 = 4p − 2 (Theorem 4 of Section 5.10) this is a contradiction. It follows that ND(X) = {0} in this case. Secondly, if 2|λ| + |ν| > 8p − 4 then deg(a) = 2|λ| + |ν| − deg δ > 8p − 4 − (4p − 2) = 4p − 2. By the same reasoning it follows that ind(a) = 0. Proof of Theorem 4. We first show that ND(X) is isomorphic to the space of symmetric meromorphic functions on 2M which are multiples of the divisor a−1 . Let Ψ be any symmetric holomorphic differential with divisor class δ and ω ∈ ND(X). Then Ψ ω is a symmetric meromorphic function on 2M , and it is a multiple of a−1 . Conversely, for every symmetric meromorphic function f on 2M which is a multiple of a−1 the negative differential ω := f Ψ −1 has the property that ω d X is holomorphic. Thus by the symmetric version of the Riemann–Roch theorem (Theorem 7 of Section 5.10) dim ND(X) = dim Ω(a) + d[2a] − genus(2M ) + 1 = ind(a) + 2|λ| + |ν| − deg(δ) − 2p + 1, where Ω(a) denotes the space of symmetric meromorphic differentials on 2M which are multiples of (a). Since deg(δ) = 4p − 2, the assertion follows.

5.15 The Corank of the Complete Conformality Operators The zeros of V C(τ ) or C(τ ) are not yet minimal surfaces. The equation V C(τ )(X) = 0 (or C(τ )(X) = 0) merely implies that X is single-valued and that the holomorphic quadratic differential (d X)2 = (dX + i ∗ dX)2 is real on ∂M , i.e. (d X)2 ∈ QR Dτ in our earlier notation. We therefore introduce the projection Pτ : T2,0 → QR Dτ as the orthogonal projection with respect to the scalar product

˜ bd dμg with (g ab ) = (gkj )−1 ˜ g := 1 g ab g cd hac h h, h 2 M ˜ from the space of all (say square integrable) symmetric (2, 0)-tensorfields h, h onto the finite dimensional subspace QR Dτ . The background metric g is the unique representative of τ in the sense of our identification of T(M ) with a submanifold Σ ⊂ MS−1 . We then define the vector field conformality operator V Kα (τ ) by

378


ˆ ˆ V Kα (τ ) : N(α)(τ ) → (T N(α)(τ ))sv × R2pn × QR Dτ , V Kα (τ )(X) = (V Cα (τ )(X), Pτ (dX + i ∗ dX)2 ), and the conformality operator by ˆ ) → H s−1 (∂M ) × R2pn × QR Dτ , Kα (τ ) : N(α)(τ Kα (τ )(X) := (Cα (τ )X, Pτ (dX + i ∗ dX)2 ). It is then obvious that the zeros of V Kα (or Kα ) are exactly the minimal surfaces spanning the curve α. The definition of V Kα and Kα is extended to ˆ ) and N ˆ in the obvious way, the bundles N(τ ˆ V K(τ )|N(α)(τ ) := V Kα (τ ),

ˆ ) := V K(τ ), V K|N(τ

and similarly for K. We now want to show ˆ Theorem 1. Let X ∈ N(α)(τ ) be a minimal surface, X|∂M = α ◦ ξ with ξ : ∂M → ∂M bijective. Then the corank of the derivative DV K(τ )(X) is νj are the total orders of the interior 2|λ| + |ν|, where |λ| = λj and |ν| = and boundary branch points of X. For the proof of the theorem we need Lemma 1. Let X be as in Theorem 1, and ω a quadratic holomorphic differ◦

ential, real on ∂M , and with zeros w1 , . . . , wk ∈M of orders λ1 , . . . , λk and ξ1 , . . . , ξ ∈ ∂M of orders ν1 , . . . , ν . Then there exists a harmonic function h : M → Rn , h|∂M ∈ H r (∂M ), such that (1)

d h · d X = ω.

Proof. Let us first remark that ω extends holomorphically onto 2M . By the theorem of Weierstrass we can choose a function f holomorphic in a neighbourhood of M which has exactly the zeros w1 , . . . , ξ of orders λ1 , . . . , ν . From the boundary regularity theorem it follows that f −1 d X is a holomorphic differential of class H r−1 (∂M ) and without zeros. Since f −1 ω is smooth, we may conclude from Corollary 2 of Section 5.12 that there exists a holomorphic differential ϕ ∈ H r−1 (∂M ) such that ϕ · f −1 d X = f −1 ω, and hence ϕ · d X = ω. Observing that d X · d X = 0 we even have (ϕ + g d X) · d X = ω for any function g. By lowering the dimension n if necessary we may assume that the differentials d X 1 , . . . , d X n are linearly independent over R (cf. proof of Theorem 3 of Section 5.14). We may therefore apply Theorem 2 of Section 5.14 and find a g holomorphic in a neighbourhood of M such that the periods of ϕ + g d X are purely imaginary. The harmonic function h = Re (ϕ + g dX ) then satisfies (1).

5.15 The Corank of the Complete Conformality Operators

379

Proof of Theorem 1. By the same kind of argument as in the proof of Theorem 3 of Section 5.14, making use of (6), we see that the corank does not depend on the smoothness of the function spaces we are working in. We may therefore compute the desired corank for the natural H 1 (M )-extension of (1) ˆ DV K(τ )(X) which is defined on TX N(α) (τ ) + LX (cf. (1) of Section 5.14). 1 As already earlier remarked, LX is H (M )-dense in the space of all τ -harmonic functions of class H 1 (M ) and since we know already from Theorem 2 of Section 5.14 that the H 1 -extension of DV K(τ )(X) has closed range in H 1 (M ) we ˆ 1 (τ ), do not change the range any more if we extend DV K(τ )(X) to all of H the set of multivalued τ -harmonic functions of class H 1 (M ). We now employ the elementary formula corank(A1 , A2 ) = corank A1 + corank A2 | Ker A1 which holds for any pair of linear operators. We must therefore compute the corank in QR Dτ of the map (2)

h → Pτ (d h · d X)

ˆ 1 (τ ). This means that h is singlewhere h is in the kernel of DV C(τ )(X) on H valued and satisfies (3)

Im(d h · d X) = 0 on ∂M,

i.e. d h · d X ∈ QR Dτ , and hence (2) reduces to (4)

h → d h · d X.

We claim that the range of this map equals the set Q0 of all those ω ∈ QR Dτ which vanish at w1 , . . . , ξ of orders λ1 , . . . , ν . It is obvious that the range of (4) is contained in Q0 and it follows from Lemma 1 that it is equal to Q0 . In order to compute dim Q0 we choose a nontrivial symmetric holomorphic differential ϕ on 2M and consider the divisor a introduced in (18) of Section 5.14 together with the space Ω(a) of symmetric meromorphic differentials which are multiples of a. Then it is easily seen that the map Q0 ω → ϕ−1 ω ∈ Ω(a) is an isomorphism and hence dim Q0 = dim Ω(a) = ind(a). It follows therefore from Theorems 1, 3 and 4 of Section 5.14 that corank DV K(τ )(X) = dim ND(X) + dim QR Dτ − dim Q0 = (2|λ| + |ν| − 6p + 3 + ind(a)) + 6p − 3 − ind(a) = 2|λ| + |ν|.

380


5.16 Manifolds of Harmonic Surfaces of Prescribed Branching Type In this section we stratify the manifold of multivalued harmonic (with respect to a fixed conformal structure) surfaces with boundary values on some embedded curve α ∈ Ar according to their branching order. For a fixed value of the Teichm¨ uller parameter τ and μ = (μ1 , . . . , μk ) ∈ Nk , ν = (ν1 , . . . , ν ) ∈ N ˆ μν (τ ) the set of all such harmonic surfaces in the bundle N(τ ˆ ) we denote by N which have exactly k interior branch points of orders μ1 , . . . , μk and boundary branch points of orders ν1 , . . . , ν . The main result in this section is that ˆ ) of codimension 2n|μ| + (n + 1)|ν| − 2k − ˆ μν (τ ) is a submanifold of N(τ N where |μ| = μ1 + · · · + μk , |ν| = ν1 + · · · + ν . The following discussion serves the purpose to make this statement more precise. A zero at p ∈ M of order ν of a holomorphic function f on M is clearly characterized by the conditions dj f (p) = 0 dwj

dν f (p) = 0, dwν

for j = 1 . . . ν − 1,

where w is any local complex coordinate around p. We can employ the same notion of order for a boundary zero p ∈ ∂M provided that f ∈ C ν (M ). There may be boundary zeros of holomorphic functions for which no order is well defined, but such functions are excluded from our discussion. We find it convenient to use the concept of a divisor class introduced at the outset: Let f be holomorphic on M and of class C m (M ), and let p1 , . . . , pk and q1 , . . . , q be pairwise different points in M and on ∂M respectively, and assume that μ1 , . . . , μk , ν1 , . . . , ν ∈ N with νj ≤ m for j = 1, . . . , . Then we say that f is in the divisor class pμ1 1 · · · pμk k q1ν1 · · · qν (which we abbreviate as pμ q ν ) provided that f has zeros p1 , . . . , q of orders μ1 , . . . , ν and no other zeros on M . The same notion applies to holomorphic differentials and to real differentiable functions and differentials on ∂M . ˆ ) is diffeomorphic to Recall that, by definition, N(τ s (∂M ) × H0 (τ ) Ar × Hid s (∂M ) where Ar are the embeddings of ∂M into Rn of Sobolev class H r , Hid s are the H -maps of ∂M to itself homotopic to the identity, and H0 (τ ) is the class of Rn -valued harmonic differentials ϕ on M that are zero along ∂M . A global diffeomorphism s ˆ ) Ψτ : Ar × Hid (∂M ) × H0 (τ ) → N(τ

is given by

Ψτ (α, ξ, ϕ) = Eτ (α ◦ ξ) +

ϕ, p0

where Eτ (f ) denotes the τ -harmonic extension of a boundary function f , and ϕ is a line integral viewed as function of its endpoint. Let now p ∈ M k , p0 q ∈ (∂M ) , μ ∈ Nk , and ν ∈ N with νj < s − 1. We then define

5.16 Manifolds of Harmonic Surfaces of Prescribed Branching Type

381

ˆ μν (τ )[p, q] = N

s (α, ξ, ϕ) ∈ Ar × Hid (∂M ) × H0 (τ ) :

d Ψτ (α, ξ, ϕ) is in the divisor class pμ q ν , dw ˆ ˆ μν (τ )[p, q]. Nμν (τ ) = N p,q

ˆ μν (τ )[p, q] is the set of Rn -multivalued harmonic branched surfaces with N fixed branch points of a specific order. Similarly we define Nμν (τ ) as those ˆ μν (τ ). single-valued members (ϕ = 0) of N The following simple lemma enables us to decompose complex derivatives into tangential and normal components. Lemma 1. Let h be real harmonic and w = u + iv be a local complex coordinate. Then for λ ≥ 1 we have ∂λh dλ h ∂λh dλ h = , Im = − λ−1 . λ λ λ dw ∂u dw ∂u ∂v Proof. The lemma is obvious for λ = 1. Since hw is holomorphic we have for λ ≥ 2: λ λ−1 λ−1 d d 1 ∂ h= hw = (hu − ihv ). dw dw 2 ∂u ˆ ) have boundary values of the form α ◦ ξ where the The surfaces in N(τ differential dα is always non singular. Hence d(α ◦ ξ) vanishes of some order ν < s − 1 if and only if dξ vanishes of the same order. Therefore it is natural to introduce the spaces Re

(1)

s s (∂M )[q] = {ξ ∈ Hid (∂M ) : dξ is in the divisor class q}, Hid,ν

and (2)

s (∂M ) = Hid,ν

s Hid,ν (∂M )[q]

q

where of course q ∈ (∂M ) , ν ∈ N , νj < s−1 as above. We need the following r Lemma 2. Let ξ ∈ Hid,ν (∂M )[q], r ≥ 2, be a homeomorphism of ∂M and let us furthermore assume that around each of the qj the representation

dξ ˜ j ) = 0 (u) = uνj ξ˜j (u), ξ˜j ∈ H r−1 , ξ(q du holds where u is some local C ∞ -coordinate on ∂M around qj with u(qj ) = 0. Then, given any h ∈ H r (∂M ) which has a corresponding representation around the point ξ(qj ), (3)

dh ˜ ˜ ∈ H r−1 , k ≥ r(νj + 2) − 3, (u) = uk h(u), h du there exists a unique β ∈ H r (∂M ) such that h = β ◦ ξ. (4)

382


Proof. Since ξ is a homeomorphism, we must have β = h ◦ ξ −1 . Obviously, β is locally of class H r away from the branch points qj . Restricting attention to one of the points qj and introducing a local coordinate around ξ(qj ) we may think of our problem as one on the real line and we may in particular assume that ξ maps a neighbourhood of the origin to itself and that ξ(0) = 0. For the purpose of this proof we replace νj by ρ. It follows then from (3) that (5) c|u|ρ+1 ≤ |ξ(u)| ≤ c−1 |u|ρ+1 , c ρ+1 |u| ≤ |ξ −1 (u)| ≤ c−1 ρ+1 |u| with some positive constant c. From (3), (4) and (5) we obtain |h() ◦ ξ −1 (u)| ≤ const |u|(k−+1)/(ρ+1) ,

1 ≤ ≤ r − 1,

˜ r−1 ◦ ξ −1 (u)|), |h(r) ◦ ξ −1 (u)| ≤ const(|u|k−r+1)/(ρ+1) + |u|k/(ρ+1) |h |(ξ −1 )() (u)| ≤ const |u|1/(ρ+1)− ,

1 ≤ ≤ r − 1,

|(ξ −1 )(r) (u)| ≤ const(|u|1/(ρ+1)−r + |u|1+(r−1)/(ρ+1)−r |ξ˜(r−1) ◦ ξ −1 (u)|). From the chain rule and using the preceding estimates we get |(h ◦ ξ −1 )(r) (u)| ≤ |h(r) ◦ ξ −1 (u)||(ξ −1 ) (u)|r + |h ◦ ξ −1 (u)||(ξ −1 )(r) (u)| + const |u|(k+2−r)/(ρ+1)+1/(ρ+1)−r (k−r)(ρ+1)+1 ≤ const( ρ+1 |u| +

+

ρ+1

+

ρ+1

(k+1−r)(ρ+1)+r−3

|u|

k+3−r(ρ+2)

|u|

k−ρr (r−1) −1 ˜ |u| |h ◦ ξ (u)|

ρ+1

|ξ˜(r−1) ◦ ξ −1 (u)|

).

It is easily seen from the transformation formula and the boundedness of ξ˜ ˜ (r−1) ◦ ξ −1 and ξ˜(r−1) ◦ ξ −1 are square integrable. All other terms are that h bounded since k ≥ r(ρ + 2) − 3 ≥ rρ + 1. One can now prove ˆ μν (τ ) be a (possibly multivalued) Theorem 1. Let X0 = Ψτ (α0 , ξ0 , ϕ0 ) ∈ N minimal surface, where νj < s−m for some m ∈ N. Then, in a neighbourhood ˆ ), the subset N ˆ μν (τ ) is a C m -submanifold of N(τ ˆ ) of X0 in the bundle N(τ ˆ μν (τ ) at of codimension 2n|μ| + (n + 1)|ν| − 2k − . The tangent space of N 0 0 ˆ X0 ∈ Nμν (τ )[p , q ] can be described as follows: Introduce local complex coordinates zj around p0j , j = 1, . . . , k, and wj around qj0 such that wj maps ∂M into the reals (j = 1, . . . , ). Then (β, w, ψ) ∈ s ˆ μν (τ ) at X0 if and only if Tα0 Ar ×Tξ0 Hid (∂M )×H0 (τ ) is a tangent vector to N the multivalued harmonic surface g = Eτ (β ◦ v0 + dα0 (v0 )w) + M ψ satisfies the relations

5.16 Manifolds of Harmonic Surfaces of Prescribed Branching Type

(6)

d dzj

d dzj

383

λ λ = 1, . . . , μj − 1,

g(p0j ) = 0,

μj

g(p0j ) + dzj (δj )

d dzj

μj +1 X0 (p0j ) = 0,

j = 1, . . . , k,

λ d g(qj0 ) = 0, λ = 1, . . . , νj − 1, dwj νj νj +1 d d g(qj0 ) + dwj (εj ) X0 (qj0 ) = 0, dwj dwj

where δj ∈ Tp0j M and εj ∈ Tqj0 ∂M are arbitrary tangent vectors. Proof. The maps of ∂M to itself which are homotopic to the identity can be identified with the maps from R to itself which are of the form identity s plus a 2π-periodic function. In particular, we may think of Hid (∂M ) as a linear space. Related to but not identical with (2), (3) we introduce the linear spaces s ˜ νs (∂M )[q] = {ξ ∈ Hid (∂M ) : ξ (λ) (qj ) = 0 for λ = 1, . . . , νj , j = 1, . . . , } H

and their disjoint union ˜ νs (∂M ) = H

˜ νs (∂M )[q]. H

q

Since the derivatives ξ (λ) (qj ) can be arbitrarily prescribed it follows directly ˜ s (∂M ) is a C m -subbundle of the trivial bundle from Lemma 1 (iii) that H ν s (∂M ) × Hid (∂M ) of codimension |ν|. Now we choose the local complex coordinates zj with domain Uj and wj with domain Vj as in the statement of the theorem. For p ∈ U := U1 × · · · × Uk and q ∈ V := (V1 ∩ ∂M ) × · · · × (V ∩ ∂M ) we consider the maps ⎛ ⎞ λ λ d h h d ⎠. χ[p, q]h = ⎝ (pj ) , Im (qj ) dzjλ ∂wjλ λ=1,...,μj λ=1,...,νj j=1,...,k

j=1,...,

It follows from Lemma 1 and the remarks following this lemma that ˆ μν (τ )[p, q] is an open subset of N ˜ s (τ )[q] × H0 (τ ) : χ[p, q]Ψτ (α, ξ, ϕ) = 0}. {(α, ξ, ϕ) ∈ Ar × H ν We therefore consider the C m -morphism of bundles over U × V , ˜ s (∂M )[V ] × H0 (τ ) → U × V × Cn|μ| × Rn|ν| , χ ˜ : U × Ar × H ν

384


where ˜ s (∂M )[V ] = H ν

˜ s (∂M )[q] H ν

q∈V

and χ(p, ˜ α, ξ, ϕ) = (p, q, χ[p, q]Ψτ (α, ξ, ϕ)). We want to show that at the given ˜ is transverse to the zero secminimal surface X0 = Ψτ (α0 , ξ0 , ϕ0 ) the map χ ˜ is surjective tion of the vector bundle (U × V ) × (Cn|μ| × Rn|ν| ), i.e. that dχ on the fibre over (p0 , q0 ). By the Weierstrass theorem on the existence of a holomorphic function (or differential) with prescribed zeros on an open Riemann surface and using Lemma 1 we can find a multivalued harmonic surface h ∈ C ∞ on M such that χ[p0 , q0 ]h = (c, d) ∈ Cn|μ| × Rn|ν| where c, d are arbitrarily given and, moreover, (7)

dλ h 0 (q ) = 0 duλj j

for λ = 1, . . . , r(νj + 2) − 3

with uj = Re wj as a local coordinate on ∂M . By the boundary regularity of minimal surfaces we have that ξ0 ∈ H r (∂M ) and that the representation (4) holds. It follows from (7) and Lemma 2 that h|∂M = β ◦ ξ0 with some β ∈ H r (∂M ), i.e. we have shown that χ[p0 , q0 ] is surjective from the space of multivalued harmonic surfaces with boundary values of the form β ◦ ξ0 , β ∈ H r (∂M, Rn ), onto Cn|μ| × Rn|ν| . This implies that χ[p0 , q0 ] is surjective from the range of dΨτ (α0 , v0 , ϕ0 ) onto Cn|μ| × Rn|ν| and hence shows the transversality of χ ˜ at the minimal surface X0 . It follows immediately from the implicit function theorem that in a neighbourhood of h0 the disjoint union ˆ μν (τ ) := ˆ μν (τ )[p, q] N N p,q

˜ νs (∂M )[V ] × H0 (τ ) of codimension is a C m -submanifold of U × Ar × H ˆ ) 2n|μ| + n|ν| and therefore of codimension 2n|μ| + (n + 1)|ν| in U × V × N(τ ˆ and that the tangent space of Nμν (τ ) is characterized by the relations (6). Formally (8)

ˆ μν (τ ) = {(s, t, g) ∈ Cn|μ| × Rn|ν| × Range dΨτ : Th0 N (s, t) arbitrary and g satisfies (6)}.

Consider now the projection ˆ ) → N(τ ˆ ) P : U × V × N(τ which is a smooth submersion of Fredholm index 2k + . As shown above, the inclusion ˆ ) ˆ μν (τ ) → U × V × N(τ ι:N

5.17 The Proof of the Index Theorem

385

has index −2n|μ| − (n + 1)|ν| and hence ˆ μν (τ )) = −2n|μ| − (n + 1)|ν| + 2k + . ind(P |N ˆ μν (τ ), the proof of the theorem will be complete once ˆ μν (τ )) = N Since P ((N ˆ we can show that P |Nμν (τ ) is a differentiable embedding. Firstly, P is one-toˆ μν (τ ), the holomorphic differential (Ψτ )w (α, ξ, ϕ) one since, for (α, ξ, ϕ) ∈ N belongs to only one divisor class pu q ν . Secondly, if a sequence of multivalued ˆ μν (τ ) converges to some h ∈ N ˆ μν (τ ) then the branch harmonic surfaces hj ∈ N points of hj converge to those of h. Finally, it follows immediately from (8) ˆ μν (τ ) is an immersion. that P |N Remark 1. Theorem 1 can be strengthened by not requiring X0 to be a minimal surface, but the proof is far more technical (as we saw in the proof for the disk case which carries over easily to the case of higher genus). Thus ˆ ) (or N(τ )) of ˆ μν (τ ) (or Nμν (τ )) is a submanifold of N(τ we obtain that N codimension 2n|μ| + (n + 1)|ν| − 2k − l.

5.17 The Proof of the Index Theorem In this final section we show that the minimal surfaces of Mμν in the bundle N of branching type (μ, ν) arise locally as subsets of local submanifolds Wμν of N. The bundle N has a natural projection onto its second factor, N(α)(τ ). Π : N → Ar , N = τ ∈T(M ) α∈Ar

We shall show that Πμν = Π|Wμν is a Fredholm map whose index is given by the formula ind Πμν = 2(2 − n)|μ| + (2 − n)|ν| + 2k + where μ = (μ1 , . . . , μk ), ν = (ν1 , . . . , ν ). From this we shall conclude that a Baire subset Ar0 of Ar contains only curves which admit no minimal surfaces with boundary branch points or multiple interior branch points and contain only isolated minimal surfaces. In addition we shall show, as in the disk case, that for ν = 0, Mμ0 = Wμ0 , i.e. Mμ0 has the structure of a submanifold of N. Remark 1. We remind the reader that we are assuming that r > s(s + 3). ˆ ) → T N(τ ) × R2pn × QR Dτ be the complete vector field Let V K(τ ) : N(τ conformality operator. Then from (1) of Section 5.15 we know that the corank ˆ μν (τ ) is 2|μ| + |ν|. of the derivative D(V K)(τ ) at a minimal surface (α, ξ) ∈ N Our first main goal is to compute the loss of rank when D(V K)(τ ) is restricted

386


ˆ μν . Like in the disk case this will be accomplished to the tangent space of N by computing dim E(α,ξ) /(E(α,ξ) ∩ T(α,ξ) Nμν (τ )), where (α, ξ) ∈ Nμν (τ ) is a minimal surface and E(α,ξ) := ker D(V K)(τ )(α, ξ). We remark that the elements in this kernel are automatically single-valued, i.e. E(α,ξ) ⊂ T(α,ξ) N(τ ). The computation of the above dimension is much more involved than in the disk case due to the lack of forced Jacobi fields which were an essential tool in that case. ˆ μν (τ ) is a submanifold of Ar × H s (∂M ) × H0 (τ ) Let us recall that N id,ν and, as we have shown in the course of the proof to Theorem 1 of Section 5.16 s s (∂M ) is of codimension |ν| − in Hid (∂M ), and the tangent space that Hid,ν s s Tξ Hid,ν (∂M ) consists of all H -vector fields ψ along ξ such that (1)

dλ ψ (qj ) = 0 for λ = 1, . . . , νj − 1; j = 1, . . . , , duλj

where q1 , . . . , q are the singular points of ξ and uj some local coordinates around qj . s (∂M )) has the same The next lemma shows that E(α,ξ) ∩ (Tα Ar × Tξ Hid,ν codimension in E(α,ξ) . Lemma 1. The map (β, ψ) →

dλ ψ (qj ) duλj

λ=1,...,νj −1 j=1,...,

is surjective from E(α,ξ) onto R|ν|− . Proof. Let X = Ψτ (α, ξ, 0) denote the minimal surface with boundary values α ◦ ξ. Fix any of the branch points qj , and let w be a meromorphic vector field in a neighbourhood U of qj which is tangent along ∂M and has a pole of order ≤ νj at qj . Let furthermore ζ ∈ C0∞ (U ) such that ζ ≡ 1 in some smaller neighbourhood of qj . Then we define (2)

ψ = ζ dξw.

By the boundary regularity of minimal surfaces, ξ is of class H r (∂M ) and dξ has zeros of order νj at qj with a representation (4) of Section 5.16. It follows therefore that dξw ∈ H r−1 (∂M ∩ U ). For the proof of the lemma it is therefore sufficient to show that any ψ of the form (2) is admissible, i.e. for any such ψ there exists β ∈ Tα Ar with (β, ψ) ∈ E(α,ξ) . We first note that by the chain rule (3)

(dα) ◦ ξψ = ζ dXw = ζ Re(d Xw).


387

We observe that ζ d Xw ∈ H r−1 (∂M ) is holomorphic in a neighbourhood (on M ) of the branch points q1 , . . . , q . If we therefore choose (using Weierstrass’s theorem) a holomorphic function f0 on M with zeros exactly at qj of order > r(νj + 2) − 3, j = 1, . . . , , then f10 d (ζ d Xw) ∈ H r−3/2 (M ), and hence there exists G0 ∈ H r−1/2 (M ) with d G0 = −

1 d (ζ d Xw). f0

It follows immediately that G1 := ζ d Xw + f0 G0 is holomorphic. Let us now write d X = fˆω where fˆ is a holomorphic function on M with zeros at the branch points of X of the corresponding orders and ω ∈ H r−1/2 (M ) a holomorphic Cn -valued differential without zeros on M . Observing that d X · d X = ω · ω = 0, we see that d G1 · ω = d ζ d Xw · ω + ζ d Xd w · ω + ζ d d Xw · ω + (d f0 G0 + f0 d G0 ) · ω = 0 + 0 + 0 + fQ for some holomorphic quadratic differential Q ∈ H r−1/2 (M ) and some holomorphic function f on M vanishing at qj of order ≥ r(νj + 2) − 3, j = 1, . . . , . According to Corollary 2 of Section 5.12 there exists a holomorphic Cn -valued differential g ∈ H r−1/2 (M ) with g · ω = −Q and hence (d G1 + f g) · ω = 0. By Theorem 2 of Section 5.14 we can find a holomorphic function f˜ on M such that the periods of f (g + f˜ω) are purely imaginary, and hence

p f (g + f˜ω) G2 (p) := Re p0

is a well defined harmonic function of class H r+1/2 (M ). It follows that G := Re G1 + G2 is harmonic, satisfies d G · ω = 0 and by (3) has boundary values of the form (dα) ◦ ξψ+ some H r -function with critical points q1 , . . . , q admitting the representation (4) in Section 5.16. From Lemma 2 of Section 5.16 we then conclude that G|∂M = dα ◦ ξψ + β ◦ ξ for some β ∈ H r (∂M, Rn ). Now we must study the additional relations which define E(α,ξ) ∩ s ). We introduce local T(α,ξ) Nμν (τ ) as a subspace of E(α,ξ) ∩ (Tα Ar × Tξ Hid,ν complex coordinates w = u + iv around each of the branch points p1 , . . . , pk , q1 , . . . , q such that w(pj ) = w(qj ) = 0 and w(∂M ) ⊂ R if w is a coordinate

388


around one of the qj ’s. We are suppressing the dependence of w on the points p1 , . . . , q . s , Let X = Ψ (α, ξ), g = dΨ(α,ξ) β, ψ with (β, ψ) ∈ E(α,ξ) ∩ Tα Ar × Tξ Hid,ν i.e. X and g are harmonic and satisfy the relations (4)

d g · d X = 0, X|∂M = α ◦ ξ,

g|∂M = β ◦ ξ + dα ◦ ξψ.

Since ξ has critical points of orders νj at qj and ψ vanishes at qj at least of order νj − 1, we clearly have ∂λX (qj ) = 0 (λ = 1, . . . , νj ), ∂uλ

(5)

∂λg (qj ) = 0 (λ = 1, . . . , νj − 1), ∂uλ νj +1 ∂ ∂ νj +1 X ξ (q ) = dα ◦ ξ (q ) , j j ∂uνj +1 ∂uνj +1 νj ∂ ψ ∂ νj g (qj ) = dα ◦ ξ (qj ) . ∂uνj ∂uνj

For interior branch points see equations (7) of Section 5.16. Let us now write

(6)

∂X = wμj H, ∂w

H(pj ) = 0,

∂X = wνj H, ∂w

H(qj ) = 0,

where we have suppressed the dependence of the locally defined function H on pj , qj . Comparing (5) and (6) and using Lemma 1 of Section 5.16 we obtain νj +1 ∂ ψ νj Re H(qj ) = dα ◦ ξ (qj ) ∂uνj +1 and hence we may rewrite the relations of (5) for g at a boundary branch point in the form ∂λg (qj ) = 0 (λ = 1, . . . , νj − 1), ∂wλ ∂ νj g Re (qj ) = c Re H(qj ) ∂wνj Re

(7)

with some c ∈ R. Let us now consider formal Taylor expansions H=

m

Hm w m ,

∂g = Gm wm . ∂w m


389

By (7) we have (8)

Re Gm = 0

(m = 0, . . . , νj − 2),

Re Gνj −1 = c Re H0

with some c ∈ R

at each of the boundary branch points. The kernel relation d X · d g = 0 can be expressed as the system H0 · G0 = 0, H0 · G1 + H1 · G0 = 0, ...

(9)

H0 · Gm + · · · + Hm · G0 = 0. From this we see that automatically H0 · G0 = 0 and, provided that G0 = 0, then automatically H0 · G1 = 0, etc. For each of the interior branch points pj we therefore introduce the Hermitian projection Pj : Cn → Cn ,

Pj z = |H0 |−2 (z · H0 )H 0

and, correspondingly, for the boundary branch points qj we let Qj : Rn → Rn denote the orthogonal projection onto the subspace spanned by the vectors Re H0 , Im H0 . Moreover, at each of the interior branch points pj we need the Hermitian projection Pˆj : Cn → Cn whose range is spanned by the vectors H0 , H 0 which by the conformality relations for X are orthogonal in Cn . Let us remark that H0 spans the tangent space of the minimal surface X at pj , qj , viewed as a complex 1-dimensional subspace of Cn or as a real 2-dimensional subspace of Rn respectively. By means of these projections we can now write s ) as the relations defining E(α,ξ) ∩ T(α,ξ) Nμν (τ ) in E(α,ξ) ∩ (Tα Ar × Tξ Hid,ν follows: Lemma 2. The relations ∂λg (pj ) = 0 ∂wλ ∂ μj g (I − Pˆj ) μj (pj ) = 0 ∂w (I − Pj )

(10)

for λ = 1, . . . , μj − 1;

imply

(11)

∂λg (pj ) = 0 for λ = 1, . . . , μj − 1, ∂wλ ∂ μj g ∂ μj +1 X (p ) = c (pj ) with c ∈ C. j ∂wμj ∂wμj +1

Similarly, the relations (12)

(I − Qj ) Im

∂λg (qj ) = 0 ∂wλ

for λ = 1, . . . , νj

390


imply

(13)

∂λg (qj ) = 0 for λ = 1, . . . , νj − 1, ∂wλ ∂ νj g ∂ νj +1 X (q ) = c (qj ) with c ∈ R. j ∂wνj ∂wνj +1

Proof. Let us first consider any of the interior branch points pj . We proceed ∂λg by induction and assume that ∂w λ (pj ) = 0 for λ = 1, . . . , m − 1 with some m, 1 ≤ m ≤ μj . We then get from (9) and (10) that H0 ·

∂mg (pj ) = 0 and ∂wm

∂mg (pj ) = dH 0 ∂wm

with some d ∈ C. It follows immediately that d = 0, proving the first μj − 1 relations in (11). The last relations (10) and (9) give ∂ μj g (pj ) = cH0 + dH 0 ∂wμj

and

H0 ·

∂ μj g (pj ) = 0 ∂wμj

with c, d ∈ C. The relation H0 · H0 = 0 implies d = 0, and hence the last relation in (11) holds. We then come to any of the boundary branch points qj . From (7) and (12) we get ∂λg (qj ) = aλ i Re H0 + bλ i Im H0 ∂wλ

with aλ , bλ ∈ R, λ = 1, . . . , νj − 1,

which, when inserted into (9), leads to the first νj − 1 relations in (13). The last relations in (7) and (12) may be written as ∂ νj g (qj ) = c Re H0 + ai Re H0 + bi Im H0 ∂wνj with a, b, c ∈ R. Inserting this into (9) we obtain ∂ νj g (qj ) ∂wνj = (Re H0 + i Im H0 ) · (c Re H0 + ai Re H0 + bi Im H0 ) = c| Re H0 |2 − b| Im H0 |2 + ai| Re H0 |2 .

0 = H0 ·

Since by the conformality of the minimal surface X we have | Re H0 |2 = | Im H0 |2 , it follows that c = b, a = 0 which is the last relation in (13). It is clear from Lemma 2 that at most 2(n − 1)|μ| − 2k + (n − 2)|ν| linear functionals are needed to define E(α,ξ) ∩ T(α,ξ) Nμν (τ ) as a subspace of E(α,ξ) ∩ s ). Concerning the linear independence of these functionals (Tα Ar × Tξ Hid,ν we prove a slightly weakened result for the moment in so far as we allow the degree of regularity to diminish.


391

ˆ μν (τ )[p, q] Lemma 3. Let X = Ψ (α, ξ) be a minimal surface with (α, ξ) ∈ N and let R(Pj ) and R(Qj ) respectively denote the ranges of the hermitian projections Pj : Cn → Cn and the orthogonal projections Qj : Rn → Rn introduced above. Then the map L defined by ∂λ (14) dΨ(α,ξ) β, 0|pj , L : β → (I − Pj ) j=1,...,k ∂wλ λ=1,...,μj ∂λ (I − Qj ) Im dΨ β, 0| qj (α,ξ) j=1,..., ∂wλ λ=1,...,νj

is surjective from E(α,ξ) ∩ (Tα Ar−s(s+2) × {0}) onto (R(P1 )⊥ )μ1 × · · · × (R(Pk )⊥ )μk × (R(Q1 )⊥ )ν1 × · · · × (R(Q )⊥ )ν . Proof. We treat interior and boundary branch points separately and start with the former ones, where a loss of regularity can still be avoided. By linearity it clearly suffices to construct a harmonic function g such that (15)

g|∂M = (dα) ◦ ξ(ψ) + β ◦ ξ

s , with (β, ψ) ∈ Tα Ar × Tξ Hid,ν

(16)

d X · d g = 0,

(17)

∂λg (pj ) = 0 for λ = 1, . . . , μj , j = 2, . . . , k, ∂wλ

(18)

∂λg (qj ) = 0 for λ = 1, . . . , νj , j = 1, . . . , , ∂wλ

(19)

∂λg (p1 ) = 0 for λ = 1, . . . , κ − 1, ∂wλ

∂κg (p1 ) = G0 , ∂wκ

where κ ∈ {1, . . . , μ1 } and G0 ∈ R(P1 )⊥ are arbitrary. Using the regularity theorem for minimal surfaces we write d X = f0 F where f0 is a holomorphic function on M and F ∈ H r−1/2 (M ) a holomorphic Cn -valued differential without zeros on M . On the domain of the local coordinate w around p1 we define the Cn -valued differential 1 wκ−1 G0 dw. G= (κ − 1)! We remark that, since G0 ∈ R(P1 )⊥ , the quadratic differential F ·G vanishes at p1 of order ≥ κ. Let then η be a smooth function on M with compact support in the domain of the coordinate w around p1 and which is identically 1 in some neighbourhood of p1 . We may also assume that η has compact support

392


in M and that none of the points p2 , . . . , pk , q1 , . . . , q meets the support of η. We then define the smooth global differential Γ = ηG and choose a holomorphic function f on M vanishing of order κ at p1 , of order ≥ μj at the other interior branch points pj , j = 2, . . . , k, and at least of order r(νj + 2) − 3 at the boundary branch points qj , j = 1, . . . , . Since f1 d Γ is of class C ∞ on M there exists a differential γ of class C ∞ on M (see Forster [1]) such that 1 d γ = − d Γ. f It follows that Γ + f γ is holomorphic, and since F · Γ vanishes at p1 of order ≥ κ we may write F · (Γ + f γ) = f Q with some holomorphic quadratic differential Q ∈ H r−1/2 (M ). By Corollary 2 of Section 5.12 there is a holomorphic differential ϕ of class H r−1/2 (M ) with F · ϕ = −Q and thus F · (Γ + f γ + f ϕ) = 0. Using Theorem 2 of Section 5.14 we may then choose a holomorphic function ζ ∈ C ∞ (M ) such that

p [Γ + f (γ + ϕ + ζ(F ))] g(p) := Re p1

is a single-valued harmonic function of class H r (M ). By construction, g fulfills (16) and, around p1 , gw = Γ + terms of order ≥ κ whereas gw vanishes of order ≥ μj at pj for j = 2, . . . , k and of order ≥ r(νj + 2) − 3 at qj for j = 1, . . . , . Hence g satisfies (17), (18) as well as (19). By Lemma 2 of Section 5.16, g|∂M = β ◦ ξ for some β ∈ H r (∂M ), which proves the first part of the lemma concerning interior branch points. Now we must show a corresponding result for any of the boundary branch points, i.e. we must find a harmonic function g, g|∂M = β ◦ ξ for some β ∈ H r−s(s+2) (∂M ), such that the relations (16), (17) (for j = 2, . . . , k), (18) (for j = 1, . . . , ) hold and such that Im

∂λg (q1 ) = 0 for λ = 1, . . . , κ − 1, ∂wλ

Im

∂κg (q1 ) = G0 , ∂wκ

(20)

where κ ∈ {1, . . . , ν1 } and G0 ∈ R(Q1 )⊥ ⊂ Rn are arbitrary. For this purpose let us consider the Taylor expansion H := w−ν1 Xw (w) of order m around q1


393

where m = s(s+2)−3 ≥ s(ν1 +2)−3. We observe that since Re H0 , . . . , Re Hm are linearly independent, the system (9) admits a real solution G0 , G1 , . . . , Gm where G0 is the given vector in (20). By means of these coefficients we define the local holomorphic differential i wκ−1 wλ Gλ (w) dw (κ − 1)! m

G(w) =

λ=0

which is an approximate solution to (16) in the sense that F · G vanishes at q1 of order ≥ m. It is important to notice that by (8) (21)

Re G|∂M = 0.

The remainder of the proof is similar to the case of interior branch points except that we encounter a loss of regularity arising from the Taylor expansion H. We define the differential Γ in an analogous way as above (replacing p1 by q1 ) and choose the holomorphic function f with zeros at pj of orders μj (j = 1, . . . , k) and at qj of order s(s + 2) − 3(j = 1, . . . , ). The differential γ is then determined as above and we can again write F · (Γ + f γ) = f Q with a holomorphic quadratic differential Q ∈ H r−1/2−m (M ). We then define ϕ ∈ H r−1/2−m (M ), ζ ∈ C ∞ (M ) and g ∈ H r+1/2−m (M ) as above, g(q1 ) = 0. It follows from (21) that Re Γ |∂M = 0, and consequently g|∂M satisfies the assumption of Lemma 2 of Section 5.16. Hence, g|∂M = β ◦ ξ for some β ∈ H r−m (∂M ). On the basis of the preceding lemmas it is not difficult to prove ˆ μν (τ ) be a minimal surface. Then we have Theorem 1. Let (α, ξ) ∈ N ˆ μν (τ )) = 2|μ|(n − 1) − 2k + |ν|(n − 1) − . dim(E(α,ξ) /E(α,ξ) ∩ T(α,ξ) N Proof. Since we assume that r ≥ s(s + 3), the derivative D(V K) is well ˆ ) := Tα Ar−s(s+2)+3 × Tξ Hid × H0 (τ ) defined and continuous on T˜(α,ξ) N(τ 2pn × QR Dτ , carrying the H s (∂M )-topology. and has values in T N(α)(τ ) × R r r−s(s+2)+3 ˆ )) is Since Tα A is dense in Tα A it follows that D(V K)(T(α,ξ) N(τ ˆ ˜ dense in D(V K)(T(α,ξ) N(τ )). However, by Theorem 1 of Section 5.14 and ˆ )) is a closed subspace of Theorem 1 of Section 5.15, Z := D(V K)(T(α,ξ) N(τ T N(α)(τ ) × R2pn × QR Dτ of codimension 2|μ| + |ν|. It follows therefore that ˆ )). Let Y be the image of the map L defined by (14). Z = D(V K)(T˜(α,ξ) N(τ On account of Lemma 3 we have dim Y = 2(n − 1)|μ| − 2k + (n − 1)|ν| − .

394


By the definition of Y and Z the map ˆ )→Z ×Y (D(V K), L) : T˜(α,ξ) N(τ is surjective. By density of Tα Ar in Tα Ar−s(s+2)+3 it is already surjective on ˆ ), and hence T(α,ξ) N(τ L : ker D(V K) → Y is also surjective. From Lemmas 2 and 3 it follows that ker D(V K)/ ker L ∼ = Eα,ξ /Eα,ξ ∩ Tα,ξ Nμν (τ ),

proving the theorem. As an almost immediate consequence we obtain Theorem 2. At a minimal surface we have ˆ μν (τ ) = 4|μ| + 3|ν|. Corank D(V K)|T N Proof. Let us consider the commutative diagram ˆ μν (τ ) ∩ E ˆ μν (τ ) T N TN

ι

ˆ ) TN ˆ μν (τ ) ∩ E T N(τ κ

D(V K)

T N(α)(τ ) × R2pn × QR Dτ

D(V K)

ˆ ) E T N(τ

where ι is the inclusion and κ the natural projection. Theorem 1 of Section 5.16 implies that ˆ N ˆ μν = 2n|μ| + (n + 1)|ν| − 2k − l. dim T N/T An easy algebraic exercise shows that ˆ N ˆ μν ∩ E)/(T N ˆ μν /T N ˆ μν ∩ E) = dim T N/T ˆ N ˆ μν . dim(T N/T Thus ind ι = −2n|μ| − (n + 1)|ν| + 2k + , and from Theorem 1 of Section 5.15 we know that ˆ )/E) = −2|μ| − |ν|. ind(D(V K)|N(τ From the functorial property of the index it follows that ˆ μν (τ )) = − ind ι − ind κ − ind(D(V K)|T N(τ ˆ )/E) codim D(V K)(T N = 2(n + 1)|μ| + (n + 2)|ν| − 2k − − dim ker κ with ker κ = E/(E ∩ T Nμν (τ )). Hence the assertion follows from Theorem 1.


395

We are now ready to proceed with the proof of the index theorem. The next lemma should be thought of as motivational; it is an immediate consequence of the implicit function theorem. Lemma 4. Let preserving map from a Hilbert Lτ : Eτ → Fτ be a smooth fibre fibre bundle τ Eτ to a Hilbert space bundle τ Fτ . Assume that at every zero (ω0 , τ0 ) of L the derivative in the fibre direction Dω Lτ is onto. Then the set of zeros of L is a submanifold of τ Eτ . If we wished to apply this lemma to the map ˆ μν (τ ) → T N(τ ) × R2pn × QR Dτ V K(τ ) : N then we must, however, realize that by Theorem 2 the derivative D(V K)(τ ) is not surjective. We therefore proceed as follows. Let X0 = (α0 , ξ0 , 0) ∈ Nμν (τ0 ) ˆ μν (τ0 )). For τ in a be a zero of V K(τ0 ) and define Y0 := D(V K)(τ0 )(TX0 N neighbourhood U (τ0 ) we may then construct a smooth Hilbert-space bundle epimorphism Pτ : T N(τ ) × R2pn × QR Dτ → Y0 × {τ } such that Pτ0 is a projection onto Y0 . We then define a map Kτ : Nμν (τ ) → Y0 × {τ } by Kτ := Pτ ◦ V K(τ ). It follows immediately that DKτ0 [X0 ] is surjective. Hence by Lemma 4 there exists a neighbourhood V (X0 , τ0 ) in Nμν such that Wμν (X0 , τ0 ) := V (X0 , τ0 ) ∩ K −1 ({0} × U (τ0 )) is a submanifold of the bundle Nμν . Moreover, we have T(X0 ,τ0 ) Wμν (X0 , τ0 ) = (DK)−1 ({0} × Tτ0 T(M )).

(22)

With the abbreviation Πμν := Π|Wμν (X0 , τ0 ) we now formulate our main result, Theorem 3. index Πμν = 2(2 − n)|μ| + (2 − n)|ν| + 2k + . Proof. Since K is fibre preserving too, the inclusion (DKτ0 (X0 ))−1 ({0}) "→ (DK(X0 , τ0 ))−1 ({0} × Tτ0 T(M ))

396


is of codimension dim T(M ). Hence, by (22), the functorial properties of the index, and the construction of K we obtain (23)

index Πμν = index DΠ| ker DKτ0 (X0 ) + dim T(M ) = index DΠ| ker D(V K)(τ0 )(X0 ) + dim T(M ).

Now we use the index formula for a product operator L = (L1 , L2 ): index L = index(L2 | ker L1 ) − dim Coker L1 .

(24)

ˆ 0 ) and L2 = D(V K)(τ0 )(X0 )|TX N(τ ˆ 0 ) and observe We set L1 = DΠ|TX0 N(τ 0 that dim Coker L1 = 0 and that L2 | ker L1 is nothing but the partial derivative Dξ (V K)α0 (τ0 ) at X0 = (α0 , ξ0 ). From the construction of V K and Theorem 1 of Section 5.13 we therefore obtain index L2 | ker L1 = index DX V Cα0 (τ0 )|X0 − dim QR Dτ0 = − dim QR Dτ0 , and hence index L = − dim QR Dτ0 . It follows immediately from Theorem 1 that (25)

ˆ μν (τ0 ) = − dim QR Dτ − 2n|μ| − (n + 1)|ν| + 2k + . index L|TX0 N 0

ˆ μν (τ0 ) and with L1 and L2 interUsing (24) again on the subspace TX0 N changed we get from (25) and Theorem 2 that index DΠ| ker D(V K)(τ0 )(X0 ) = − dim QR Dτ0 − 2n|μ| − (n + 1)|ν| + 2k + + dim Coker D(V K)(τ0 )(X0 ) = − dim QR Dτ0 − 2(n − 2)|μ| − (n − 2)|ν| + 2k + . Combining this with (23) and observing that dim T(M ) = dim QR Dτ0 we arrive at the formula of the theorem. For minimal surfaces which are immersed on the boundary (i.e. ν = 0) we obtain a stronger result: Theorem 4. The set of minimal surfaces Mμ0 ⊂ Nμ0 is a submanifold such that the projection map Πμ0 = Π|Mμ0 is of index 2(2 − n)|μ| + 2k. Proof. We remind the reader of the definition of the conformality operator ˆ ) → H s−1 (∂M ) × R2pn × QR D(τ ), K(τ ) : N(τ


397

given at the beginning of Section 5.15. Let X = (α, ξ, τ ) be a minimal surface in Mμ0 . Since ξ is immersed, the maps of the form β ◦ ξ, β ∈ H r (∂M, Rn ) are dense in H s (∂M, Rn ), and it follows then from the proof of Theorem 1 ˆ μ0 (τ ) is dense in the space of all multivalued τ of Section 5.16 that T(α,ξ) N harmonic maps of class H s (∂M ) which satisfy the relations (7) of Section 5.16 with ν = 0. Let us denote this space with Hs [X]. Since by Corollary 1 of Section 5.13 the operator DK(τ )(X) has closed range, it follows therefore that ˆ μ0 (τ )) = DK(τ )(X)(Hs [X]). DK(τ )(X)(T(α,ξ) N Denoting by h the τ -harmonic extension of α ◦ ξ, we immediately see from (7) of Section 5.16 that all functions of the form g = Re f d h where f is an arbitrary holomorphic function on M belong to Hs [X] and, moreover, they satisfy d g · d h = 0. We conclude therefore from Theorem 2 of Section 5.14 that the period map is surjective from the kernel of the two other components of DK(τ ), namely, (26)

g →

∂g ∂h ∂g ∂h · + · ∂N ∂T ∂T ∂N

and (27)

g → Pτ (d g · d h),

g ∈ Hs [X],

with Pτ : QD(τ ) → QR D(τ ) the orthogonal projection. It follows therefore that DK(τ )(X)(Hs [X]) = (range of (26), (27)) × R2pn . Since the right-hand side of (26) may be rewritten as Im(d g d h|∂M )/ds2 , where ds2 is the arc-length form on ∂M , and since for any quadratic differential q QD(τ ) q → (Im(q|∂M )/ds2 , Pτ (q)) is a one-to-one map, we see that the range of (26), (27) is isomorphic to the range of Hs [X] g → d g · d h ∈ QD(τ ). In view of (7) of Section 5.16 and Corollary 2 of Section 5.12 this range consists of all quadratic holomorphic differentials of class H s−1 (M ) which have zeros at the branch points pj of order μj , j = 1, . . . , k. It follows therefore that DK(τ ) ˆ μ0 (τ ) in a neighbourhood of any zero of K(τ ). By has constant corank on N ˆ μ0 (τ ) the rank theorem, the zeros of K(τ ) form a submanifold Mμ0 (τ ) ⊂ N whose tangent space is the kernel of DK(τ ). It is, however, obvious that ker DK(τ ) = ker DV K(τ ). Hence we may directly apply (25) in the special case ν = 0. Observing that Mμ0 = τ Mμ0 (τ ), this proves the theorem.

398


Remark 2. Using weighted Sobolev spaces like in U. Thiel [3] one can probably prove a manifold structure also for Mμν with ν = 0. By means of the Sard–Smale theorem we immediately obtain from Theorem 4: Corollary 1. Let N by a positive integer and choose r > N (N + 3). Then there is a subset Ar1 of first Baire category with Ar1 ⊂ Ar such that any curve α in Ar \Ar1 has the following property: If X : M → Rn is a minimal surface with the topological boundary map X : ∂M → α(∂M ) and with a total branching order |μ| + |ν| ≤ N , then we have (i) X has at most simple interior branch points and no boundary branch points; (ii) X is immersed provided that n ≥ 4; (iii) X is non-degenerate for Π, i.e. Π : Mμ0 → Ar is a local diffeomorphism in a neighbourhood of X ∈ Mμ0 ; (iv) X is an isolated point of |μ|+|ν|≤N Mμν ∩ Π −1 (α); (v) If M is a disk, then the a priori estimates imply that for n ≥ 3 the set of all such minimal surfaces X is finite (the generic finiteness result of B¨ ohme and Tromba [1]). We remark that (iv) follows from (iii) since a surface with k interior branch points of orders μ1 , . . . , μk cannot be the limit of surfaces with more branch points nor with branch points of higher order. Conversely, for n = 3 one can see from the Weierstrass representation that a minimal surface with k simple interior branch points cannot be the limit of minimal surfaces with less than k branch points. Remark 3. The Gauss–Bonnet formula for branched surfaces (cf. Section 2.11 of Vol. 2) gives an estimate for the total branching order of a minimal surface in terms of the total curvature of its boundary curve. Hence, if we restrict to Jordan curves of total curvature < κ where κ is an arbitrary positive number, then in Corollary 1 the number r can be chosen in terms of κ, and the statements of the corollary hold for all minimal surfaces spanned by a generic Jordan curve α ∈ Ar of total curvature < κ. From this corollary, (iv), we can of course conclude that a generic Jordan curve bounds only finitely many minimal surfaces within some compact set of surfaces. We give two examples; the first one refers to the well known Douglas condition, the second one to incompressible surfaces. Corollary 2. (i) Let κ be a bound for the total curvature of α, and choose the number r accordingly such that Corollary 1 is applicable. For α ∈ Ar let δ(α, p) denote the infimum of the area of all surfaces of genus less than p which span α. Then, for any ε > 0, a generic curve in Ar of total curvature < κ spans only finitely many genus p-minimal surfaces with area less or equal to δ(α, p) − ε.

5.18 Scholia

399

(ii) Let K be a compact body in R3 whose boundary has a mean curvature ≥ 0 with respect to its interior normal, and let κ and r be as above. Then a ◦

generic curve α ∈ Ar inside K and of total curvature less than κ spans only finitely many minimal surfaces inside K which induce a monomorphism of the fundamental group of M into that of K.

5.18 Scholia The theory of linear Fredholm operators began at the turn of the twentieth century with the work of E.I. Fredholm and D. Hilbert. The famous Fredholm–Alternative Theorem is, at its roots, the calculation of the index of a linear operator which is a perturbation of the identity by a compact integral operator. The first high points of this theory were Hilbert’s six publications “Grundz¨ uge einer allgemeinen Theorie der linearen Integralgleichungen” (1904–1910; a collected edition was published in 1912). In 1927, the encyclopedia article by E. Hellinger and O. Toeplitz, “Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten” appeared. This reported on the state of the theory of linear integral equations. At that time the development of the abstract linear and nonlinear functional analysis was already in full swing. Besides the work of F. Riesz and S. Banach, the basic contributions of J.P. Schauder and later the seminal paper by J. Leray and J.P. Schauder showed how functional analysis can be used for the study of problems for partial differential equations. The treatise of T. Kato (1966) on the “Perturbation theory for linear operators” provides an excellent introduction into the theory of linear Fredholm operators and their index. The theory of linear Fredholm maps culminated with the proof of the very general Atiyah–Singer Index Theorem (first proposed to Atiyah by Smale), linking the index of an elliptic system of partial differential equations on sections of a vector bundle with a topological index. The definition of a nonlinear Fredholm map and its index was first given by S. Smale. The index theorems in this chapter concerning minimal surfaces which solve Plateau’s problem are the first application of Smale’s theory to global analysis. Besides the work of R. Böhme and A. Tromba [2] and F. Tomi and A. Tromba [6] we refer to the index theorems of K. Sch¨ uffler [1–4,6,8] and U. Thiel [1–3]. The work of Sch¨ uffler is restricted to minimal surfaces without boundary branch points. In [1] he derives a simple explicit sufficient criterion that a k-fold connected minimal surface with only simple interior branch points is isolated and stable with respect to perturbations (cf. Theorem 9 in Section 5.8). The paper [2] furnishes the same result for disk-type H-surfaces. In [3], Jacobi fields are characterized by decomposing them into tangential and normal parts. For minimal surfaces of genus 1 with k boundary components,

400


without boundary branch points, and with interior branch points of a given multiplicity, Sch¨ uffler in [6] defines a manifold structure and computes the associated index, and in [8], the corresponding result for minimal surfaces on Möbius strips is obtained. Ursula Thiel [3] shows that the set of disk-type minimal surfaces with interior and boundary branch points of given multiplicities has the structure of a differentiable manifold. Earlier, Böhme and Tromba had proved such a structure only for surfaces without boundary branch points, while the set of minimal surfaces with boundary branch points was only understood as a subset of certain local manifolds without a specific meaning, and with respect to these the Fredholm index could be computed. In her paper [2], U. Thiel defines a differentiable manifold structure on the set of k-fold connected minimal surfaces which only have interior branch points of given multiplicities, and the associate index is computed. A corresponding result is derived in Sch¨ uffler’s paper [4], which presents a global-analytic treatment of the Douglas problem. This method works only in R3 , but has the advantage to yield an explicit criterion for the nondegeneracy of minimal surfaces.

Chapter 6 Euler Characteristic and Morse Theory for Minimal Surfaces

The main goal of this chapter is to define an Euler characteristic for minimal surfaces solving Plateau’s problem in Rn , n ≥ 3, and to derive its main properties. The most interesting case n = 3 is much more difficult than the case n ≥ 4 since, according to the index theorem, for n = 3 the nondegenerate minimal surfaces are not generic. Nevertheless it will be useful to have a theory for n ≥ 4 as one can connect R3 -contours via homotopies in Rn with Rn -contours, n ≥ 4, bounding only nondegenerate minimal surfaces. In Section 6.1 we recall some properties of Fredholm vector fields W : M → T M on a Hilbert manifold M, and we define a special class of Palais–Smale vector fields. For such a vector field V one can define an Euler characteristic χ(V ) if its zeros are nondegenerate, and for two vector fields V, W of this kind one has χ(V ) = χ(W ) if V and W are properly homotopic. In the following section these ideas are applied to sufficiently smooth embeddings α : S 1 → Rn realizing a closed Jordan curve Γ = α(S 1 ). This leads to the definition of gradient vector fields Wα : Nα → T Nα defined on the homotopically nontrivial component Nα of sufficiently smooth mappings ∂B → Γ such that the zeros of Wα correspond to the boundary maps X : S 1 → Rn of minimal surfaces bounded by Γ . The gradient is taken in a natural way with respect to Dirichlet’s integral Eα . The first problem consists in proving regularity for Wα . Then it can be shown that Wα is a Palais–Smale vector field. Next, in Section 6.3, the Euler characteristic χ(Wα ) for a vector field is defined, first for a “generic” curve α : S 1 → Rn , n ≥ 4, which bounds only finitely many minimal surfaces, and it is shown that χ(Wα ) = 1 in this case. This is Morse’s last (in)equality for Plateau’s problem. This approach cannot be applied to R3 since in this case the generic nondegeneracy of minimal surfaces breaks down. In Sections 6.4–6.7 it is shown how this difficulty can be overcome. This leads in Section 6.7 to the definition of a local winding number deg(Wα , X, 0) of the minimal surface vector field ˆ with the boundary map X : S 1 → Γ . If X ˆ Wα at a generic minimal surface X


401

402

6 Euler Characteristic and Morse Theory for Minimal Surfaces

is immersed then deg(Wα , X, 0) = ±1. This leads to a Morse Equality: If α is a generic curve in R3 , then (by the Index Theorem) the curve Γ = α(S 1 ) is ˆ1, . . . , X ˆ k satisfying a three-point spanned by finitely many minimal surfaces X ˆ k |S 1 satisfy condition, and their boundaries Xk := X k

deg(Wα , Xj , 0) = 1.

j=1

This implies a version of the mountain-pass theorem presented in Section 6.7 of Vol. 1: If a curve Γ = α(S 1 ) is spanned by two strict, local minimizers of Dirichlet’s integral and α is generic in R3 , then there is a third, unstable, immersed minimal surface spanning Γ . In Section 6.8.1 of the Scholia we give a brief report on the development of a Morse theory for minimal surfaces in Rn . It will be pointed out that there is such a theory for n ≥ 4, but not for n = 3. However, in Section 6.8.2 we prove generic nondegeneracy for closed two-dimensional minimal surfaces of genus greater than one in an n-dimensional manifold of strictly negative sectional curvature. Together with the equivariant Morse theory developed by N. Hingston [1] this leads to a Morse theory in that context.

6.1 Fredholm Vector Fields Let H be a separable Hilbert space. By Lc (H) we mean the linear operators T = I + K which are of the form identity + compact, and by GLc (H) we mean those operators in Lc (H) which are invertible. Let π0 (GLc (H)) denote the set of components of GLc (H), necessarily itself a group. Although the full general linear group is contractible, the following result permits a considerable amount of finite dimensional topology to be carried over to Hilbert (or Banach) manifolds. Theorem 1. π0 (GLc (H)) = Z2 (the group of 2-elements). Proof. It will only be necessary to show that GLc (H) has two components. Let I be the identity map. Write H = H1 ⊕ R where R is a one-dimensional subspace. Define J ∈ GLc (H) by J(x, y) = (x, −y),

(x, y) ∈ H1 ⊕ R.

We will show that the two components of GLc (H) are determined by I (the identity component) and J. Let {πn } be an increasing sequence of orthogonal projection maps, projecting H onto n-dimensional subspaces En (En ⊂ En+1 · · · ) and which converge pointwise to the identity (πn x → x) with E1 identified with R above. Let T ∈ GLc (H). Then T = I + K, K compact. A standard result in functional analysis states that πn ◦ K converges “in norm” to K. We first

6.1 Fredholm Vector Fields

403

show that there are at least two components by proving that I and J are in different components of GLc (H). So suppose the opposite. Then there is a continuous path T (t), 0 ≤ t ≤ 1, with T (t) = I + K(t), T (0) = I, T (1) = J. Assume now that, given > 0, for N sufficiently large πN K(t) − K(t) < for all t ∈ R. Choose so small that I + πN ◦ K(t) ∈ GLc (H). Then I + πN K(t) |EN ∈ GL(EN ), where GL(EN ) denotes the general linear group of EN . Clearly I and J restrict to elements of GL(EN ) with I + πN ◦ K(0) = I, I + πN ◦ K(1) = J, a contradiction, since on EN , det I = 1, det J = −1. We now show that an arbitrary T ∈ GLc (H) can be path-wise connected to either I or J, and this will complete the proof of Theorem 1. Let λi , i = 1, . . . , n be the finite number of points in the spectrum of K which are less than −1. The following facts are well known: For each i there is a least integer ni < ∞ with Ei = Range(K −λi I)ni +k = Range(K − λi I)ni and Fi = Null(K − λi I)ni +k = Null(K − λi I)ni for all integers k. Here Null(K − λi I)ni denotes the null space or kernel of (K − λi I)ni . In addition, Ei ⊕ Fi = H, and Ei , Fi are invariant subspaces under the action of K, with dim Fi < ∞. Let πi : H → Ei be the natural projection given by this decomposition. Then πi commutes with every operator which commutes with K, in particular πi πj = πj πi for all i, j. We note that P = Πj πj is a projection operator which commutes with T . Let E = Null(I − P ) and F = Null P . Then H =E⊕F with E and F invariant subspaces of T and dim F < ∞. On E, the operator K has no eigenvalue which is less than minus one so that the path (I +τ K)|E, τ ∈ [0, 1], is a path in GL(E) between (I + K)|E and the identity operator on E. Since dim F < ∞ there is either a path between (I + K)F and the identity of F or a map which is the identity on a subspace of F of codimension one and minus the identity on a complement of this subspace in F . Putting these facts together we see (I + K) can either be connected to the identity operator or to an operator J.

404


Denote the component group of the identity by GL+ c (H) and its comple(H). ment in GLc (H) by GL− c Let M be a Banach (or Hilbert) manifold and W : M → T M (its tangent bundle) a C 1 -vector field. W is said to be Fredholm if, whenever W (p) = 0, its derivative DW (p) : Tp M → Tp M is Fredholm, and Palais–Smale if it belongs to Lc (Tp M). W is proper if the zero set {p : W (p) = 0} is compact. A proper homotopy between vector fields W and V is a C 1 -map F : M × [0, 1] → T M such that for each t ∈ [0, 1], the map p → F (p, t) is a C 1 -vector field on M, the set of zeros {(p, t) : F (p, t) = 0} is compact, and F (·, 0) = W,

F (·, 0) = V.

The vector fields W and V are then said to be properly homotopic. On a finitedimensional compact manifold any two vector fields W and V are properly homotopic via the linear homotopy tW + (1 − t)V . This is why the Euler characteristic of a vector field on a compact manifold without boundary is a topological invariant. Its equality with the Euler characteristic defined via homology follows from a Morse-theoretic argument. We now define the Euler characteristic of a Palais–Smale vector field whose zeros are nondegenerate. A zero p of V is nondegenerate if DV (p) ∈ GLc (Tp M). Nondegenerate zeros must be isolated. We define the sign of DV (p) by 1 if DV (p) ∈ GL+ c (Tp M), sgn DV (p) = −1 if DV (p) ∈ GL− c (Tp M). If V is proper and the zero set Zero(V ) consists of only nondegenerate zeros, then Zero(V ) is, in fact, a finite set. We then define the Euler characteristic χ(V ) by sgn DV (p). (1) χ(V ) := p∈Zero(V )

In Tromba [7] the following result is shown: Theorem 2. If the two Palais–Smale vector fields V and W both have nondegenerate zeros and are properly homotopic then χ(V ) = χ(W ). We will be adapting arguments in the proof of Theorem 1 in order to define the Euler characteristic for a gradient vector field associated to Plateau’s problem.

6.2 The Gradient Vector Field Associated to Plateau’s Problem

405

6.2 The Gradient Vector Field Associated to Plateau’s Problem As in Chapter 6 of Vol. 2 we again take the point of view of Jesse Douglas and consider minimal surfaces as critical points of Dirichlet’s integral within the class of harmonic surfaces B → Rn , n ≥ 2, that are continuous on B and map the boundary ∂B = S 1 of the unit disk B homeomorphically onto a closed Jordan curve Γ of Rn . Therefore the admissible boundary maps ∂B → Γ are viewed as the primary objects while their harmonic extensions B → Rn are of secondary nature; hence we again change notation as in Vol. 2, Chapter 6: An admissible boundary map is denoted by X : ∂B → Γ , whereas its harmonic ˆ i.e. X ˆ ∈ C 0 (B, Rn ) ∩ C 2 (B, Rn ) solves extension to B is called X, ˆ = 0 in B, ΔX

ˆ X(w) = X(w)

for w ∈ ∂B.

More generally, the symbol ∧ will denote the harmonic extension to B of any continuous mapping defined on ∂B. In the sequel we assume that Γ is the image of a C r+7 -embedding α : S 1 → Rn , r ≥ 2. According to Section 6.3 of Vol. 2 the mappings X ∈ HΓ2 (∂B, Rn ) ˆ : B → Rn spanning Γ is a C r -manifold, and their harmonic extensions X 5/2 r n form a C -manifold denoted by HΓ (B, R ). Let Nα be any homotopically nontrivial component of X ∈ HΓ2 (∂B, Rn ), i.e. X ∈ Nα if X is not homotopic (as a map from S 1 = ∂B to Γ ) to a constant. Then Dirichlet’s integral Eα : Nα → R, defined by 1 ˆ u |2 + |X ˆ v |2 ) du dv, Eα (X) := (|X 2 B is of class C r . Now we wish to define a gradient vector field Wα : Nα → T Nα whose zero set consists of “boundaries” of minimal surfaces, and then we shall define and calculate its Euler characteristic χ(Wα ). In order to discuss a “gradient” we need a Riemannian metric; the natural choice is the one induced by Dirichlet’s integral. We define

·, ·X : TX Nα × TX Nα → R by (1)

V, W X :=

ˆ du dv ∇Vˆ · ∇W B

where ˆ u + Vˆv · W ˆv ˆ := Vû · W ∇Vˆ · ∇W

406


ˆ and W ˆ u, W ˆ v are defined as and W ˆ u := ∂ W (X), W ∂u

ˆ := W W (X),

ˆ v := ∂ W W (X). ∂v

ˆ (X) and DW ˆ (X) instead of W ˆ and DW ˆ if we Later on we shall also write W want to stress the dependence of W on X. Here V (w), W (w) ∈ TX(w) Γ , i.e. ˆ are the harmonic extensions of V, W . We call (1) V, W : S 1 → Rn , and Vˆ , W a weak Riemannian metric, i.e. ·, ·X induces a weaker H 2 -topology on Nα than the defining H 2 -topology. We now have the following Theorem 1. There is a uniquely defined C r−1 -smooth vector field Wα : Nα → T Nα which is a gradient of Eα with respect to (1); i.e.: For each X ∈ Nα we have ˆ α ) du dv for all V ∈ TX Nα . (∇Vˆ · ∇W (2) DEα (X)V = B

Proof. Fix X ∈ Nα and define the functional Φ : TX Nα → R by ˆ − ∇Vˆ ) · (∇X ˆ − ∇Vˆ ) du dv. (3) Φ(V ) := (∇X B

The functional Φ is weakly lower semicontinuous and therefore Φ has an absolute (necessarily unique) minimizer Wα (X) ∈ H 1 (B, Rn ) (with H s := H2s ) with Wα (X)(w) ∈ TX(w) Γ for almost all w ∈ S 1 . Uniqueness follows from Lemma 1 below. / H 2 (S 1 , Rn ), First we only have Wα (X) ∈ H 1/2 (S 1 , Rn ), and if Wα (X) ∈ then X → Wα (X) would not be a vector field on Nα . The goal now is to show: (i) Wα (X)|S 1 ∈ H 2 (S 1 , Rn ) (regularity), (ii) X → Wα (X) is C r−1 -smooth, implying that Wα is in fact a vector field on Nα . Fix p ∈ Γ . Let Ω(p) : Rn → Tp Γ denote the orthogonal projection of Rn onto the tangent space of Γ at p; p → T1 (p), a unit tangent field to Γ , and p → Tj (p), j = 1, . . . , n, an orthonormal frame field along Γ . Lemma 1. The minimizer Wα weakly satisfies the mixed elliptic Neumann– Euler–Lagrange system ˆ α = 0. 1. ΔW 2. Tj (X(w)) · Wα (X(w)) = 0, j = 2, . . . , n (this just says that Wα (w) ∈ TX(w) Γ for almost all w ∈ S 1 ). ˆ

ˆ

Wα X 3. Ω(X(w)) ∂∂r = Ω(X(w)) ∂∂r on S 1 , where

A solution to 1.–3. is unique.

∂ ∂r

denotes the radial derivative.


407

Proof. Since Wα is defined by harmonic extension, 1. is automatically satisfied. Since Wα (X) yields the minimum (in H 1/2 (S 1 , Rn ) over V ∈ TX Nα ), 2. is ˆ α , as a minimum must satisfy the variational condition satisfied. Finally, W ˆ du dv = 0 ˆ − ∇Wα ) · ∇h (∇X B

for all h ∈ H 1/2 (S 1 , Rn ), h(w) ∈ TX(w) Γ for a.e. w ∈ S 1 . Integrating the above by parts (formally) yields 3. Now if Wα is another solution then we obtain ∇(Wα − Wα )(X) · ∇(Wα − Wα )(X) du dv = 0. B

Hence Wα − Wα (X) is a constant function in H 1 (B, Rn ) over a homotopically nontrivial map X, and thus must be zero. Our goal now is to prove regularity, i.e. that Wα ∈ H 2 (S 1 , Rn ). Towards this goal we have: Lemma 2. Suppose X ∈ H s (S 1 , Rn ) ∩ Nα , 2 ≤ s ≤ r. Denote by (TX Nα )k the H k -vector fields over X, 1 ≤ k ≤ s, (TX Nα )k := V ∈ H k (S 1 , Rn ) : V (w) ∈ TX(w) Γ for all w ∈ S 1 . Then the linear operator LX (V ) := Ω(X)

(4)

∂ Vˆ ∂r

is an isomorphism from the H k -vector fields over X to the H k−1 -vector fields over X. Proof. First note that the range of LX is dense in the H 0 (:= L2 )-vector fields over X; for if not, there is an h ∈ (TX Nα )0 with h = 0 and

∂ Vˆ · h dθ = 0 for all V ∈ (TX Nα )1 , Ω(X) ∂r S1 and by integrating by parts we get ˆ du dv = 0 for all V ∈ (TX Nα )1 ∇Vˆ · ∇h B

ˆ is constant. Since h(w) ∈ TX(w) Γ for a.e. w ∈ S 1 , we obtain implying that h ˆ = 0, a contradiction. h Let U ∈ (TX Nα )k−1 be arbitrary. Now choose a sequence V m ∈ (TX Nα )k with

408


(5)

Ω(X)

∂ Vˆ m →U ∂r

in L2 (S 1 , Rn ).

ˆ · Vˆ m − Vˆ m . Clearly ψm = 0 on S 1 . Define ψ m := Ω(X) Moreover · ∇Vˆ m Δψ m = 2∇Ω(X)

(6)

denotes the gradient of the matrix entries of Ω(X), the matrix where ∇Ω(X) defined with respect to the standard basis of Rn . Now (5) implies that {Vˆ m } is a bounded sequence in H 1 (B, Rn ). To see this note that (5) implies that ˆ du dv ≤ const ˆ · ∇U ˆ du dv ∇Vˆ m · ∇U ∇U B

B

yielding that

∇Vˆ m · ∇Vˆ m du dv ≤ const

B

ˆ · ∇U ˆ du dv < ∞. ∇U B

Thus, by the compact inclusion of H 1/2 (S 1 , Rn ) into L2 (S 1 , Rn ), we may assume that V m |S 1 converges in L2 . We also have, by (6), that ψ m is bounded in H 2 (B, Rn ), and thus we may assume that ψ m |S 1 converges in H 1 (S 1 , Rn ). Now on S 1 , (7)

ˆ ˆm ˆm ∂Ω ∂ψ m ˆ ∂V − ∂V , = ·Vm+Ω ∂r ∂r ∂r ∂r ˆm

V converges in L2 , and hence V m converges and this converges in L2 . Thus ∂ ∂r 1 1 n ∗ 1 1 now in H (S , R ) to V ∈ H (S , Rn ), where

Ω(X)

∂V ∗ = U ∈ (TX Nα )k−1 . ∂r

Denote the limit of ψ m by ψ ∗ . Then ⎧ ∗ ˆ ⎪ · ∇Vˆ ∗ , ⎨ ∇ψ = 2∇Ω(X) (8) ˆ ∂Ω ∂ψ ∗ ∂V ∗ ⎪ ⎩ = ·V∗+U − , ∂r ∂r ∂r and (7) now implies that if k = 1 we are done (V ∗ ∈ H 1 (S 1 , Rn )). If k > 1 we can boot-strap to get V ∗ ∈ H k (S 1 , Rn ). This proves that LX is surjective. But LX is also injective (a constant vector cannot be tangent to a non-trivially homotopic X). Thus, by the open mapping theorem, LX is an isomorphism, and the proof of the lemma is complete. Corollary 1. Wα (X) ∈ H 2 (S 1 , Rn ).


Proof. We have

(9) ˆ ∂X ∂r

Wα (X) =

L−1 X

ˆ ∂X Ω(X) ∂r

409

,

∈ H 1 , and Lemma 2 yields the result.

To complete the proof of Theorem 1, we need to show that Wα as a function of X is smooth. Since Γ is C r+7 -smooth, Ω is C r+6 as a function of p ∈ Γ . Thus, X → Ω(X) is C r+2 . This gives that X → LX is smooth. Then (9) and the fact that LX is an isomorphism immediately implies that X → Wα (X) is C r -smooth. We are now ready for Theorem 2. The zeros of Wα on Nα are minimal surfaces. Furthermore Wα is Palais–Smale. Proof. Suppose Wα (X) = 0. Then, since (10)

Ω(X)

ˆ α (X) ˆ ∂W ∂X = Ω(X) ∂r ∂r

ˆ ˆ X X ˆ w } = 0 and ˆw · X · ∂∂r = 0 on S 1 . This implies that Im{w2 X we see that ∂∂θ ˆ ˆ ˆ thus Xw · Xw = 0, and X is a minimal surface. For the proof of the second assertion we need

Lemma 3. If Wα (X) = 0, X is C r+6,σ , 0 < σ < 1.

Proof. See Chapter 2 of Vol. 2.

Continuing with the proof of Theorem 2 we differentiate (10) in the direction of h ∈ TX Nα obtaining DΩ(X)[h]

ˆα ∂ ∂W + Ω(X) DWα (X)[h] ∂r ∂r

= DΩ(X)[h]

ˆ ˆ ∂h ∂X + Ω(X) . ∂r ∂r

Since Wα (X) = 0 we see that Ω(X)

ˆ ∂ ∂X DWα (X)h − h = DΩ(X)[h] ∈ H 2. ∂r ∂r

Thus

ˆ ∂X ∈ H 3 (S 1 , Rn ). ∂r Since H 3 is compactly included in H 2 it follows that DWα (X) ∈ LX (TX Nα ), i.e. Wα is Palais–Smale. DWα (X)h − h = L−1 X DΩ(X)[h]

410


We now conclude this discussion with a look at the conformal group of the disk and its relation to Wα . The conformal group G of the disk is the set of biholomorphic maps of the disc to itself. All elements of G are of the form g(w) = c

w−a , 1 − aw

a, c ∈ C, |a| < 1, |c| = 1.

Thus G is a noncompact three-dimensional Lie Group. G acts continuously (but not differentiably) on Nα via the map Lg ˆ = X(g(w)). ˆ Lg (X) 5

ˆ → Lg (X) ˆ is linear) on H 2 (B, Rn ) as well. We also Clearly Lg acts linearly (X Γ have the map g → ϕX (g) = Lg (X) whose image OG (X) is the orbit of G through X. If X ∈ H 3 (S 1 , Rn ), OG (X) is a smooth, immersed submanifold of Nα and an elementary calculation, carried out in Section 6.1 of Vol. 2, shows that the tangent space Tid OG (X) to OG (X) at the identity is spanned by (11)

Xθ , Xθ cos θ, Xθ sin θ.

If X ∈ Nα , but not in H 3 , then OG (X) ⊂ Nα is defined but Tid OG (X) is now only a subspace of H 1 (S 1 , Rn ) and not of TX Nα , as is clear from (12). Nevertheless Xθ , Xθ cos θ, and Xθ sin θ are still vector fields over X. It is well known that Dirichlet’s integral is invariant under G, i.e. (12)

Eα (ϕX (g)) = Eα (Lg (X)) = Eα (X);

implying that Eα is constant along orbits. Thus, if we have one critical point X, we have a whole three-dimensional family of critical points, namely OG (X). Similarly, the Riemannian metric is invariant under G. Noting that, since X → Lg (X) is linear, we may identify Lg with its derivative. Thus if U, V ∈ TX Nα , (13)

Lg U, Lg V Lg (X) = U, V X ,

cf. (1), and (14) says that G acts as a group of isometries of Nα with respect to the weak Riemannian structure. Let [Tid OG (X)]⊥ denote the orthogonal complement of Tid OG (X) in TX Nα . Then we have: Theorem 3. At a minimal surface X ∈ Nα , the derivative DWα (X) maps [Tid OG (X)]⊥ into itself, and Tid OG (X) is contained in the kernel of DWα (X) as well as in the cokernel. Proof. For any X ∈ Nα , g → Eα (ϕX (g)) is constant and so the derivative at id is 0. Thus

6.3 The Euler Characteristic χ(Wα ) of Wα

DEα (X)h = 0

411

for all h ∈ Tid OG (X).

Consequently (14)

DEα (X)h = Wα (X), hX = 0.

Since X → Wα (X) is smooth, it necessarily follows that at a minimal surface X D2 Eα (X)(v, h) = DWα (X)v, hX for all v. As the Hessian (and hence DWα (X)) is symmetric, the result follows. Note that (14) implies that Wα (X) is “orthogonal” to the orbits OG (X) at the identity of G, and the same calculation shows that it is always orthogonal. This, and the fact that G acts as a group of isometries, could lead one to conjecture that it might be possible to develop a Morse theory for minimal surfaces on the quotient manifold Nα /G using the flow of Wα . Such an approach seems problematic since it is not clear that the flow of Wα goes on for all time, or whether under the flow the monotonicity of an initial value X is preserved. Critical points that are not monotonic are beyond the scope of the index theorem in Chapter 5 and are thus not approachable by methods of Morse theory. Using a different approach, Struwe [4] was able to set up a Morse theory for minimal surfaces in Rn with n ≥ 4 while the case n = 3 cannot be handled by his methods, since in R3 the generic curve does not bound only nondegenerate surfaces. This follows from the index theorem for genus zero surfaces, taking Sch¨ uffler’s observation into account; see Chapter 5. We now develop a “degree theory” which works for all n ≥ 3.

6.3 The Euler Characteristic χ(Wα ) of Wα Let X ∈ Nα be a critical point of Eα . Then, by Lemma 3, X is of class C r+6,σ , and the orbit OG (X) of the conformal group through X is an immersed submanifold of Nα . By Theorems 2 and 3 in the last section we know that DWα (X) ∈ LC ([Tid OG (X)]⊥ ). We say that X is a nondegenerate critical point if DWα (X) ∈ GLC ([Tid OG (X)]⊥ ). From Chapter 6 in Vol. 2 we know that forced Jacobi fields are always in Ker DWα (X), and thus nonmonotonic X can never be nondegenerate since they must have boundary branch points. However, when solving Plateau’s problem we always assume that X be monotonic as a map from S 1 to α(S 1 ).

412


For this reason we shall restrict ourselves to the set ∗ Nα := {X ∈ Nα : X is homotopic to α, X : S 1 → α(S 1 ) is monotonic}. ∗ Nondegenerate critical points will then be in the interior of the closed set Nα . If n = 3, the index theorem implies that nondegenerate minimal surfaces are not generic. If n ≥ 4 there is an open dense set of curves with the property ∗ are in fact nondegenerate. that if α is in this set, all minimal surfaces X ∈ Nα ∗ We say that a minimal surface X ∈ Nα is isolated if its orbit OG (X) is isolated in Nα . If we impose a three-point condition then X is isolated in the ⊂ Nα be the set of X ∈ Nα such that X traditional sense. To this end let Nα maps three pre-assigned distinct points ζ1 , ζ2 , ζ3 to three pre-assigned distinct points p1 , p2 , p3 in α(S 1 ). For the record we state is a submanifold of Nα “transverse” to the orbits of the Theorem 1. Nα conformal group at each minimal surface X that is free of boundary branch points, i.e.: = TX Nα . Tid OG (X) ⊕ TX Nα

(1)

Proof. If we “define” TX Nα = {Y ∈ TX Nα : Y (ζi ) = 0} then the construction with the structure of a submanof the manifold structure on Nα provides Nα ifold with TX Nα as its tangent space. Since TX Nα has codimension three and Tid OG (X) has dimension 3, we need only that Tid OG (X) ∩ TX Nα = {0}.

But by (12) in Section 6.2 we have: If Y is in this intersection then Y = aXθ + bXθ cos θ + cXθ sin θ. If ζj = eiθj , then (2)

0 = Y, Xθ Rn = a Xθ 2 + b Xθ 2 cos θj + c Xθ 2 sin θj

must vanish for j = 1, 2, 3. For convenience, by invariance under G, we may assume {θj } = (0, π/4, π/2) then (2) immediately implies that a = b = c = 0, completing the proof. has been called S. Remark. Note that in Chapter 5 the submanifold Nα

For n ≥ 3, the index theorem implies that the generic curve α has the ˆ spanning α(S 1 ) are free of boundary property that all minimal surfaces X ∗ ˆ with X ∈ Nα are diffeobranch points, and therefore all minimal surfaces X 1 1 morphisms of S onto α(S ). The inverse function theorem implies that all nondegenerate minimal surfaces are isolated. In defining the Euler characteristic we begin with the simpler case n ≥ 4. Let α : S 1 → Rn be a generic curve. Then by the index theorem and a priori


413

ˆ1, . . . , X ˆ l (actually estimates there are a finite number of minimal surfaces X ∗ a finite number of orbits) with boundary values X1 , . . . , Xl in Nα . Define the sign of DWα (Xj ) or ⊥ +1 if DWα (Xj ) ∈ GL+ C ([Tid OG (Xj )] ), sgn DWα (Xj ) = ⊥ −1 if DWα (Xj ) ∈ GL− C ([Tid OG (Xj )] ). We then introduce the Euler characteristic χ(Wα ) by (3)

χ(Wα ) :=

l

sgn DWα (Xj ).

j=1

As G acts as a group of isometries with respect to the Riemannian metric defining Wα as a gradient, and since Eα is invariant under G, (3) does not depend on the choice of Xj in any given orbit. Our first main results in this section are an extension of the Euler characteristic theory of vector fields on finite-dimensional manifolds. Theorem 2. χ(Wα ) does not depend on the choice of generic α. Theorem 3. χ(Wα ) = 1. We begin to outline the proof of these two theorems. It is well known that in Rn , n ≥ 4, there are no knots. Thus any two curves α0 and α1 are isotopic, that is homotopic through embeddings. Assume α0 and α1 are generic and that αt is an isotopy between them. We would like to “adjust” this isotopy somewhat, but before doing this we need some results. Let Nα , N := α

N∗ :=

∗ Nα ,

α

N :=

α

Nα ,

A=

α,

α

α ∈ H r+8 (S 1 , Rn ) an embedding. Then N, N∗ and N are trivial fibre bundles over A. (For this and the following see Chapter 5.) The projection map π : N → A is defined by π(X) = α

if and only if X ∈ Nα .

Let M ⊂ N∗ , M ⊂ N ∩ N∗ , be the set of all minimal surfaces. Then X ∈ M ˆ 1 is a homeomorphism. The minimal surfaces in M arise from implies that X|S taking the orbits of those in M . One can think of M as a global cross section of M transverse to the action of the conformal group. The index theorem implies that M and M are stratified as follows. Let λ = (λ1 , . . . , λp ), ν = (ν1 , . . . , νq ) be p- and q-tuples of natural numbers respectively. As in Chapter 5, define Mλν , Mλν to be the set of all minimal

414


surfaces in N∗ (resp. N∗ ∩ N ) with p (arbitrarily located) interior branch points of orders λ1 , . . . , λp and q arbitrarily located boundary branch points of orders ν1 , . . . , νq . = π|Mλ0 has Fredholm index Both Mλ0 and Mλ0 are manifolds and πλ0 2(2 − n)(Σλj ) + 2p =: I(λ, 0), and πλ0 := π|Mλ0 has index I(λ, 0) + 3, the number 3 arising from the conformal group. Furthermore, for ν = 0, each Mλν and Mλν are locally con ˜ λ ) such that tained in manifolds Σλν and Σλν (in Chapter 5 called Wλν and W ν πλν := π|Σλν , πλν := π|Σλν have, respectively, Fredholm indices: I(λ, ν) := 2(2 − n)(Σλj ) + (2 − n)(Σνj ) + 2p + q and I(λ, ν) + 3. Clearly π is constant under the action of the conformal group. Thus π(M) = π(M ). For n ≥ 4 I(λ, ν) ≤ −4(Σλj ) − 2(Σνj ) + 2p + q ≤ −4p − 4q + 2p + q = −2p − 3q. Thus, if either p or q is non-zero, then I(λ, ν) ≤ −2, telling us that for n ≥ 4 the set of curves bounding a minimal surface is rather sparse, in fact, locally “looks like” a space of codimension at least two, and suggests that the set of curves bounding a branched minimal surface does not disconnect the set of curves α. The technical details proving this fact will now be presented. Theorem 4. Let Σ, A be two Banach manifolds, M a closed subset of Σ and f : Σ → A a C 1 -Fredholm map of index −l such that f |M is proper. If −l ≤ −2 then f (M ) does not disconnect any component of A. Corollary 1. Let Σ and A be Banach manifolds as above with f : Σ → A a C 1 -map which is proper on the closed subset M . Suppose that for m ∈ M there exist a neighbourhood Um of m in Σ, a submanifold Σm of Σ, and a homeomorphism ϕm of Um (which may in latter applications be the identity) such that ϕm (Um ∩ M ) ⊂ Σm , f (Um ∩ M ) ⊂ f (Σm ), and f |Σm is Fredholm of index less than or equal to −2. Then f (M ) does not disconnect any component of A. For a proof of this theorem we need a few preparatory lemmas. Lemma 1. Let f : U → F be a Fredholm map of index −l 0 with U open in a Banach space E. Let 0 ∈ F, F also a Banach space and x ∈ f −1 (0). Then there exist splittings of F = F0 × Rn+l and E = E0 × Rn , a neighbourhood Vx of x, and a diffeomorphism G−1 of Vx and a ball B0 in F0 × Rn such that (4)

f ◦ G(z, w) = (z, ψ(z, w))

for (z, w) ∈ B0 .


415

Proof. See Section 5.1.

Lemma 2. Let X be a closed subset of an open connected set V ⊂ Rm . Assume that the (m − 1)-dimensional Hausdorff measure of C is zero. Then X does not disconnect V . Moreover, given any two points v0 , v1 in V \ C and a line L joining them in V there exist points v0 , v1 arbitrarily close to v0 , v1 in V \ C such that the straight line L joining v0 and v1 is parallel to L and does not intersect C. Proof. Pick a plane P perpendicular to L through v0 and let π : Rm → P be the orthogonal projection. Now assume the contrary, i.e. suppose that there exist no points v0 , v1 with the above properties; then π(C) covers a neighbourhood of v0 in P. Therefore since π has norm 1, it follows that the Hausdorff (m − 1)measure of π(C), satisfies Hm−1 (π(C)) Hm−1 (C). Therefore Hm−1 (C) > 0. This contradiction proves the lemma. Lemma 3. Let g be a map of an open neighbourhood U of a closed box B = I1 × I2 ⊂ F0 × Rn ⊂ E into F = F0 × F1 which is Fredholm of index −l −2 and assume that g is in normal form (4). i.e. g(z, w) = (z, ϕ(z, w)). ˜ Then the image of B under g does not disconnect any open connected set B whose projection onto F0 is contained in the projection of U to F0 . Proof. As we observed in Corollary 1 of Section 5.1 it follows from represen˜ be some open set in F0 × Rn+l whose tation (4) that g(B) is closed. Let B projection onto F0 is contained in the projection of U onto F0 . ˜ Let Y = g(B). Let v, w be two points in B \ Y . We can connect v and w with broken line segments L = Li with vertices v = v0 , v1 , . . . , vk not in Y . It follows that if π is a the projection onto the first factor, π(Li ) is the straight line joining π(vj−1 ) to π(vj ) in F0 . By density we can assume that π(vj−1 ) = π(vj ). By the closedness of Y it clearly suffices to show that for and vj arbitrarily close to vj−1 and vj such that each j there exist points vj−1 the line segment joining them does not intersect Y . So assume j = 1. Let J1 be a small open extension of π(L1 ) contained in π(U ) such that J1 contains π(L1 ). Define a map Φ : J1 × I2 → J1 × Rn+l by Φ(z, w) = (z, ϕ(z, w)). Let us first note that π(L1 ) × Rn+l can be canonically embedded in F0 × Rn+l in an obvious way. Further note that if y ∈ / image g then y viewed as an element of F0 × Rn+p is also not an element of image g. Identify J1 as a subset of R1 . Then Φ : J1 × I2 → R1 × Rn+l has finite (n + 1)-measure in R1 × Rn+l (e.g., see Cor. 2.10.11, p. 176, in Federer [1]). This implies that the image of Φ has zero (n + 2)-measure and consequently zero (n + )-measure in R1 × Rn+l ∼ = Rn+l+1 . Thus if m = n + l + 1, the fact

416


that l 2 implies that the image of Φ has zero (m − 1)-Hausdorff measure in Rm , and by Lemma 2 we can conclude that we can find points v0 and v1 in R1 × Rn+l arbitrarily close to v0 and v1 such that π(v0 ) and π(v1 ) ∈ J1 and the line segment joining them does not intersect the image of Φ and hence does not intersect the image of g. This concludes the proof of the lemma. Definition 1. Let C be a locally path-connected topological space. A set A in C is locally non-disconnecting if for every x ∈ C there exists a basis of neighbourhoods B(x) such that the set U (x) \ A is nonempty and connected for every U (x) ∈ B(x). Lemma 4. Let g be as in Lemma 3. Then the image of B under g is locally non-disconnecting. Proof. Obvious.

Lemma 5. If A ⊂ C is closed and locally non-disconnecting with C connected then C \ A is connected.Moreover if {Ai } is a finite collection of closed nondisconnecting sets then Aj is locally non-disconnecting. Proof. Left to the reader.

We are now ready to complete the proof of Theorem 4. Proof of Theorem 4. Let f : Σ → A be as in the statement of the theorem and let y ∈ A. Assume f −1 (y) = ∅. It suffices to assume that A is connected and since, by the Sard theorem (cf. Milnor [2]), A \ f (Σ) is dense it actually suffices to show that for any point y ∈ A there is an open connected set Vy about y such that Vy \ f (M ) is connected. Since A is a manifold and we are considering only a local result we may assume that A is open in a Banach space F. From the fact that f is proper on M and from Lemma 1 we can find a connected neighbourhood V of y and a finite number of coordinate neighbourhoods Uj , Uj∗ covering f −1 (y) such that (a) Uj ⊂ Uj∗ , (b) Uj ⊃ f −1 (V ) ∩ M , (c) with respect to some suitable coordinate change f on Uj∗ (again identifying the Uj∗ with open subsets of a Banach space E) has the normal form (6). Applying Lemma 4 to f on each Uj we see that Cj = f (Uj ) is closed and locally nondisconnecting. Applying Lemma 5 wesee that Cj is locally nondisconnecting. Hence, again by Lemma 5, V \ Cj and V \ f (M ) are connected. This concludes the proof of the theorem. We end this discussion by remarking that Corollary 1 follows directly from the methods employed in the proof of Theorem 4. We now return to the proof of Theorem 2. Let α0 and α1 be generic curves and αt an embedded path in A joining α0 and α1 . By the index theorem and


417

Corollary 1 we may assume that no αt bounds a branched minimal surface. By Smale’s transversality Theorem 3 of Section 5.1 we may assume that π00 is transverse to the image (a one-dimensional submanifold) of αt in A. −1 ) (αt ) is a one-dimensional submanifold P of M00 effectThus (π00 −1 −1 ) (α0 ) and (π00 ) (α1 ), i.e. the finite set ing a “cobordism” between (π00 −1 −1 (π00 ) (α0 ) ∪ (π00 ) (α1 ) is the boundary of P. −1 −1 Now let (π00 ) (α0 ) = X1 , . . . , Xl and (π00 ) (α1 ) = X1 , . . . , Xl . The fact that χ(Wα0 ) = χ(Wα1 ) is now a consequence of: Theorem 5. (A) Suppose Xj0 and Xj1 are endpoints of a single component of P. Then sgn DWα0 (Xj0 ) = − sgn DWα0 (Xj1 ). (B) If Xi and Xj are endpoints of a single component of P then sgn DWα0 (Xj ) = sgn DWα1 (Xk ). Proof. Weprove (A); then the proof of (B) follows in an analogous manner. Think of t Nαt as Nα0 × I (identify (t, p) ∈ I × Nα0 with αt (α0−1 (p))). Let s → σ(s), 0 ≤ s ≤ 1 be a parametrization of a path joining Xj0 and Xj1 . Let us consider a point on this path, say σ(s0 ). Using a local coordinate chart, we may assume that σ(s0 ) is in an open neighbourhood of a Hilbert space H × R. Then H ×R∼ = (Tσ(s0 ) σ) ⊕ (Tσ(s0 ) σ)⊥ where again ⊥ means “orthogonal complement”. For convenience assume that σ(s) is parametrized by arc length and that a neighbourhood V of 0 ∈ H × R is taken so that for h ∈ V the following map ψ can be defined by ψ(h) = h − h, σ (s0 )σ (s0 ) + σ(s0 + λ), λ = h, σ (s0 ), ·, · the inner product on H. Then Dψ(0) is the identity. Thus ψ is locally invertible and since ψ : Tσ(s0 ) σ → image σ, its inverse is a locally flattening map for σ. Thus, via ψ, we may identify a neighbourhood of σ(s0 ) with a neighbourhood of 0 ∈ H × R. Then if p ∈ V is sufficiently close to 0 Dψ(p) ∈ GL+ c (H × R). Let W (t, ·) := Wαt (·) be the (vertical) vector field on Nα0 × I. From the above discussion we may assume that we are working in a neighbourhood of σ(s0 ) where the image of σ (locally) lies in a one dimensional subspace H1 . We must also consider the action of G. Through every point σ(s), s near s0 , OG (σ(s)) is a smooth and locally embedded submanifold. At σ(s0 ) =: v0 we may locally flatten OG (v0 ). Moreover for every v = σ(s) we have a diffeomorphism Lv : OG (v) → OG (v0 ) given by (v, Lg (v)) → (v, Lg (v0 )). Putting these together we may assume that we have a coordinate neighbourhood with coordinates (u, w, v), v ∈ σ, w ∈ OG (v).

418


From our constructions in the proof of the index theorem it follows that the subspace spanned by v is the (one-dimensional) kernel of the vector field family W (t, ·). First, since W vanishes along σ(s), it follows immediately that Tσ(s0 ) σ ∈ Ker DW . For the benefit of the reader we review the linear algebra ideas behind these constructions. So, let E and F be Banach spaces and L : E × F → F a surjective linear map (here we are thinking of L as DW ). Define L1 (e) = L(e, 0), L2 (f ) = L(0, f ). We assume L2 is Fredholm of index 0. Define M := Ker L, π : E × F projection onto the first factor, and πM := π|M . Then Ker πM ∼ = Ker L2 (∼ = = isomorphism) and Coker πM = Coker L2 . Thus πM is also Fredholm of index zero. In the case at hand, the transver to αt implies that dim Ker πM ≤ 1, so dim Ker L2 ≤ 1. Thus sality of π00 dim Coker πM ≤ 1. Let {λα}, λ ∈ R, represent a one-dimensional subspace of E. If Ker L2 = {0}, then clearly (5)

S

(λ, f ) → L(λα, f )

is surjective. If {λα} is transverse to πM , i.e. {λα}∩range πM = {0}, then S is still surjective. This follows from the fact that if β ⊂ Coker πM , β → L(β, v), v ∈ (Ker L2 )⊥ is a bijective map onto Coker L2 . To see this, note first that β → L(β, v) does not depend on the choice of v; for if L(β, v) = L(β, v2 ), then v1 − v2 ∈ Ker L2 and v1 = v2 . Second, if L(β, v) = 0, (β, v) ∈ M and β ∈ range πM ; thus β = 0. Consequently S is a surjective Fredholm map of index one, implying that Ker S (representing the kernel of the linearization of W ) is one-dimensional. In a neighbourhood of σ(s0 ) where σ and the orbits of G through σ are flattened, define the map ϕ0 (z, w, v) := (W (z, w, v0 ) + w, v), where

. σ(s) ⊂ {z = 0, w = 0} ⊂ M00

Then Dϕ0 (σ(s0 )) = I, the identity, and for any s near s0 we have Dϕ0 (σ(s)) ∈ GLC (H × R). We may assume that ∂v ∂s > 0 (otherwise replace the v on the right-hand side of (8) by −v). By the inverse function theorem ϕ0 is locally (differentiably) invertible and thus represents a local coordinate chart about σ(s0 ) ∈ Nα0 ×I. By n the compactness of the unit interval we may find a finite set of points {sj }j=1 , s0 = 0, sn = 1 and coordinate neighbourhoods Uj , Uj covering σ and associated coordinate maps ϕj , ϕj (zj , wj , vj ) ∈ H × R. We may assume that Uj are small enough so that for σ(s) ∈ Uj , Tid OG (σ(s)) ⊕ range DW (σ(s)) = H. The crux of the matter is now Lemma 6. If σ(s) ∈ Ui ∩ Uj , then (6)

+ D(ϕj ◦ ϕ−1 j−1 )(σ(s)) ∈ GLc (H × R).


Proof. The matrix of (6) has the form: I +K ...

0 ∂vj ∂vj−1

419

.

The operator I + K is the identity on the range of DW (σ(s)). For each σ(s), there is a preferred basis or orientation for Tid OG (σ(s)), namely X(s)θ , X(s)θ cos θ, X(s)θ sin θ if X(s) := σ(s). The linear map I + K then takes X(sj−1 ) to X(sj ), X(sj ) cos θ to X(sj−1 ) cos θ, X(sj−1 ) sin θ to X(sj ) sin θ. We can then find a one-parameter family I + Kμ which at μ = 0 is I + K and I +Kμ fixes the range of DW (σ(s)) and maps X(sj−1 ) to X(μsj−1 +(1−μ)sj ) ∂vj and so forth. Clearly K1 = 0 and thus I + K ∈ GL+ c (H). Since ∂s > 0 for ∂vj all s, ∂vj−1 > 0 and the lemma is proved. Now we complete the proof of (A) in Theorem 5. Since α0 is a regular maps σ(s), 0 ≤ s < ε, and σ(s), 1 ≥ s > 1 − ε2 diffeomorphically value, π00 into α0 (t), 0 ≤ t < δ. Represent coordinates in Nα0 about σ(0) by (u0 , w0 ), w0 ∈ OG (σ(0)) and about σ(1) by (u1 , w1 ). About σ(0) we have the map ϕ˜0 (u0 , w0 , s) = (Wα0 (u0 , w0 ), s) (s the locally flattening variable for σ near s = 0) and since t decreases as s increases towards 1 we know that for t near ∂t < 0. Thus about σ(1) we have 1, ∂s ϕ˜1 (u1 , w1 , t) = (Wα0 (u1 , w1 ), 1 − t) ∂ which is admissible, since ∂s (1 − t) > 0. Hence if DWα0 (σ(0)) ∈ + ⊥ GLc (Tid OG (σ(0)) ) then Dϕ˜0 (σ(0)) ∈ GL+ c (H × R). Using the above collection of coordinate charts and the last lemma we know that

Dϕ˜1 (σ(1)) ∈ GL+ c (H × R); ⊥ but the only way this can happen is if DWα0 (σ(1)) ∈ GL− c (Tid OG (σ(1)) ). This completes the proof of (A). (B) follows exactly along the same line with the exception that σ(1) ∈ Nα1 and the chart ϕ˜1 is defined with t replacing 1−t. The conclusion of (B) follows.

Remark 1. The previous argument in this infinite-dimensional setting is a formalization of the “picture” argument in the finite-dimensional case presented by Milnor in his treatise [1]. Thus, we have established the independence of the Euler characteristic χ(Wα ) on the choice of a regular value α : S 1 → Rn , n ≥ 4. Let us move on to the definition of the Euler characteristic χ(Wα ) for the generic curve α in R3 . In this case either α bounds only nondegenerate minimal surfaces X (DWα (X) ∈ GLc (Tid OG (X)⊥ ), and then the Euler characteristic is defined as before, or α bounds one or more minimal surfaces which are simply branched

420


and nondegenerate, in the sense that if M00 ∪ Mλ0 , λ = (1, . . . , 1) represents the union of any two strata in the stratification of minimal surfaces in R3 and α is a regular value for π|M00 ∪ Mλ0 , then there are only finitely many minimal surfaces bounded by α. In this situation, by the index theorem there is a generic β : S 1 → R4 such that β is isotopic to α, and that β bounds only finitely many nondegenerate minimal surfaces. Define χ(Wα ) := χ(Wβ ). By Theorem 2 this does not depend on the choice of β. We now obtain the assertion of Theorem 3 for n ≥ 4: Theorem 6. We have χ(Wα ) = 1 for any α. Using the fact that there are no knots in Rn , n ≥ 4 (i.e. all α are isotopic to the standard embedding of S 1 into R2 ⊂ Rn ), this will follow from: Lemma 7. The map X(w) = w of the unit disk to itself is a nondegenerate minimal surface spanning S 1 . Proof. Let α : S 1 → S 1 be the standard embedding. Taking the Fréchet derivative of both sides of formula (10) in Section 6.2 at the zero X of Wα we obtain for k ∈ TX Nα that ˆ ˆ ∂ k ∂ ∂ X ˆ α (X)k = Ω(X) + DΩ(X)(k) . Ω(X) DW ∂r ∂r ∂r From Ω(X)kˆr = Xθ , kˆr |Xθ |−1 Xθ it follows that (7)

∂ ˆ α (X)k = |Xθ |−2 Ω(X) DW ∂r

ˆ ∂k ∂X , ∂r ∂θ

+

∂X ∂k , ∂θ ∂r

∂X , ∂θ

·, · = the R2 -inner product. Suppose that DWα (X)k = 0; then taking the inner product with k = λ(θ)Xθ and integrating over S 1 we see that (8)

λ S1

Now

ˆ ∂X ∂ kˆ ∂ X ∂k , + , dθ = 0. ∂r ∂θ ∂θ ∂r ∂k ∂X ∂2X = λθ +λ 2 ∂θ ∂θ ∂θ

ˆ

∂X and since ∂X ∂θ , ∂r = 0 we see that (8) equals


421

ˆ 2 ˆ ∂X ∂ X ∂k λ , , k dθ dθ + ∂r ∂θ2 ∂r 2

=0 ˆ ˆ ∂ ∂k ∂X , k dθ − λ , k dθ. = ∂r ∂θ ∂r Writing Xθ = equations

∂X ∂θ ,

ˆr = X

ˆ ∂X ∂r ,

we have for X = (X 1 , X 2 ) the Cauchy-Riemann

ˆ r = (X ˆ 1, X ˆ 2 ) = (−X ˆ 2, X ˆ 1 ). X r r θ θ So ˆ r = (−λXθ2 , λXθ1 ) = (−k 2 , k 1 ) λX and (8) reduces to: 1 ˆ2 1 2 ˆ (kr , kr ), (k , k ) dθ − (−kθ2 , kθ1 ), (k 1 , k 2 ) dθ 0= S1

=

S1

=

S1

ˆ1

∂k · k 1 dθ + ∂r S1 1 ˆ ∂k · k 1 dθ + ∂r S1

S1

∂k ∂k 2 1 ∂k 1 2 · k 2 dθ + · k dθ − k dθ ∂r ∂θ S 1 ∂θ ∂ kˆ2 2 ∂k 1 2 · k dθ − 2 · k dθ. ∂r S 1 ∂θ ˆ2

Let k 1 (θ) = Σcν eiνθ , cν = c−ν and k 2 (θ) = Σdν eiνθ , dν = d−ν . Using this Fourier representation this chain of equalities transforms to |ν|{cν |2 + |dν |2 } − 4π iνcν d−ν . (9) 0 = 2π ν

By the Cauchy–Schwarz inequality Σiνcν d−ν ≤ Σ|ν||cν |2 · Σ|ν||dν |2 . Let p = Σ|ν||cν |2 , q = Σ|ν||dν |2 . Then (10) Let

√ √ p + q ≥ 2 p · q. √ i νcν , aν := −i |ν|cν , √

and bν :=

ν dν , |ν| dν ,

ν ≥ 0, ν≤0 ν ≥ 0, ν ≤ 0.

Then Σiνcν d−ν = {aν } · {bν }, where the dot product of sequences is defined by {aν } · {bν } = Σaν bν . Again applying the Cauchy–Schwarz inequality we have

422


(11)

|{aν } · {bν }| ≤ |{aν }| 2 · |{bν }|l2

with equality holding if {bν } is a scalar multiple of {aν }. In this inequality | · | 2 denotes the sequential l2 -norm. Thus (11) implies that √ √ p + q = 2|{aν } · {bν }| ≤ 2|{aν }|l2 · |{bν }|l2 = 2 p · q. √ √ From (9) and (10) we have p + q = 2 p · q, and so |{aν } · {bν }| = |{aν }|l2 |{bν }|l2 . By the Schwarz inequality we can conclude that the sequence }. It is easy {bν } is a scalar multiple of {aν√ √ to see that the only √ possible√such scalar multiple is 1. Therefore, ν dν = i νcν for ν ≥ 0 and ν dν = −i νcν for ν ≤ 0, whence (12)

cν = −idν ,

(13)

cν = idν ,

ν ≥ 0, ν ≤ 0.

These equalities imply that kˆθ1 = kˆr2 , or more importantly that the harmonic extension kˆ of k : S 1 → R2 , k(θ) ∈ TX(θ) Γ , to the disk B is a holomorphic map on B. t ˆ The vector field kˆ generates a 1-parameter flow ϕt (z), dϕ dt = k(ϕt (z)) which is necessarily a 1-parameter family of automorphisms of the disk to itself. This immediately implies that k ∈ Tid OG (X). Thus DWα (X) ∈ GLc (Tid OG (X)⊥ ) and X is nondegenerate. Finally, the Hessian satisfies D2 Eα (X)(k, k) = DWα (X)k, k dθ |ν|{|cν |2 + |dν |2 } − 4π iνcν d−ν ≥ 0, = 2π ν

and it equals zero only if k ∈ Tid OG (X). It follows that DWα (X) ∈ ⊥ GL+ c (Tid OG (X) ). The proof of Theorem 6 now follows from the fact that by the Riemann mapping theorem X(z) is the unique (up to conformal reparametrizations) minimal surface spanning α. We may use the last theorem to prove a uniqueness result, namely: Theorem 7. Let α be a plane curve, and let X(α) be the “unique” minimal surface (holomorphic map) bounded by α. Then for any embedding β : S 1 → Rn , n ≥ 2, “sufficiently close” to α, there is a “unique” minimal surface X(β) close to X(α) bounded by β. (“Uniqueness” means: “Uniqueness after fixing a three-point condition”.)

6.4 Sard–Brown Theorem for Functionals

423

Proof. (Sketch; a detailed presentation is given in Tromba [5].) The last proof, with only minor modifications, shows that X(α) is nondegenerate. For any β close to α, the inverse function theorem, suitably modified to account for the action of the conformal group, implies that there is a minimal surface X(β) near X(α) bounded by β with β → X(β) C r -smooth. Uniqueness follows from the a priori estimates: Suppose that there is a sequence βj ∈ H r+8 (S 1 , Rn ), βj → α such that each βj bounds a minimal surface X j outside a fixed ball in H 2 (S 1 , Rn ) centered at X(α) and containing X(βj ). After imposing a threepoint condition onto X(α) and X j , we may conclude that X j are bounded in H r+8 , and therefore, after taking a convergent subsequence we may assume that X j converges to a planar minimal surface bounded by α different from the unique Riemann map X(α), a contradiction. Remark. The a priori estimates for minimal surfaces of disk type and the a priori nondegeneracy of any planar solution to Plateau’s problem provide the basis of a “continuity” proof of the existence of the Riemann mapping theorem in case one requires a conformal map to be bounded by a smooth curve in the plane. Specifically, if one has an isotopy αt , 0 ≤ t ≤ 1, from α0 , the standard embedding of S 1 into the plane, then the “unique” planar solution X(w) = w bounded by α0 can be smoothly continued to a conformal map X1 bounded by α1 .

6.4 The Sard–Brown Theorem for Functionals Let M be an infinite-dimensional Banach or Hilbert manifold of class C ∞ . Then it is known that Sard’s theorem does not hold for arbitrary real valued functions f ∈ C ∞ (M). In this section we prove a Sard theorem for such functions under an assumption valid for many cases in the calculus of variations, and in particular to the problem of geodesics and Plateau’s problem. A C 1 -vector field W : M → T M is said to be associated to f in case that W (p) = 0 if and only if Df (p) = 0. Thus the critical point set of f coincides with the zero set of W . Theorem 1 (Sard Theorem for functionals). Let M be a second countable Banach manifold modelled on E, f ∈ C ∞ (M), and W an associated Fredholm vector field to f . Then the regular values of f are of second category. If the critical points of f form a compact set in M then the regular values are open and dense. Proof. Again, by second countability, we may work locally. We may assume that E = E0 × Rn and that locally W has the form W (z, w) = (z, ϕ(z, w)) ∈ E0 × Rn , ϕ a C ∞ -map from E0 × Rn to Rn (see Section 5.1, Theorem 2).

424


We have W (z, w) = 0 if and only if z = 0, ϕ(0, w) = 0. Thus, locally, the set of critical points of f lie in 0 × U , U an open set in Rn . We have no a priori bound on n, but if f is C ∞ this does not matter and we may apply the classic Sard theorem to conclude that the image of the critical points (locally) is of first category. This immediately implies the assertion of the theorem. As a consequence, we have: ∗ ∩ Nα → R be Dirichlet’s integral where α is a Corollary 1. Let Eα : Nα ∞ C -curve. Then the set of regular values of Eα is open and dense. ∗ ∩ Nα are, by a priori bounds, a compact Proof. The minimal surfaces in Nα ∞ ∞ set. Since α is C , Eα is C (see Chapter 6 of Vol. 2). The fact that the associated vector field Wα is Fredholm concludes the proof.

6.5 The Morse Lemma It is a classical consequence of Taylor’s theorem that if U ⊂ Rn is an open subset, q ∈ U a critical point for a C 2 -mapping f : U → R and the second derivative of f at q or “Hessian” of f , D2 f (q) : Rn × Rn → R is a positive quadratic from, then q is a strict minimizer for f . If U ⊂ H is open in a Hilbert space H, q a critical point for f : U → R, and D2 f (q)(v, v) ≥ c v 2 , c > 0, then q is also a strict minimizer for f . If we merely have D2 f (q)(v, v) > 0 for all v = 0, the implication that q is a strict minimizer is, in general, false. In this section we prove an abstract Morse lemma in a Banach space and then apply it to Plateau’s problem. We begin with The Local Theory in Banach Space For clarity of exposition of our ideas we shall start first with a discussion of functionals defined on open subsets of certain Banach spaces, and then only later globalize to the somewhat more complicated case of functionals defined on open subsets of Banach manifolds. Let E be a real Banach space with norm and suppose

·, · : E×E → R is an inner product on E, i.e.

·, · is continuous, symmetric, bilinear and satisfies

u, u ≥ 0 and equals 0 if and only ifu = 0. The inner product

·, · induces a weak norm on E given by u w =

u, u with u w ≤ const u . We can complete E in this norm obtaining a Hilbert space Ew with a natural inclusion map j : E → Ew . By construction j will have a dense range in Ew . Definition 1. A function f ∈ C r (U ), r ≥ 1, with U ⊂ E convex, E a Banach space with an inner product

·, ·, is called C r -acceptable with respect to

·, ·, if there is a C r−1 -vector field W : U → E such that the differential, df satisfies df (p)h =

W (p), h

6.5 The Morse Lemma

425

for all h ∈ E and p ∈ U . Such a W can be thought of as the gradient of f with respect to

·, ·. Suppose we have an inner product

·, · on a Banach space E. Let A denote the class of continuous linear maps A from E to itself which have continuous linear adjoints A∗ : E → E with respect to

·, · with A∗ = A. The set A is clearly a closed subspace of the ring of all linear maps. We shall call the members of A self-adjoint linear maps. Thus A ∈ A if and only if

Au, v =

u, Av for all u, v ∈ E. Lemma 1. If f : U → R is a C 2 -acceptable functional then the Fréchet derivative at any point p ∈ U of the gradient W of f with respect to

·, · is self-adjoint. Proof. Since d2 f (p)(u, v) =

DW (p)u, v, the result follows immediately from the symmetry of second derivative of f . Theorem 1. Let f : U → R be a C r -acceptable mapping on a Banach space E which is convex respect to an inner product structure

·, · on E. Suppose that q is a critical point of f (i.e. df (q) = 0, W (q) = 0) with D2 f (q)(v, v) > 0 for v = 0 and the Fréchet derivative DW (q) : E → E an isomorphism. Then q is a strict minimizer for f . This result will follow directly from the next lemma, a generalization of the Morse lemma. The essential computational ideas are due to Palais. Theorem 2 (Morse Lemma). Let f : U → R be a C r -acceptable mapping with a critical point q such that DW (q) : E → E is an isomorphism. (Such a critical point will be said to be nondegenerate.) Then there exists a C r−2 diffeomorphism ϕ of a neighbourhood of q such that 1 f ◦ ϕ(η) = f (q) + D2 f (q)(η, η). 2 Proof. For simplicity let us assume that q = 0. By the fundamental theorem of integral calculus we have that 1 1 df (ty)y dt = f (0) + W (ty) dt, y . f (y) = f (0) + 0

0

Again applying the fundamental theorem this equation becomes 1 f (y) = f (0) +

A(y)y, y, 2 where

1

A(y) =

1

DW (sty) ds dt. 0

0

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Since DW (p) ∈ A for each p ∈ U it follows that A(y) ∈ A for y ∈ U . Moreover A(0) is an isomorphism and y → A(y) is C r−2 . Let Q(y) = A−1 (y)A(0). Then for each y, y → Q(y) is C r−2 and Q(0) = I, the identity. Thus for y sufficiently close to 0, Q(y) has a “square root” S(y) which is a power series in Q with S(y)2 = Q(y), y → S(y) being of class C r−2 . Now Q has an adjoint Q∗ with respect to

·, · defined by Q∗ (y) = A(0)A−1 (y), and Q∗ (y)A(y) = A(y)Q(y) = A(0). Moreover, locally Q∗ has a square root S˜ with S˜ = S ∗ , and since S(y) is given as a power series in Q(y) we have that S ∗ (y)A(y) = A(y)S(y), and so S ∗ (y)A(y)S(y) = A(y)S 2 (y) = A(0). Let R(y) = S(y)−1 . Then we obtain 1 f (y) = f (0) +

R(y)∗ A(0)R(y)y, y, 2 where R(0) = I and y → R(y) is of class C r−2 . Let Ψ (y) = R(y)y. It follows immediately that DΨ (0) = I, and thus Ψ has a local inverse ϕ. Clearly ϕ satisfies the obstruction of the lemma.

The Global Theory We next prove manifold versions of the last two theorems. Let M ⊂ E be a submanifold. The pairing on E induces a pairing on the tangent space Tp M ⊂ E for each p ∈ M . Since M is a submanifold of E there is for each q ∈ M a diffeomorphism of a neighbourhood V of q in E onto a neighbourhood of 0 which locally “flattens” M , i.e. ϕ(M ∩ V ) ⊂ Tq M ⊂ E, an open subset of the tangent space at q. Definition 2. A C r -submanifold M ⊂ E will be called a C r -weak Riemannian submanifold if a collection of locally flattening diffeomorphisms {ϕ, V } as above can be found such that (i) {ϕ|V ∩ M, V ∩ M } is a coordinate cover for M . (ii) For each p ∈ M,

·, · is an inner product (although not necessarily complete) for Tp M ⊂ E.

6.5 The Morse Lemma

427

If (ϕ, V ) is a coordinate chart about q ∈ M then for each p ∈ V , by definition, Dϕ(p) : E → E. Let Dϕ(p) := Dϕ(p)|Tp M , the restriction of Dϕ(p) to Tp M . So Dϕ(p) : Tp M → Tq M . (iii) For each p we require that Dϕ(p) be continuous in the topology induced by

·, · and to have a continuous inverse Dϕ(p)−1 . (iv) For each p ∈ V we require that Dϕ(p) have an adjoint with respect to

·, · such that Dϕ(p)∗ : Tq M → Tp M , isomorphically. Consequently Dϕ(p)∗ ∈ L(Tq M, E), the Banach space of linear maps from Tq M to E. We shall require that p → Dϕ(p)∗ is of class C r−1 . A maximal atlas of such ϕ’s will be called a weak Riemannian structure for M . Definition 3. Let U ⊂ M be open and f ∈ C r (U ), r 1, M ⊂ E as above. f is called C r -acceptable with respect to the induced inner product structure on the tangent bundle T M , if for each p ∈ U there is a C r−1 -vector field W : U → T M |U such that the differential of f satisfies df (p)[h] =

W (p), h for all h ∈ Tp M and p ∈ U ; W is then the gradient of f with respect to

·, ·. Our goal is to prove versions of Theorem 2 for such an f about a critical point q ∈ U ⊂ M under the assumption that DW (q) : Tq M → Tq M is an isomorphism. At first glance it may seem that we could do this without any special coordinate structures on M . We would begin by choosing any coordinate chart (ϕ, V ), V ⊂ U , about q to “pull things back” to V˜ = ϕ(V ) ⊂ Tq M , and then apply these theorems. However, we would not be in the same situation as in these earlier theorems, since the pull back of a vector field W , ˜ , is not the gradient of f with respect to the “constant” inner product say W

·, · on Tq M or, in the language of differential geometry, with respect to the flat metric on V˜ . Fortunately, the assumptions that M is a smooth weak Riemannian submanifold of E and that f is C r -acceptable guarantee that ˜ on V with respect to the flat or constant f does indeed have a gradient W metric

·, · on Tq M . ˜ (˜ For p˜ ∈ V˜ , W p) will be given by the formula (1)

˜ (˜ p)), W p) = [Dϕ(p)∗ ]−1 W (ϕ−1 (˜

where Dϕ∗ (p) : Tq M → Tp M is the assumed adjoint. To see that this formula ˜ is indeed correct note that for W (2)

df (p)[h] =

W (p), h.

428


˜ ∈ Tq M Therefore for h ˜ = p)[h] d(f ◦ ϕ−1 )(˜

˜ W (ϕ−1 (˜ p)), Dϕ−1 (˜ p)[h]

˜ p)), [Dϕ(p)]−1 [h] W (ϕ−1 (˜ ˜ . p)), h = [Dϕ(p)∗ ]−1 W (ϕ−1 (˜

=

˜. This verifies the formula for W Applying these remarks and our previous theorems we obtain Theorem 3 (Morse Lemma). Let f ∈ C 3 (U ), where U is an open subset of a weak Riemannian submanifold M of E. Assume that f is C 3 -acceptable and that q is a critical point of f with DW (q) : Tq M → Tq M an isomorphism. Then for any weak Riemannian coordinate chart ϕ about q, there exists a C 1 diffeomorphism Ψ of a neighbourhood of ϕ(q) with a neighbourhood of ϕ(q) in Tq M ⊂ E, preserving ϕ(q) and such that (f ◦ ϕ−1 )(Ψ (y)) = f (q) + D2 (f ◦ ϕ−1 )(ϕ(q))(y, y). This immediately implies Theorem 4. Let f : U → R be a C 3 -acceptable function on an open subset U of a weak-Riemannian submanifold M of E. Let q be a nondegenerate critical point of f (that is, DW (q) : Tq M → Tq M is an isomorphism), where W is the weak gradient of f . If D2 f (q)(v, v) > 0 for all v ∈ Tq M , v = 0, then q is a strict local minimizer for f . We now proceed to apply this to Plateau’s problem. The Problem of Plateau Let Nα be the manifold of harmonic surfaces introduced in Section 6.3. We then have: Theorem 5. Nα is a weak Riemannian submanifold of H 2 (S 1 , Rn ). Proof. Let X0 ∈ Nα . Then we have that Φ(Y )(s) := expX0 (s) Y (s) is a C r coordinate chart for Nα , mapping the H 2 -vector fields Y over X0 onto Nα , with (3)

DΦ(0) = id.

Then h → DΦ(Y )h is a map of the form h → v(Y )(eiθ )h(eiθ ), where for each θ v(Y )(eiθ ) ∈ L(Rn , Rn ),

6.5 The Morse Lemma

429

and L(Rn , Rn ) is the set of the linear maps of Rn to itself. Note that Y → v(Y ) is C r−1 . For fixed Y , h → v(Y )(eiθ )h is a continuous linear map from H 2 to H 2 or from H 1 to H 1 , implying that Y → v(Y ) is a C r−1 map into L(H s (S 1 , Rn ), H s (S 1 , Rn )), s = 1, 2. By (3), Φ has a local inverse ϕ with Dϕ(X) ∈ L(H s (S 1 , Rn ), H s (S 1 , Rn )) for s = 1, 2. We would now like to compute the adjoint of Dϕ(X) = Dϕ(X)|TX Nα . For X near X0 , Dϕ(X) also has the form h → λ(X)(eiθ )h(eiθ ), λ(X)(eiθ ) ∈ L(Rn , Rn ). Consistent with our previous notation let

·, · denote the H 1 inner product, a weak inner product on the H 2 -Hilbert manifold Nα . For h ∈ TX Nα , k ∈ TX0 Nα we have

Dϕ(X)h, k =

λ(X)h, k n

(λh)r , kR dθ =

h, λ∗ kˆr Rn dθ = S1

S1

where for each θ, λ∗ is the Rn -adjoint of λ. Now for k ∈ H s (S 1 , Rn ) λ∗ kˆr ∈ H s−1 (S 1 , Rn ). Define

∗ˆ Dϕ(X)∗ k := L−1 X λ kr

where LX was defined by formula (4) in Section 6.2. This is the adjoint we are seeking and clearly satisfies the desired properties, concluding the proof of Theorem 5. However, we cannot directly apply the abstract Morse lemma to our situation because, due to the action of the conformal group, the derivative of the vector field Wα : Nα → T Nα always has a kernel. In this equivariant situation, we called (Section 6.3) a zero X0 nondegenerate, if DWα (X0 ) : [Tid OG (X0 )]⊥ → [Tid O(X0 )]⊥ is an isomorphism, where ⊥ represents the H 1 (B)-orthogonal complement. What we need is a local submanifold (or slice) transversal to the action of the conformal group and a gradient vector field for Eα restricted to this slice. Consider again the map Y → Φ(Y ) mapping a neighbourhood U of 0 ∈ TX0 Nα onto a neighbourhood of X0 in Nα . We choose the obvious local slice to be V = [Tid OG (X0 )]⊥ ∩ U. ˜ α on U, Pulling back Wα via (1) to U, we obtain a vector field, W (4)

˜ α (Y ) = ((Dϕ∗ )−1 )Wα (ϕ−1 (Y )). W

˜ α , however, need not be a vector field on V. To rectify this we define W

430


(5)

˜ α (Y ), V˜α (Y ) = π0 W ∗ −1

V˜α (Y ) = π0 ((Dϕ )

Y ∈V )Wα (ϕ

−1

or π0 (Y )),

π0 : TX0 Nα → [Tid OG (X0 )]⊥ the H 1 (B)-orthogonal projection. We now need ˜ α , and V˜α is Lemma 2. The zeros of V˜α on V are identical with those of W the gradient of Dirichlet’s integral restricted to V. ˜ α on V is a zero of V˜α . Suppose now that Proof. Clearly from (5) a zero of W ˜ Vα (Y ) = 0. Then ((Dϕ∗ )−1 )Wα (ϕ−1 π0 (Y )) ∈ [Tid OG (X0 )] or Wα (ϕ−1 (Y )) ∈ Dϕ∗ [Tid OG (X0 )]. Let (TX0 Nα ) be the H 1 -closure of TX0 Nα and π1 (X) : (TX0 Nα ) → Tid OG (X) the H 1 -orthogonal projection. Then X → π1 (X) is continuous in the H 2 topology. Thus X → π1 (X)Dϕ∗ (X) is continuous with π1 (X0 )Dϕ∗ (X0 ) the identity. Hence, for X close to X0 , the mapping π1 (X)Dϕ∗ (X) : Tid OG (X0 ) → Tid OG (X) is an isomorphism. But, by (14) of Section 6.2, Wα (X) = Wα (ϕ−1 (Y )) is orthogonal to Tid OG (X). Therefore, π1 (X)Wα (X) = π1 (X)Dϕ∗ (X)[Z] = 0, Z ∈ Tid OG (X0 ). Thus Z = 0 and hence Wα (X) = 0 and

˜ α (Y ) = 0. W

Moreover, ˜ α (Y ), h DEα (Y )h =

W ∼ means “identifiable with”), ∼ TY V for any Y (= for all h. If h ∈ [Tid OG (X0 )]⊥ = then ˜ α (Y ), h =

V˜α (Y ), h. DEα (Y )h =

π0 W This proves that V˜α is the gradient of Eα restricted to V and ends the proof of the lemma. Theorem 6. Let Eα : Nα → R be Dirichlet’s integral with X0 a nondegenerate critical point. Then in a local coordinate neighbourhood there is a neighbourhood V ⊂ [Tid OG (X0 )]⊥ and a C r−2 -diffeomorphism ψ of V into Nα such that (6)

1 Eα (ψ(w)) = Eα (X0 ) + D2 E α (0)(w, w) 2

6.5 The Morse Lemma

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where X0 corresponds to 0 ∈ TX0 Nα , and E α denotes the restriction of the pull back of Eα to V. Theorem 7. If X0 is a minimal surface such that D2 Eα (X0 ) > 0 on [Tid OG (X0 )]⊥ \ {0}, then modulo the action of the conformal group (Eα is constant on orbits), X0 is a strict local minimizer. Proof. If D2 Eα (X0 ) > 0 on [Tid OG (X0 )]⊥ \ {0} then DWα (X0 ) has a trivial kernel on [Tid OG (X0 )]⊥ , and since it is of the form identity plus compact linear, it is an isomorphism (recall that DWα (X0 ) : [Tid OG (X0 )]⊥ → [Tid OG (X0 )]⊥ ). Thus X0 is nondegenerate. The result on a slice ψ(V) in Nα follows immediately from (7). The difficult question is whether or not ev˜ for a unique g, ery X ∈ Nα in a neighbourhood of X0 is of the form Lg X ˜ ∈ ψ(V), i.e. the orbits Lg ψ(V) uniquely cover a neighbourhood of X0 . The X difficulty lies in that the conformal group G does not act smoothly on Nα . If Lg ψ(V) covered a neighbourhood of X0 the Morse lemma would read (7)

1 Eα (Lg ψ(w)) = Eα (X0 ) + D2 E α (0)(w, w) 2

reflecting the constancy of Eα on orbits, and Theorem 7 would follow immediately. The key to resolving this difficulty is to use another exponential map of this weak metric, other than the exponential map of a totally geodesic metric 1 on Rn , but the exponential map of the weak H 2 (S 1 , Rn ) metric which is G-invariant (G acts as a group of isometries). The difficulty with this approach is that a priori this exponential map is not a map on H 2 (i.e. on Nα ). This is a regularity question quite analogous to what we had to cope with in defining the gradient vector field Wα in Section 6.2. We proceed to construct: The Spray of the Weak Metric In this section we prove the existence of a smooth spray and exponential map for the weak Riemannian structure on T Nα . These are used to construct local codimension-three submanifolds transverse to the orbit of a zero of Wα . Such local submanifolds (or slices) are used in showing that nondegenerate minimal surfaces are isolated, and are also crucial to several theorems in the next section. To begin with, let us briefly review the notion of a spray and exponential map on a Banach manifold. Definition 4. A spray on a manifold M is a vector field ξ on the tangent bundle π : T M → M which satisfies (i) π∗ ξ(v) = v, (ii) ξ(sv) = s∗ sξ(v),

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where π∗ : T 2 (T M ) → T M is the Fréchet derivative of the projection map π and s∗ denotes the Fréchet derivative of the bundle isomorphism from T M to itself which is fibrewise multiplication by the real number s. The following theorem characterizes how sprays look locally, i.e. in some coordinate chart. Theorem 8. Let U be open in the Banach space E viewed as a C r -manifold, and let T (U) = U × E be the tangent bundle. A C r−2 -map f : U × E → U × E × (E × E) is a local representation of a spray on T (U) if and only if f is of the form f (x, v) = (x, v, f2 (x, v)) where f2 (x, sv) = s2 f2 (x, v) for all real numbers s. If M is a C r -Riemannian manifold the Riemannian structure induces a -spray on M , known as the geodesic spray. Generally this is constructed C as follows: On any C r -Hilbert manifold M there is a canonical closed nondegenerate two-form ω on T M . In particular, ω is a C r−1 -section of the tensor bundle over T M whose fibre over p ∈ T M consists of the real valued, antisymmetric, bilinear real valued maps on Tp (T M ). Therefore for each p ∈ T M we have an antisymmetric bilinear map ωp : Tp (T M ) × Tp (T M ) → R. Let I : T M → R be defined by 1 I(v) = v, vx , 2 r−2

where ·, ·x : Tx M × Tx M → R is the Riemannian structure on T M . Then −dI is a section of T ∗ (T M ). Let np ∈ Tp (T M ). Since the two-form ω is nondegenerate the map np → ωp (np , ·) induces a bijective correspondence between the one forms and vector fields on T M . The geodesic spray is then defined to be the vector field on T M corresponding to the one-form −dI. Suppose that M is a Riemannian manifold, but also has an inner product

·, · : T M × T M → R which is weaker than the Riemannian inner product. Moreover, suppose one needs to construct a spray corresponding to the weak metric

·, ·. Then the above construction breaks down. However, the appearance of weak metrics and their associated geodesic sprays (when they exist) in fundamental problems in nonlinear analysis is not uncommon; in fact a detailed study of the spray of a weak metric is the key to the solutions of the Euler and Navier–Stokes equations of hydrodynamics by Ebin and Marsden. We shall now give another, more intuitive construction of the geodesic spray in the special case that our Hilbert manifold M is isometrically embedded in a Hilbert space H. This alternate construction will enable us to see how to construct a spray on Nα associated to the weak Riemannian structure introduced in (1) of Section 6.2.

6.5 The Morse Lemma

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So, let us suppose M is a C r -closed Riemannian submanifold of a Hilbert space H, with M given the Riemannian structure induced by the Riemannian structure on H. For each m ∈ M we get a map Πm : H → Tm M which is the orthogonal projection of H onto Tm M . The map m → Πm is a C r−1 map of M into L(H). The tangent bundle T (H) and second tangent bundle T 2 (H) over H and T (H) respectively are canonically isomorphic to H × H and H × H × (H × H). Define the map Z : T (H) → T 2 (H) by Z(x, v) = ((x, v), v, 0)). Let J : M → H be the inclusion. Then T M is a subbundle of the pullback bundle J∗ [T (H)]. Z can also be viewed as a map from J∗ [T (H)] to T (J∗ T (H)) since these bundles are equivalent to M ×H and (M ×H)×(H ×H) respectively. Define the bundle map Π : J∗ [T (H)] → T M by Π (m, v) = (m, Πm v). ˜ : J∗ [T (H)] → H, given by Π(m, ˜ The map Π v) = Πm v, is the “principal part” of Π . Let DΠ : T (J∗ [T (H)]) → T 2 M denote the tangent map of Π . Now (DΠ ) ◦ Z : J∗ [T (H)] → T 2 M restricts to a map from T M to T 2 M . Clearly DΠ (m, v, h, k) = (m, v, h, DΠm (h)(v) + Πm k) where DΠm denotes the Fréchet derivative of m → Πm . Therefore (DΠ ◦ Z)(m, v) = (m, v, v, DΠm (v)(v)). One now easily checks that ξ : T M → T 2 M defined by (8)

ξ(m, v) = (DΠ ◦ Z)(m, v)

is a spray on M , and in fact is the geodesic spray induced by the Riemannian structure of M inherited from H. Using this construction we now show that the weak Riemannian structure

·, · : T Nα × T Nα → R on the C r manifold Nα induces a C r−2 -spray on T Nα . Let X ∈ Nα , ΠX : H 2 (S 1 , Rn ) → (TX Nα )1 be the weak orthogonal projection and LX : TX Nα → (TX Nα )1 the isomorphism introduced in Section 6.3. Then (9)

ΠX h = L−1 X Ω(X)

ˆ ∂h . ∂r

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It follows that the weak orthogonal projection ΠX : H 2 (S 1 , Rn ) → (TX Nα )1 and that X → ΠX is C r−1 smooth. Let H = H 2 (S 1 , R3 ) and M = Nα ⊂ H. Then formula (9) defines the geodesic spray ξ associated to the weak Riemannian structure. As a vector field on T M , ξ is a C r−2 -map, and so even though

·, · is weaker than the H 2 -inner product on T Nα we still obtain a C r−2 -spray. To each spray ξ : T M → T 2 M on a manifold M there is an exponential map exp : U → M which is defined on an open neighbourhood U of the zero section of T M and which is constructed as follows. Let t → βv (t) be the flow on T M of the vector field ξ with initial condition v ∈ T M . Let U be the set of vectors v on T M such that βv is defined at least on the unit interval [0, 1]. It is well-known that U is an open neighbourhood of the zero section of T M . If π : T M → M denotes the tangent bundle projection map we define a new map exp : U → M by exp(v) := πβv (1). If M is a C r -manifold exp is a C r−2 -map. For m ∈ M define expm = exp |Tm M . The map expm is a local diffeomorphism from a neighbourhood V0 of the zero 0m ∈ Tm M to a neighbourhood of m with expm (0m ) = m and D[expm ](0m ), the Fréchet derivative of expm at 0m , the identity map of Tm M into itself. If σ(t) = πβv (t) one can show from the definition of a spray that σ satisfies the differential equation (10)

σ (t) = DΠσ (σ )(σ ),

where DΠm again denotes the Fréchet derivative of m → Πm . Moreover for such a σ, expm v = σ(1), σ(0) = m, σ (0) = v. Let us formally state the main parts of the preceding results as Theorem 9. The weak Riemannian structure on Nα induces a smooth C r−2 geodesic spray. The corresponding exponential map restricted to TX0 Nα gives a diffeomorphism of a neighbourhood of zero in TX0 Nα onto a neighbourhood of X0 in Nα . We now come to the principal reason for using the geodesic spray of the weak metric; namely the construction of a local C r−2 codimension-three submanifold of Nα transverse to the orbit of a minimal surface X0 . Now restrict this exponential map to a neighbourhood V ⊂ [Tid OG (X0 )]⊥ where it remains a diffeomorphism onto its image. This image E is a local exponential surface containing X0 in its interior. ˆ ˆ = X(g(z)) be the G-action on Nα , and let Let g ∈ G and (Lg X)(z) U ⊂ TX0 Nα be a neighbourhood of 0 where this exponential map is defined. Then, since G acts as a group of isometries

6.5 The Morse Lemma

(11)

435

Lg expX h = expLg X Lg h

and L−1 g = Lg −1 . The next result is critical in dealing with the non-smooth action of G. Theorem 10. The map (g, Y ) → expLg X0 ΠLg X0 Y is a diffeomorphism of a neighbourhood of the identity of G cross a neighbourhood V of 0 in [Tid OG (X0 )]⊥ onto an open neighbourhood of X0 in Nα , and (12)

Lg (expX0 ΠX0 L−1 g Y ) = expLg (X0 ) Y.

Proof. The fact that (g, Y ) → expLg X0 ΠLg X0 Y is a local diffeomorphism follows from the fact that X0 ∈ C r+6,σ (Lemma 3 of Section 6.2) that g → Lg X0 is a local diffeomorphism onto its image, the standard properties of the exponential map, and finally from the fact that G acts as a group of isometries we have (13)

ΠLg X0 Lg h = Lg (ΠX0 h)

yielding (12). Formula (13) shows that, even though for X ∈ E, g → Lg X is not smooth, the orbits of X ∈ E uniquely cover an open neighbourhood of X0 ∈ Nα . Recall that our goal is to use the Morse lemma (Theorem 6) to prove Theorem 7. In proving the Morse lemma we needed another slice arising from the finite dimensional exponential map Φ. For this slice we have to show that every point in a neighbourhood of X0 can be uniquely written as a point in the orbit of an element in such a slice. Thus if W is a local slice arising from the image of the map Φ, then every point w ∈ W lies on the orbit of a point on E, i.e. w = Lg e, for a unique e = e(W) and g = g(W). We need to show the converse, namely that each point in E lies on a unique orbit of a point in W. The mapping W → E is a local diffeomorphism, and E = L−1 g w. Hence the orbits of points on W uniquely cover a neighbourhood of X0 in Nα and the proof of Theorem 7 is complete.

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6.6 The Normal Form of Dirichlet’s Energy about a Generic Minimal Surface in R3 If X is a minimal surface such that DWα (X) is an isomorphism on Tid [OG (X)]⊥ , then the Morse Lemma provides a normal form for Dirichlet’s integral in a neighbourhood of X. However if X is simply branched and a generic minimal surface, DWα (X) is no longer an isomorphism, and thus the Morse lemma no longer applies. Therefore we need a new result which we prove in this section. We begin with The Splitting Lemma of Gromoll–Meyer We adapt the methods of Gromoll and Meyer [1] to our setting. As in the beginning of the section, we assume that f : U → R, U ⊂ E is a C r -acceptable map with a C r−1 -gradient vector field W . Suppose that the origin is a critical point and that the Hessian of f has a finite dimensional kernel J0 ⊂ E. Then J0 has a weak orthogonal complement J1 , such that E = J1 ⊕ J0 . Clearly DW (0) : J1 → J1 is a self-adjoint injective map which we assume to be an isomorphism. This will automatically be the case if DW (0) is Fredholm of index 0, as it is in Plateau’s problem. For z ∈ E, write z = x + y, x ∈ J1 , y ∈ J0 . Then: Theorem 1. Under the above conditions there is a C r−2 -diffeomorphism Φ preserving the origin, and a C r−2 -map h : V0 → J1 from a neighbourhood V0 of 0 in J0 such that (1)

f ◦ Φ(x, y) =

1 2 D f (0)(x, x) + f (h(y), y). 2

This clearly generalizes the Morse lemma and will be critical to our computation of the winding number of the minimal surface vector field about a generic minimal surface in R3 . Proof. Let V1 := U ∩ J1 . We begin by defining a map ϕ : V1 × V0 → J1 × J0 by ϕ(x, y) = (P W (x, y), y) where P : E → J1 is the (weak) orthogonal projection. Here ϕ(0, 0) = 0, and Dϕ(0, 0) is an isomorphism and thus has a local inverse φ, with φ(0, y) = (h(y), y). This defines h, and Dh(0) = 0. Moreover, for each y, (h(y), y) is a critical point of x → f (x, y). Let g(x, y) := f (x, y) − f (h(y), y)

6.6 The Normal Form of Dirichlet’s Energy about a Generic Minimal Surface in R3

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and ρ(x, y) := (x − h(y), y) a local diffeomorphism whose inverse we denote by γ. Then κ(x, y) := g(γ(x, y)) = f (x + h(y), y) − f (h(y), y). We now define a continuous bilinear form Bxy on J1 , by:

1

(1 − t)D2 f (tx + h(y), y)(u, v) dt

Bxy (u, v) := 0

1 (1 − t)DW (tx + h(y), y)[u] dt, v = P 0

=:

Axy u, v defining Axy . Since DW is self-adjoint with respect to ·, · , it follows that Axy is self-adjoint for each (x, y). By Taylor’s theorem, κ(x, y) =

Axy x, x. For (x, y) close to 0, Axy is an invertible map of J1 to itself. To simplify computations, let R := Axy , T = A00 . Then for (x, y) sufficiently close to (0, 0), R−1 T has a unique square root S, S 2 := R−1 T . Moreover, since A00 is self adjoint, we have ∗ R(R−1 T ) = R(R−1 T ) = (R−1 T )∗ R∗ . Since DW is self-adjoint, all adjoints exist. Furthermore, since S is defined as a power series in R−1 T , the adjoint of S exists and satisfies RS = S ∗ R∗ = S ∗ R. Therefore Let Q := S −1 . Then whence

T = RS 2 = S ∗ R∗ S = S ∗ RS. R = Q∗ T Q,

κ(x, y) =

Rx, x =

Q∗ T Qx, x =

A00 Qx, Qx. −1/2 For small y, x → Qx = A−1 x has a derivative at 0 which is an xy A00 isomorphism of J1 to itself. Thus the map (x, y) → (Qx, y) is a local diffeomorphism in a neighbourhood of 0 ∈ E. Denote the J1 -component of the inverse by H. Then

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κ(H(x, y), y) =

1 2 D f (0)(x, x), 2

and so

1 2 D f (0)(x, x) + f (h(y), y). 2 Setting Φ(x, y) := (H(x, y) + h(y), y) we have the result. f (H(x, y) + h(y), y) =

Using the methods of the Morse lemma we may immediately conclude ˆ surface with the boundary values X0 such Theorem 2. Let X0 be a minimal 2 that D Eα (X0 ) [Tid OG (X0 )]⊥ has a finite dimensional kernel J0 . Then there is a local slice W ⊂ Nα centered at X0 transverse to the action of the conformal group G and a local diffeomorphism Φ : [Tid OG (X0 )]⊥ → W such that for Z = x + y, x ∈ J1 , y ∈ J0 (J1 the orthogonal complement to J0 in [Tid OG (X0 )]⊥ ) (2)

Eα ◦ Φ(x, y) =

1 2 D E α (0)(x, x) + E α (h(y), y), 2

E α = the pull back of Eα to a neighbourhood of 0 ∈ [Tid OG (X0 )]⊥ ⊂ TX0 Nα . Moreover, every X ∈ Nα in a sufficiently small neighbourhood of X0 can be uniquely written as X = Lg Φ(x, y), and (3)

Eα (Lg Φ(x, y)) =

1 2 D E α (0)(x, x) + E α (h(y), y). 2

ˆ =X ˆ 0 is a generic We would like to study this result further in the case X 3 branched minimal surface in R . In this case the kernel of the Hessian of Eα at X consists only of forced Jacobi fields (which include all elements of uffler proved that X is generic if each branch point w0 Tid OG (X0 )), and Sch¨ ˆ is an interior branch point and simple, and furthermore of X (4)

ˆ w (w) = (w − w0 )F (w) X

where F (w0 ) and F (w0 ) are linearly independent over C. We would now like to use (4) and the intrinsic third derivative of Dirichlet’s integral ∂ 3 Eα (X) ˆ ww τ ρλ dw ˆ ww · X = −4 Re w3 X ∂h∂k∂l S1 where ˆ w ) = λ(eiθ )Xθ , h = Re(iwλ(w)X ˆ w ) = ρ(eiθ )Xθ , k = Re(iwρ(w)X ˆ w ) = τ (eiθ )Xθ . l = Re(iwτ (w)X Here λ, ρ, τ are meromorphic functions on the disk B, real on S 1 , with poles at the branch points, such that the order of the poles does not exceed the order of the branch points.


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ˆ has only one simple branch point, which For the moment, assume that X we may assume is at the origin 0 ∈ B. B 2 D 3 C 2 ˆ Xw = Aw + w + w + · · · , w + · · · 2 3 2 where A = (A1 , A2 ), A = 0, B = (B1 , B2 ), D = (D1 , D2 ) are vectors in C2 ˆw · X ˆ w = 0 immediately implies and C ∈ C. The minimal surface equation X 2 2 that A · A = 0, A · B = 0. Thus A1 + A2 = 0 and A1 B1 + A2 B2 = 0. Hence A1 = ±iA2 which implies that B1 = ∓iB2 or that B · B = 0. Now ˆ ww = (A + Bw + Dw2 + · · · , Cw + · · · ) X and ˆ ww = A · A + 2A · Bw + 2A · Dw2 + B · Bw2 + C 2 w2 + · · · ˆ ww · X X = (2A · D + C 2 )w2 + · · · . But 2 ˆw · X ˆ w = A · Aw2 + B · B w4 + 2 A · D w4 + A · Bw3 + C w4 + · · · 0=X 4 3 4 2 C2 4 = A·D+ w + ··· . 3 4

Therefore, 2A · D = − 34 C 2 , hence we have the expansion (5) Again

2 ˆ ww · X ˆ ww = C w2 + (· · · )w3 . X 4

ˆ w = w A + B w + · · · , C w + · · · = wF (w). X 2 2

Sch¨ uffler’s nondegeneracy condition, mentioned in the previous chapter, that F (0), F (0) be linearly independent is the condition that the vector (A, 0) and ( B2 , C) in C3 are linearly independent over C. This immediately implies that ˆ ww · X ˆ ww has an expansion about 0 of the form (5) where C = 0, and hence X in the nondegenerate case the coefficient of w2 is nonzero. This fact now puts us in a position to compute the intrinsic third derivatives in the directions of Jacobi fields. If w0 = 0 is a simple branch point the two dimensional (over R) space of Jacobi fields arising from this branch point, λwθ , on ∂D where λ(w) is a meromorphic function on D, real on ∂D with a pole of up to order 2 (if w0 = 0, the pole w0 of λ(w) would have at most order 1). The space of such λ’s is spanned by the meromorphic functions 1, w + w1 , i(w − w1 ), w2 + w12 , i(w2 − w12 ). The first three of these correspond to a basis for the orbit of the conformal

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group. Since, in our normal form (3) we have factored out the conformal group we shall be interested only in the two dimensional space of Jacobi fields spanned by w2 + w12 and i(w2 − w12 ). Consider the Jacobi field corresponding to the linear combination 1 1 1 2 2 a w + 2 − ib w − 2 = (a − ib)w2 + (a + ib) 2 . w w w The third intrinsic derivative of Eα is ∂ 3 Eα (X) ˆ ww · X ˆ ww dw. = −4 Re w3 λ3 (w)X ∂l3 1 S By the Cauchy integral theorem the value of the integral is equal to (a + ib)3 C 2 2 w dw, w3 −4 Re w6 4 S1 therefore (6)

∂ 3 Eα (X) = −4 Re ∂l3

S1

(a + ib)3 w

C2 dw. 4

By the residue theorem we obtain ∂ 3 Eα (X) = Re{−2πi(a + ib)3 C 2 }. ∂l3 We now wish to find a R-linearly independent set of λ’s, say λ, τ , so that ∂ 3 Eα (X) = 1, ∂l3 and

∂ 3 Eα (X) = 0, ∂l2 ∂h

∂ 3 Eα (X) =0 ∂h3 ∂ 3 Eα (X) = −1. ∂h2 ∂l

Pick a cube root of −2πiC 2 , say, ε, and set a1 + ib1 =

1 , ε

i a2 + ib2 = . ε

Then −2πiC 2 (a1 + ib1 )3 = 1,

−2πiC 2 (a2 + ib2 )3 = −i.

Let λ and τ be the linearly independent meromorphic functions corresponding to these choices of a’s and b’s. Then ∂ 3 Eα (X) = 0, ∂l3

∂ 3 Eα (X) = 1. ∂h3


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Similarly we see that ∂ 3 Eα = −1, ∂l2 ∂h

∂ 3 Eα = 0. ∂h2 ∂l

ˆ spanning a Suppose a nondegenerate minimal surface X one simple branch point and that α is of class C r+7 , r by Theorem 2 and Taylor’s theorem we know that there variables, Φ, so that Dirichlet’s integral Eα restricted to to the action of the conformal group has the form:

contour α has only ≥ 3 smooth. Then is a local change of a slice S transverse

1 1 Eα ◦ Φ(x, y, v) = Eα (X) + D2 E α (0)(v, v) + (x3 − 3xy 2 ) + O( (x, y) 4 ) 2 6 where v → D2 E α (0)(v, v) is a nondegenerate quadratic form, and E α := Eα |S. ˆ Now let us consider the case where a generic branched minimal surface X 3 in R has p simple interior branch points w1 , . . . , wp . Then the space of forced Jacobi fields has dimension 2p (over the reals). For each wj , j = 1, . . . , p, we obtain, as above, a two dimensional space of Jacobi fields represented by hj := λj Xθ , lj := τj Xθ , λj , τj meromorphic functions on B. Now, if j = k, any Jacobi field will have the property that the mixed derivatives, ∂ 3 Eα (X) =0 ∂h2j ∂lk

(7) and similarly

∂ 3 Eα (X) = 0. ∂lk2 ∂hj

(8)

This follows immediately from the fact that ˆ ww · X ˆ ww λ2j τk X

and

ˆ ww · X ˆ ww τk2 λj X

will have no poles at all, and hence by Cauchy’s integral theorem we obtain (7) and (8). We summarize our results by: ˆ 0 be a generic branched minimal surface in R3 spanning Theorem 3. Let X a curve α, and having p simple interior branch points. Then there is a local slice W transverse to the orbits of the conformal group and a local change Φ of variables so that the image of Φ is W, and Eα ◦ Φ(x1 , y1 , . . . , xp , yp , v) ˆ 0 ) + 1 D2 E α (0)(v, v) + 1 = Eα ( X (x3 − 3xj yj2 ) + O( (x, y) 4 ). 2 6 j=1 j p

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If we write ξj = xj + iyj , then we have Eα ◦ Φ(x1 , y1 , . . . , xp , yp , v) ˆ 0 ) + 1 D2 E α (0)(v, v) + 1 Re ξj3 + O( ξ 4 ), = Eα ( X 2 6 j=1 p

ξ = (ξ1 , . . . , ξp ). By methods of singularity theory, one can show that if p = 1, the term O( ξ 4 ) may be eliminated. If we wish to include the action of the conformal group, we have that each point X in a neighbourhood of X0 can be uniquely written as ˜ X = Lg X,

˜ ∈ W, X

and (4) becomes Eα (X) = Eα (Lg Φ(x1 , y1 , . . . , xp , yp , v)) 1 1 ξj3 + O( ξ 4 ) = Eα (X0 ) + D2 E α (0)(v, v) + Re 2 6 which is our normal form.

6.7 The Local Winding Number of Wα about a Generically Branched Minimal Surface in R3 We will define and calculate the local winding number deg(Wα , X0 , 0) about ˆ 0 in R3 with p simple branch points. a generically branched minimal surface X As a consequence of Theorem 3 in the last section it will follow that this winding number must be ±2p . We begin with The Local Degree We begin by slightly extending the results of Section 6.1. Let E be a Hilbert or Banach space and S(E) := T ∈ L(E), tT + (1 − t)I ∈ GL(E) i.e. S(E) are all those invertible linear operators which can be connected to the identity through invertible maps by a linear path. Theorem 1. S(E) is open in GL(E) and T ∈ S(E) if and only if T −1 ∈ S(E). Proof. Let T ∈ S(E). Then (tT x + (1 − t)x) ≥ δ x for all t ∈ [0, 1] and some δ > 0. Suppose that T˜ ∈ GL(E) with T˜ − T < δ/2. Then for all t ∈ [0, 1], tI + (1 − t)T˜ is invertible. Thus T ∈ S(E). This proves openness.

6.7 The Local Winding Number of Wα about a Generically Branched

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Suppose now that (tT + (1 − t)I) ∈ GL(E) for all t ∈ [0, 1]. Then T −1 (tI + (1 − t)T ) ∈ GL(E), which implies (after setting δ = 1 − t) that T −1 ∈ S(E), concluding the proof of the theorem. It is not hard to see that S(E) is contractible to the identity. Define Rc (E) := A ∈ L(E) : A = T + K, T ∈ S(E), K compact linear and GRc (E) := Rc (E) ∩ GL(E). Clearly, GLc (E) ⊂ GRc (E). What is critical for us is: Theorem 2. π0 (GRc (E)) = 2, i.e. GRc (E) has two components. We name these components GRc+ (E) and GRc− (E). Then A ∈ GRc+ (E) if and only if − −1 K ∈ GL− I + T −1 K ∈ GL− c (E), and A ∈ GRc (E) if and only if I + T c . Proof. A straight-forward consequence of the fact that π0 (GLc (E)) = 2.

We will be encountering special elements of Rc (E), namely those of the form B ∗ (I + K)B where B ∗ is the adjoint of B with respect to a weak inner product. Again, if B is invertible, B ∗ (I + K)B ∈ GRc+ (E) if and only if ∗ − (I + K) ∈ GL+ c (E) and B (I + K)B ∈ GRc (E) if and only if (I + K) ∈ − GLc (E). Let Ω ⊂ E be a bounded open set with boundary ∂Ω. Let f : Ω → E be a proper C 1 -map such that Df (p) ∈ Rc (E) for all p ∈ Ω. We shall call such an f a C 1 -Rothe mapping. From the previous remarks it follows that Rothe / f (∂Ω). maps f are proper C 1 -Fredholm maps of index zero. Assume that p ∈ Then, by properness, it follows that f (x) − p ≥ δ > 0 for all x ∈ ∂Ω, and E \ f (∂Ω) is open. By the Smale–Sard theorem there is a regular value q of f in the same component of E \ f (∂Ω) as p. Then either f −1 (q) consists of finitely many points in Ω, say x1 , . . . , xk , or f −1 (q) is empty. From the definition of regular value and the fact that f is C 1 -Fredholm of index zero it follows that Df (xj ) ∈ GRc (E) for all j. Now define 0 if f −1 (q) = ∅, (1) deg(f, Ω, p) = ! −1 (q) = {x1 , . . . , xk }, k sgn Df (xk ) if f where (2) Then we obtain

sgn Df (xk ) =

1

if Df (xk ) ∈ GRc+ (E),

−1

if Df (xk ) ∈ GRc− (E).

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Theorem 3. The function deg(f, Ω, p) does not depend on the choice of q. Proof. Let q˜ be another regular value in the component of p in E −f (∂Ω) and α(t), 0 ≤ t ≤ 1, a C 1 -path connecting q and q˜. By the Smale-transversality theorem of Section 5.1 we may assume that f is transverse to α. Then f −1 (α) q ). is a 1-dimensional manifold effecting a cobordism between f −1 (q) and f −1 (˜ q ) = (˜ x1 , . . . , x ˜l ). Suppose that f −1 (q) = (x1 , . . . , xk ) as before and f −1 (˜ Now essentially the same agreement as used in the proof of Theorem 5 in Section 6.3 (see also Elworthy and Tromba [1,2]) shows that if xk and xj belong to the same component of f −1 (α), then sgn Df (xj ) = − sgn Df (xk ), and similarly if x ˜j and x ˜k belong to the same component sgn Df (˜ xj ) = − sgn Df (˜ xk ). ˜k belong to the same component of f −1 (α), then However, if xj and x sgn Df (xj ) = sgn Df (˜ xk ). This immediately implies the independence of deg(f, Ω, p) on the choice of regular value q. We say that two Rothe maps f and g are properly homotopic if there is a proper C 1 -map F : I × Ω → E such that for each (t, x) → F (t, x), 0 ≤ t ≤ 1, is a C 1 -Rothe mapping, with F (0, x) = f (x), F (1, x) = g(x) (I the unit interval). Assume now that we have two proper C 1 -Rothe maps f and g and a point p∈ / F (I × ∂Ω). We then obtain Theorem 4 (Invariance of Degree under Homotopy). If f and g are properly homotopic, as above, then deg(f, Ω, p) = deg(g, Ω, p). Proof. This again follows using exactly the same ideas as in the proof of Theorem 5, Section 6.3 (see also Elworthy and Tromba [1,2]). We also have the following stability result: Theorem 5. Suppose f : Ω → E is Rothe and that on Ω the normal form f ◦ G(z, w) = (z, ψ(z, w)) holds. Suppose further that g : Ω → E is sufficiently C 1 -close to f . Then g is a Rothe map and if p ∈ / f (∂Ω), then p ∈ / g(∂Ω) and finally deg(f, Ω, p) = deg(g, Ω, p).

6.7 The Local Winding Number of Wα about a Generically Branched

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Proof. Since S(E) is open in GL(E), it follows that any g sufficiently C 1 -close to f is Rothe. If f ◦ G(z, w) = (z, ψ(z, w)) then ˜ w)) ˜ w) = (z, ψ(z, g ◦ G(z, ˜ sufficiently close to G. It follows that g is proper and that for G (t, x) → (tf + (1 − t)g)(x) is a proper homotopy between f and g. Since f is proper, it follows that f (x) − p ≥ δ > 0 for all x ∈ ∂Ω. If the C 1 -norm satisfies f − g < δ/2, then p ∈ / g(∂Ω). The conclusion of the theorem now follows from Theorem 3. We would now like to apply these ideas to the definition of the local winding number of a generic simply branched minimal surface in R3 . In (5) of Section 6.5 we showed that the boundary X0 a minimal surface ˆ 0 is a zero of a vector field V˜α on an open subset V ⊂ H, V representing a X coordinate neighbourhood of a slice to the action of the conformal group. If X0 is generic it is isolated and in this neighbourhood we identify X0 with 0 ∈ H. Thus 0 becomes an isolated zero of the vector field V˜α . Now DV˜α (0) is of the form B ∗ (I + K)B, and thus DV˜α (0) ∈ Rc (H). By Theorem 1, DV˜α (Y ) ∈ Rc (H) for all Y close to 0. Thus in an open neighbourhood about 0, V˜α is a Rothe map and thus a Fredholm map of index zero. Since Fredholm maps are locally proper (Corollary 1 of Section 5.1) we may find an open connected / V˜α (∂Ω). neighbourhood Ω of 0 in H such that V˜α |Ω is proper and 0 ∈ We now define the local winding number of our minimal surface vector field Wα by (3)

deg(Wα , X0 , 0) := deg(V˜α , Ω, 0).

We would like to compute this degree. Consider for the moment the vector field V on R2p defined by (4)

V (x1 , y1 , . . . , xp , yp ) := (x21 − y12 , 2x1 y1 , . . . , x2p − yp2 , 2xp yp )

or in complex notation zj = xj + iyj (5)

V (z1 , . . . , zp ) = (z12 , . . . , zp2 ).

We may perturb V slightly so that it has 2p nondegenerate zeros ξ1 , . . . , ξ2p such that det DV (ξi ) > 0 (e.g. consider the perturbation Vε (z1 , . . . , zp ) := (z12 − ε, . . . , zp2 − ε). Thus the local degree about 0 of V (and Vε ) is 2p . By the normal form Theorem 3, Section 6.6, we have a coordinate system in which H = H1 ⊕ R2p and

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(6)


V˜α (v, z1 , . . . , zp ) = (Av, V (z1 , . . . , zp )) + a small perturbation

where A ∈ GRc (H1 ). Therefore, by Theorem 5, for sufficiently small ε > 0, the local winding number (i.e. the local degree) is (7)

deg(V˜α , Ω, 0) = deg((A, Vε ), Ω, 0) = ±2p

the sign depending on whether A ∈ GRc± (H1 ). If X0 is immersed (the case of the Morse lemma), V˜α (v) = Av and (8)

deg(V˜α , Ω, 0) = sgn A = sgn DWα (X0 )|[Tid OG (X0 )]⊥ .

Thus if X0 is nondegenerate in the sense that DWα (X0 )|[Tid OG (X0 )]⊥ is an isomorphism, the local degree and sgn DWα (X0 ) agree, as they must. Let us state this formally as: Theorem 6. If X0 denotes a generic minimal surface in R3 , then deg(Wα , X0 , 0) = ±2p where p is the number of branch points. Thus if X0 is immersed the degree is ±1, otherwise it is always even. We now have the following result, which is the Morse Equality for minimal surfaces in R3 : Theorem 7 (Morse Equality). Let α represent a generic curve in R3 . Then by the index theorem for disk surfaces, there are finitely many minimal surfaces ˆ k spanning the image of α (finitely many if one imposes a three point ˆ1, . . . , X X condition, or finitely many orbits under the action of the conformal group, if one does not), and their boundaries Xj satisfy deg(Wα , Xj , 0) = 1. (9) j

Proof. About each Xj , there is a neighbourhood Ωj such that V˜α |Ω j has a local normal form (and thus is proper) and V˜α |∂Ωj = 0 (cf. Section 5.1). For α close to αwe can consider the vector field V˜α also to be defined on Ω j (the bundle Nα is locally, in fact globally trivial). By taking α sufficiently close we can guarantee that V˜α |∂Ωj = 0 and that deg(V˜α , Ωj , 0) = deg(V˜α , Ωj , 0). If we take α to be a generic curve in R4 , then for each j the zeros of V˜α in Ωj , say Xkj , 1 ≤ k ≤ lj , are classically nondegenerate, in that DWα (Xij )|[Tid OG (Xij )]⊥ are isomorphisms. Then 0 is a regular value of V˜α and

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(10)

deg(V˜α , Ωj , 0) =

lj

447

sgn DWα (Xij ).

j=1

The result now follows from Theorem 3 of Section 6.2.

Corollary 1. For a generic curve α in R3 , if there are two minima which span α(S 1 ), there must be a third, unstable, immersed surface. Remark 1. Compare this result with Section 6.7 of Vol. 1. In the generic case, one can conclude that a third surface is unstable and immersed. Proof. The local winding number about the minima must be 1 and their sum contributes a 2 to (9). If all other surfaces were branched the contribution to the left side of (9) would be congruent to 0 mod (2). This would then contradict Theorem 7, concluding the proof of this corollary. Note that this corollary can easily be generalized to the fact that the existence of an even number of minima implies the existence (in the generic case) of an unstable immersed surface. Corollary 2. The number of solutions to Plateau’s problem can be arbitrarily large. Proof. Using the Weierstrass representation theorem and Sch¨ uffler’s criterion for nondegeneracy we can construct a nondegenerate minimal surface X in R3 with p interior branch points bounding an embedded smooth Jordan curve α. The winding number deg(Wα , X, 0) = ±2p . Perturbing α slightly to a curve β in R4 where all the minimal surfaces are nondegenerate in the sense of Section 6.3, we see that β must bound at least 2p minimal surfaces.

6.8 Scholia 6.8.1 Historical Remarks and References to the Literature Morse theory has had a long and fascinating history since the pioneering work of G.D. Birkhoff and M. Morse. Their results apply to geodesics in Riemannian manifolds and, more generally, to one-dimensional variational problems. The attempts of Morse to develop an analogous theory for higher-dimensional variational problems, say, for minimal surfaces, were fruitless since his topological assumptions cannot be checked in case of concrete problems. For instance, it cannot be checked when a given minimal surface is a “homotopy critical point” in the sense of Morse. Such a topological notion of a critical point was crucial around 1940 since at this time one did not have tools available such as “differentiable manifolds of mappings”, “differentiable functions

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on such manifolds”, nor their “Morse indices”. Even in the case of finitedimensional differentiable manifolds, whose development was well underway, Morse was unaware of the relation between homotopy critical points and true critical points. His topological definition was just designed to give a tautological proof of the Morse inequalities; therefore his “topological approach” is nowadays never referred to. However, his results concerning the existence of unstable minimal surfaces obtained in cooperation with C.B. Tompkins [1–5] as well as the parallel work by M. Shiffman [2,4,5] and the new approach by Courant [12,15] were formidable achievements. These results were based on various variants of the mountain pass lemma (or, in Courant’s notation, the theorem of the wall ). We have described this theory in Chapter 6 of Vol. 1; a brief history is given in the Scholia 6.7.1–6.7.4, and an overview of the work of Marx, Shiffman, Heinz on quasi-minimal surfaces (with later contributions by F. Sauvigny and R. Jakob) is presented in 6.7.5. After J. Eells’s introduction of infinite-dimensional manifolds in the late nineteenfifties it seemed feasable to generalize Morse’s complete finite-dimensional theory to infinite dimensions. An abstract theory was developed in the sixties independently by R. Palais and S. Smale. Palais and Smale introduced the Condition (C) on the variational functional, often referred to as the Palais–Smale Condition, that replaced the compactness of the underlying space. Smale was the first to show that Morse theory did not require the Morse lemma. However, the full application of the Palais–Smale method was possible only for one-dimensional variational problems such as geodesics, whereas only a few significant applications to multi-dimensional problems were possible, and interesting geometric problems such as minimal surfaces or surfaces of prescribed mean curvature seemed to be excluded. It was Smale, in 1970, who pointed out to his student Tromba that the existence of a differentiable Morse theory for minimal surfaces was still an open question, the resolution of which would need new ideas that went beyond infinite-dimensional manifolds and the Palais–Smale condition. The index theorems presented in the preceding chapter were the first step in this direction. Both Tromba and R. B¨ ohme were aware that their index theorem ruled out a full Morse theory for disk-type minimal surfaces in R3 ; but it left open the possibility for such a theory in Rn , n ≥ 4, since in these dimensions all critical points of Dirichlet’s integral are generically nondegenerate. As there is no full Morse theory for minimal surfaces in dimension n = 3, the Morse number computed in Section 6.7 forms a certain substitute. These results obtained by A. Tromba in his papers [10, 11] are based on his earlier papers [1–7] and on the Elworthy–Tromba degree theory. In 1984 M. Struwe, building on the work of Smale and Palais, Böhme and Tromba, proved the marvellous result that the full Morse inequalities hold in Rn for n ≥ 4 (cf. Struwe [4], and [11], Chapter II; for a correction of II.5 see Imbusch and Struwe [1]). Finally, the Morse lemma was generalized by Palais to Hilbert manifolds and later by Tromba to general manifolds of maps. His techniques were used here to prove general versions of both the Morse lemma and the splitting theorem of Gromoll/Meyer.

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In the following subsection we present a version of an index theorem for 2dimensional minimal surfaces without boundary and of genus ≥ 1 in Riemannian manifolds of strictly negative sectional curvature developed by A. Tromba [25]. In this context we also refer to the thesis of J. Hohrein [1]. 6.8.2 On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory Now we introduce a finite dimensional manifold T (a Teichm¨ uller space) such that minimal surfaces of a fixed genus in a hyperbolic manifold N can be ˜ on T with described as the critical points of a proper real valued function E only nondegenerate critical points. Let (N n , G) be a C ∞ -Riemannian manifold of dimension n, with Riemannian metric G, and let M be a 2-surface. A “branched” minimal surface of the topological type of M is a pair (S, g), where g is a C ∞ -Riemannian metric on M and S : (M, g) → (N, G) is harmonic and conformal. Harmonicity means that the nonlinear Laplacian of S is 0. In local coordinates this means ∂ 1 D jk √ g g kS = 0 √ g ∂uj ∂u where D is a covariant differentiation on N with respect to G along S. Conformal means that the pull back S ∗ G of the metric G is pointwise conformal to g, i.e. S ∗ G = μg for some positive real-valued C ∞ -function μ on M . Since M is two-dimensional, there is another way to characterize such a minimal immersion. Suppose that N n ⊂ Rd is isometrically embedded in some Euclidean space. Associated to the metric g on M is a complex structure induced by taking conformal coordinates about each point x ∈ M . A harmonic map S : (M, g) → (N, G) is in fact a map into Rd , S = (S 1 , . . . , S d ). With respect to the induced complex structure we can consider the complex 2-form (1)

2 d ∂S j j=1

∂w

dw2 .

Since S is harmonic, this 2-form is a holomorphic quadratic differential on M which vanishes if and only if S is conformal. What is significant here is that, given a fixed Riemannian manifold (N, G), then finding a minimal branched immersion of M into N is tantamount to producing not only a harmonic map S : (M, g) → (N, G) but also a metric g (equivalently, a complex structure) with respect to which S is conformal. This fact was pointed out by Jesse Douglas in his paper [28] on higher genus minimal surfaces in N . Our goal is to prove an index theorem for two-dimensional minimal surfaces in (N, G) of some fixed genus greater than one. We begin by describing a functional whose critical points are minimal surfaces in N .

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To this end we introduce the generalized Dirichlet integral E(X, g) as a functional on mappings X : M → N and on metrics g. The functional E is defined by 1 g(u)(∇g X j , ∇g X j ) dμg , u = (u1 , u2 ), (2) E(X, g) = 2 M where g(p) : Tp M × Tp M → R is a Riemannian metric on M , ∇g X j is the g-gradient of the j th component function of X, and μg is the classical volume measure induced by the metric g. In local coordinates u = (u1 , u2 ) on M the functional E is given by 1 g αβ Xuα , Xuβ dμg (3) E(X, g) = 2 M where dμg (u1 , u2 ) = det(gαβ ) du1 du2 , ·, · the inner product on Rd , and we are using the standard conventions of differential geometry. Obviously, the functional E(X, g) depends only on the conformal class of g, i.e., E(X, g) = E(X, λg) for any positive C ∞ -function λ on M . One also readily verifies the invariance of E under conformal diffeomorphisms f of M , (4)

E(X, g) = E(X ◦ f, f ∗ g).

Let Hp3 (M, N ), p ≥ 2, be the manifold of all Sobolev Hp3 -smooth maps from M into N . If p = 2, Hp3 (M, N ) is a Hilbert submanifold of all H3p -maps of M into Rd , and if p > 2, Hp3 (M, N ) is a Banach submanifold of all Hp3 -maps of M into Rd . In this section we consider the function E to be defined on Hp3 (M, N ) × M−1 , M−1 the C ∞ -metrics on M of negative scalar curvature (cf. Section 4.2). We show that all critical points of E are two-dimensional minimal surfaces of N . Suppose that for fixed g0 , X0 ∈ Hp3 (M, N ) is a critical point of E with ∂E respect to the X variable, i.e. ∂X (X0 , g0 ) = 0. Then it is standard that X0 is ∞ harmonic, C -smooth and that 2 d ∂X0i dw2 ξ := ∂w i=1 is a holomorphic quadratic differential on M . We would now like to state Theorem 1. If (X0 , g0 ) is a minimum (or in fact critical) for the functional (X, g) → E(X, g),

(X, g) ∈ Hp3 (M, N ) × M−1

then X0 : (M, g0 ) → Rd is harmonic and conformal, i.e. 2 d ∂X0j ξ := dw2 = 0. ∂w j=1

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Proof. See Section 4.5.

Properties of the Flow of a Vector Field Transverse to Dirichlet’s Integral for Fixed Conformal Structures Let (M, g) again denote a compact Riemann surface without boundary and (N, G) an n-manifold isometrically embedded in Rd , with (N, G) having negative sectional curvature. Let Eg (X) := E(g, X) denote Dirichlet’s integral where we now consider the variable g as fixed. If X ∈ Hp3 (M, N ), the Sobolev embedding theorem implies that X and all its first derivatives are continuous, and if p > 2, the second derivatives are continuous. Let Σ be a component of Hp3 (M, N ). The tangent space of Hp3 (M, N ) at X consists of the Hp3 -vector fields W over X, i.e. those W ∈ Hp3 (M, Rd ) such that W (p) ∈ TX(p) N for all p ∈ M . For q ∈ M let Π(q) : Rd → Tq N be the orthogonal projection of Rd onto Tq N . We note that S : (M, g) → (N, G) is harmonic if and only if (5)

(ΔS)(p) := Π(S(p))(Δg S)(p) = 0,

where Δg is the Laplace–Beltrami operator. Define the nonlinear Laplacian Δ by equation (5); S is therefore harmonic if and only if ΔS = 0. This means in local conformal coordinates (u1 , u2 ) on (M, g) that (6)

1 D ∂S α ΔS = √ , g ∂ul ∂ul

α = 1, . . . , d.

For vector fields W over X we can define the linear Laplacian Δ (X is assumed fixed) by 1 D DW . ΔW := √ g ∂ul ∂ul

(7)

Here we are using the Einstein summation convention of summing over repeated indices, and we note that the k th component of the covariant derivais given by tives of W along X, DW ∂ul (8)

DW ∂ul

k =

∂W k + Γjlk (X)W j ∂ul

where Γjlk are the Christoffel symbols of g. We are now ready to define the vector field W on Σ whose trajectories will lead us to a harmonic map. Again fix X ∈ Σ and consider the linear partial differential operator (R := Riemann curvature tensor on (N, G)) ∂X ∂X 1 − W =: EX (W ). (9) ΔW + R W, l λ ∂u ∂ul

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Here ul are conformal coordinates such that gjk = λδjk . One checks easily that W → EX (W ) maps Hp3 -vector fields over X ∈ Hp3 (M, N ) to Hp1 -vector fields over X. Moreover, W → EX (W ) is a linear self-adjoint second order operator, and therefore, by standard elliptic theory, the Fredholm alternative holds, i.e. W → EX (W ) is surjective if and only if it is injective. Another way of saying this is that the operator W → EX (W ) is a linear Fredholm operator of index zero. Theorem 2. For X ∈ Σ, the map W → EX (W ) is an isomorphism of the Hp3 -vector fields over X to the Hp1 -vector fields over X. Proof. By the previous remarks we need only show that EX (W ) = 0 implies W = 0. Suppose EX (W ) = 0. Then, denoting the Rd -inner product by ·, ·, we have

EX (W ), W dμg = 0. M

Integrating by parts and using the fact that the curvature is negative we see that W 2 dμg = 0 M

and therefore W = 0. We now define the vector field W on Σ by W (X) := E−1 X ΔX.

Since X → EX and X → Δ(X) are smooth, X → W (X) is a smooth C ∞ vector field on Σ. Moreover it is easy to see that W (X) = 0 if and only if X is harmonic. Thus the zeros of W are precisely the harmonic maps in Σ. Theorem 3. If S ∈ Σ is harmonic, the Fréchet derivative DW (S) of W at S is a linear map of the form (Identity + Compact), mapping TS Hp3 (M, N ), the space of Hp3 -vector fields over S, to itself. Proof. The Fréchet derivative DW (S) of W at S satisfies the equation ∂S ∂S 1 ΔDW (S)h + R DW (S)h, l − DW (S)h λ ∂u ∂ul ∂S ∂S 1 = Δh + R h, l . λ ∂u ∂ul The last calculation is standard, and we leave its verification to the reader. Let ϕ := DW (S)h − h. Then we have ∂S ∂S 1 − ϕ = h. Δϕ + R ϕ, l λ ∂u ∂ul Therefore

ϕ = −E−1 S h

which is a compact linear map on the Hp3 -vector fields over S.

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Theorem 4. The derivative of Dirichlet’s energy Eg in the direction W is positive except at a critical point, i.e. DEg (X)W (X) ≥ 0 and equals zero if and only if X is harmonic. Proof. DEg (X)W = − M

ΔX, W Rd dμg

= −

ΔW + M

∂X ∂X 1 R W, l , W dμg λ ∂u ∂ul

W 2 dμg ≥ 0.

+ M

Thus if DEg (X)W (X) = 0 then W (X) = 0 and hence X is harmonic.

However, since the existence theory of harmonic maps is now well-known, we would like to know more, namely, that there is a unique energy minimizing harmonic map in each homotopy class which is also nondegenerate. For such a result we need the additional assumption of the injectivity of the induced map on the fundamental group, i.e. the incompressibility condition of Schoen and Yau [2] which is a variant of Courant’s classical condition of cohesion. We first state the relevant classical results on existence (due to Eells and Sampson [1]) and uniqueness (due to Hartman [1]). Theorem 5 (Eells–Sampson and Hartman). Let (M, g) be a compact Riemannian manifold and (N, G) a complete Riemannian manifold of strictly negative sectional curvature satisfying the existence property (E) that in every homotopy class there is a harmonic map S. If S is not unique then the image of M under S is either a point or a closed geodesic. This manifold (N, G) satisfies property (E) if N is compact or if, with respect to some embedding η into Euclidean space Rl , there is a compact set K outside of which the quadratic form G + η, βRl is positive semi-definite where β denotes the second fundamental form of the embedding. Using this we now have the following result due to F. Tomi [unpublished]. Corollary 1. In the case at hand where M is a Riemann surface of genus greater than zero, let Σ be a component of Hp3 (M, N ), where (N, G) has strictly negative sectional curvature and such that X ∈ Σ implies that the induced map on π1 , X∗ : π1 (M, p) → π1 (N, X(p)), is injective. Then for each metric g on M there is a unique harmonic map S ∈ Σ which is absolutely energy minimizing and nondegenerate. Proof. Let S be an absolute minimizer of energy. We wish to show that S is unique in its homotopy class. By Hartman [1], uniqueness fails if the image

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of S is either a point or a closed geodesic. The injectivity assumption on π1 clearly rules out the case that S(M ) is a point. It remains to show that S(M ) cannot be a closed geodesic. Suppose S(M ) = γ a closed geodesic. Let q ∈ γ be a regular value of S as a map from M to γ. Then S −1 (q) must consist of smooth Jordan curves which must be null homotopic. Each one of these curves must bound a disk. Choose one such curve C and let D be the disk in M that it bounds. Define a new map S˜ : M → N by ˜ S(u) =

S(u)

if u ∈ / D,

q

if u ∈ D.

Since the exponential map expq : Tq N → N is a covering map, the homotopy lifting property implies that the second homotopy group π2 (N ) = 0. This implies that S˜ is in that same homotopy class of S, but clearly has a lower Dirichlet integral, a contradiction. Thus S(M ) cannot be a closed geodesic, and hence S is unique in its homotopy class. We now show that S is nondegenerate. Let W∗ (S) be the derivative of the vector field W at its zero S with h in the kernel of W∗ (S). Then h satisfies a linear partial differential equation of the form (in local conformal coordinates gjk = λδjk ) (10)

Δh +

∂S ∂S 1 R h, l = 0. λ ∂u ∂ul

This is the defining equation of a Jacobi field over S. It implies that ∇

∂ ∂ul

h=0

and hence that h 2 is constant. Again, by Hartman, the rank of the Jacobian of the harmonic map S at some point must be maximal. Thus (10) implies that h vanishes at this point and hence h is identically zero. Therefore DW (S) is injective and by Theorem 3 an isomorphism. This is the meaning of the nondegeneracy of S. Again let M now be a fixed Riemann surface with genus (M ) > 1. Assume that the complete manifold (N, G) has strictly negative sectional curvature and satisfies (E). Let Σ be a component of Hp3 (M, N ) such that X ∈ Σ implies that the induced map of π1 , X∗ : π1 (M, p) → π1 (N, X(p)) is injective. From the last corollary it follows that, for each g, the critical points of Eg are nondegenerate and are the absolute minimizers of Eg . Now the nondegeneracy of the unique minimizer for Eg implies that this minimizer S(g) is a smooth function of g, as g varies. This follows from the fact that the vector field W is also a smooth function of g and thus its nondegenerate zero (by the inverse function theorem) depends smoothly on the parameter g.

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˜ : M−1 → R, defined by Consider the function E ˜ E(g) = E(g, S(g)). The uniqueness of S(g) in each component of Hp3 (M, N ) implies that f ∗ S(g) = S(g) ◦ f = S(f ∗ g) for all f ∈ D0 , where D0 denotes the set of diffeomorphisms that are homotopic (equivalently, isotopic) to the identity. This follows from the fact that f ∗ S(g) is harmonic with respect to f ∗ g and f ∗ S(g) is homotopic to S(g) if ˜ passes to a C ∞ -function E ˜ on f ∈ D0 . Thus, by the D-invariance of E, E Teichm¨ uller space T(M ). ˜ represent miniFrom our discussion it follows that critical points g of E mal submanifolds of N . We will show that for the generic negative sectional ˜ are nondegenerate in the curvature metric G on N , the critical points of E sense of classical finite dimensional Morse theory. In order to acknowledge the ˜ on G, we shall now denote E ˜ by E ˜G . For a full Morse theory dependence of E ˜ for EG (and hence for branched minimal surfaces) one needs to show that ˜G : T(M ) → R is proper. In order to prove this, it unfortunately appears E that we need a condition stronger than injectivity on the fundamental group and even stronger than injectivity on the free homotopy classes of loops. We summarize this as: ˜G : T(M ) → R yield branched minimal Theorem 6. The critical points of E ˜G is determined by a component surfaces in N of the same genus as M . If E Σ whose elements S(g) induce an isomorphism S(g)∗ : π1 (M ) → π1 (N ), then ˜G is proper. the C ∞ -function E uller space T(M ). Proof. Let g ∈ M−1 and let [g] denote its class in Teichm¨ ˜G [gn ] = E(gn , S(gn )) is bounded then [gn ] has a converWe must show: If E gent subsequence. By (now) standard arguments (cf. Chapters 4 and 5), the injectivity assumption on π1 , the boundedness of E and the Mumford compactness Theorem imply that there exists a subsequence, again say gn , and a sequence of diffeomorphisms fn ∈ D such that fn∗ (gn ) converges C ∞ to a metric g0 ∈ M−1 . By the D-invariance of Dirichlet’s integral, the sequence of numbers E(fn∗ gn , fn∗ S(gn )) is also bounded. Let us now denote the Dirichlet energy density by eg (X), thereby indicating the role of the metric g. By invariance, ef ∗ g (f ∗ X) = f ∗ eg (X). Thus a supremum bound on eg (X) implies a supremum bound on Ef ∗ g (f ∗ X). From Eells and Sampson [1] we know that 1 Δg eg (X) ≥ K(g)eg (X), 2 K(g) the Gauss curvature of g. This inequality implies that if gn converges to a metric g, we can bound the supremum norm of egn (S(gn )) in terms of its L1 norm (cf. F. Tomi [12])

456


and hence the supremum norm of egn (S(gn )) can be bounded by the energy. We thus have a supremum bound on efn∗ gn (fn∗ S(gn )). But fn∗ gn converges to g0 in the C ∞ -sense. We thus have a supremum bound on eg0 (fn∗ S(gn )). This implies that the fn∗ S(gn ) are equicontinuous and therefore have a convergent subsequence, again say fn∗ S(gn ). Thus for large n, the fn∗ S(gn ) are all in the same homotopy class. We claim that for −1 large m and n, fn ◦ fm are homotopic to the identity. In the sequel we follow the ideas of J¨ urgen Hohrein [1]. We note that Hohrein was able to develop a Lyusternik–Shnirelman theory for the general Plateau problem in strictly negatively curved manifolds. Denote the limit of fn∗ S(gn ) = S(gn ) ◦ fn by S. Pick a base point p0 ∈ M and set q0 = S(p0 ). Since S(gn )◦fn converges to S, we can find, for sufficiently large n, a sequence of diffeomorphisms ψn of N homotopic to the identity such that ψn ◦ S(gn ) ◦ fn (p0 ) = q0 . We will be working on the level of homotopy, and will therefore assume that ψn is the identity for all n. On a surface, one can clearly find a diffeomorphism homotopic to the identity taking any given point to p0 . We may therefore assume that fn (p0 ) = p0 for all n, and hence S(gn )(p0 ) = q0 . Now the induced maps S∗ , S(gn )∗ [fn ]∗ , and S(gm )∗ [fm ]∗ , will agree on the fundamental group π1 (M, p0 ). Hence S(gn )∗ = S∗ [fn−1 ]∗ ,

−1 S(gm )∗ = S∗ [fm ]∗ .

But S(gn ) and S(gm ) are freely homotopic. Let Hmn : M × [0, 1] → N be such a homotopy. The path γmn (t) := Hmn (p0 , t) is a loop at q0 . The loop γmn induces a map on π1 (N, q0 ) by conjugacy, namely a class [α] is mapped to [γmn ][α][γmn ]−1 . Denote by γmn conjugation by γmn . It now follows from elementary topology that the induced maps S∗ [fn−1 ]∗ and −1 ]∗ on π1 (M, p0 ) are conjugate, i.e., S∗ [fm −1 ]∗ .

γmn S∗ [fn−1 ]∗ = S∗ [fm

Since S ∈ Σ induces an isomorphism S∗ , by assumption of the theorem, there has to exist some loop σmn about p0 in M such that S∗ [σmn ] = [γmn ], which leads us to −1

σmn [fn−1 ]∗ = [fm ]∗ . However, by a well-known theorem in topology (Whitehead [1], Theorem 4.3 and Cor. 4.4, p. 225), the free homotopy classes of selfmaps are in one-to-one correspondence with the conjugacy classes of endomorphisms of π1 (M ). (For surfaces, this result is originally due to Nielsen, who proved that, for surfaces, D/D0 is isomorphic to the group of outer automorphisms of the fundamental group.) −1 , and hence fm ◦ fn−1 is homotopic In any case, fn−1 is homotopic to fm to the identity as claimed. Therefore fn and hm are in the same homotopy

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class as some f ∈ D. Thus in Teichm¨ uller space [fn∗ gn ] = [f ∗ gn ]. Hence the ∗ convergence of fn gn implies the convergence of [gn ] ∈ T(M ). This shows that ˜G is proper. E ˜G with Respect to the Weil–Petersson Metric The Gradient of E ˜G [g]. Let ρ ∈ T[g] T(M ) be a trace-free Theorem 7. Consider the map [g] → E divergence-free (0, 2)-tensor with respect to the metric g. Then the derivative ˜ is given by DE[g] ˜ DE[g]ρ = −2

Re ξ, ρg

(11)

where

·, · denotes the L2 -inner product introduced in the last section, and ξ :=

2 d ∂S j j=1

∂w

1 ∂ := ∂w 2

dw2 ,

∂ ∂ −i 2 ∂u1 ∂u

,

u1 , u2 the local conformal coordinates on M , and S := S(g). Since S is harmonic it follows that ξ is a holomorphic quadratic differential on M with respect to the complex structure it inherits from g (via local conformal coordinates). Its real part is then a trace-free (0, 2)-tensor. Proof. From ˜G [g] = E(g, S(g)) = 1 E 2 we infer

g M

jk

∂S ∂S , ∂uj ∂uk

dμg Rd

∂S ∂S dμg ρ , ∂uj ∂uk M ∂S ∂S 1 jk (trg ρ) dμg + g , 4 M ∂uj ∂uk

1 ∂E (g, S(g))ρ = − ∂g 2

jk

where trg ρ denotes the trace g jk ρjk of ρ with respect to g. This calculation follows from the fact that the derivative of g → g jk in the direction ρ is ρ → −ρjk , and the derivative of μg is 12 (trg ρ)μg . Since ρ is trace-free we have ∂S ∂S 1 jk ˜ dμg . ρ , (12) DEG [g]ρ = − 2 M ∂uj ∂uk Writing this out in conformal coordinates, gjk = λδjk , using the fact that ∂S ∂S ρ11 = −ρ22 (trace-free), and setting lαβ := ∂u α , ∂uβ we obtain 1 ˜ (13) DEG [g]ρ = − λ−2 {ρ11 (l11 − l22 ) + 2ρ12 l12 } dμg . 2 M

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Now viewing Re ξ as a symmetric (0, 2)-tensor we have d

1 Re ξ = Re 4 j=1 =

∂S j ∂S j − i ∂u1 ∂u2

2 dz 2

1 1 (l11 − l22 ) du1 du1 + l12 du1 du2 + (l22 − l11 ) du2 du2 . 4 4

Therefore (Re ξ)11 = −(Re ξ)22 =

1 (l11 − l22 ), 4

(Re ξ)12 = l12 ,

and thus the Weil–Petersson inner product is 1 (Re ξ), ρ = g αβ g γδ (Re ξ)αγ ρβδ dμg 2 M g 1 = λ−2 ρ11 /2(l11 − l22 ) + 2ρ12 l12 dμg 2 M 1 ˜ = − DE G [g]ρ. 2

Corollary 2. After identification of the tangent space of Teichm¨ uller space with real parts of holomorphic quadratic differentials we see that the gradient ˜G with respect to the Weil–Petersson metric on T(M ) is ˜G of E ∇E ˜G = −2(Re ξ). ∇E There is another way to look at these computations. Expression (13) can be rewritten as 1 1 ∂E (g, S(g))ρ = − ρjk − g jk trg ρ ljk dμg . (14) ∂g 2 M 2 Now the symmetric tensor ρT , (ρT )jk := ρjk − 12 (trg ρ)gjk , is the trace-free part of ρ. Thus, the same computation as above shows that (15)

ˆG [g]ρ = −2

Re ξ, ρT g . DE

The transversality theorem and generic nondegeneracy We will be needing: Theorem 8. Let N be a smooth, finite-dimensional manifold, A a smooth second-countable Banach manifold, and X : A × N → T N be a C 1 -smooth family of vector fields, i.e. for each a ∈ A, X(a, ·) : N → T N is a C 1 vector field on N. Suppose that whenever X(a, n) = 0, the total derivative DX : Ta A × Tn N → Tn N is surjective.

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Then there is a Baire subset A0 ⊂ A such that for each a ∈ A0 , the zeros of the vector field Xa := X(a, ·) are nondegenerate (i.e., if n is a zero, DXa (n) : Tn N → Tn N is an isomorphism). If the zero set of X is compact, A0 may be taken as open and dense. Proof. The details can be found in A. Tromba [27], but the argument has been used in Chapter 5. Basically the assumptions imply that the set of zeros M ⊂ A×N is a submanifold. Then a ∈ A0 if and only if a is a regular value for the projection map π : A × N → A restricted to M which is of index zero. ˜G Our goal now is to show that this transversality criteria holds for ∇E where the “parameter” is G itself. We will now assume that N is compact, without boundary, and admits a metric with strictly negative sectional curvature. Define A as the set of all metrics on N with strictly negative sectional n + 2, so curvature, topologized by some H s -Sobolev topology with s > dim 2 s that A is open (and non-empty) in the space of all H -(0, 2) tensors. With this definition A is endowed with a Hilbert manifold structure.1 Since S is a nondegenerate minimizer of Dirichlet’s integral EG (g, ·) with G and g fixed, it follows that the unique minimizer S is a smooth C ∞ -function of G (as well as of g). From formula (18) it then becomes clear that ∇E˜G is a smooth function of G as well. For simplicity of notation let us introduce the symbol XG for ∇E˜G . Thus XG : T(M ) → T T(M ) is a smooth family of vector fields on Teichm¨ uller’s moduli space. We have the identity (16)

˜G [g]ρ =

ρ, XG [g]. DE

Let H be any (0, 2)-tensor on N , thought of as a linearization of G (i.e., a tangent vector to A at G) and h a linearization of g. Denote by DX(G, [g]) the total derivative of X at (G, [g]). We must then show that (17)

(H, h) → DX(G, [g])(H, h)

is surjective. Thinking of X as a D0 -invariant vector field on M−1 , this is equivalent to showing that if ρ is a trace-free, divergence-free, symmetric (0, 2)tensor orthogonal to the range of DX then ρ = 0 i.e. if (18)

ρ, DX(G, [g])(H, h) = 0

for all (0, 2)-tensors H on N and all trace-free, divergence-free (0, 2)-tensors on (M, g) then ρ = 0. One could of course pick any Sobolev model Hpk as long as the Hpk -tensors are contained in the C 2 -tensors.

1

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Our penultimate goal is to derive a formula for (18) from which it will become apparent that DX(G, [g]) is surjective. We begin with equation (16). Since we are calculating the derivative of X at a zero, it follows that calculating a formula for (18) is equivalent to calculating the total derivative of (G, [g]) → DE˜G [g]. Clearly this total derivative will be the sum of the partial derivatives first with respect to [g] and then with respect to G. We must keep fixed in mind the fact that S is a smooth function of both g and G as they vary. We will use the notation: Vh := DS(g)h,

Vρ := DS(g)ρ

and

VH := DS(G)H

to denote the derivatives of S with respect to g and G in the directions h, ρ, H respectively. Clearly, Vh , Vρ and VH are all smooth vector fields over S, i.e. Vh : M → T N , Vh (x) ∈ TS(x) N , and analogously for Vρ and VH . In addition to this, if S is conformal, then we know that S is an immersion except for finitely many branch points. Therefore locally (except for finitely many points) the image of S is a smooth submanifold S of N . Now recall from formula (14) (19)

˜G ([g])ρ = DEG (g, S(g))ρ DE 1 ∂S α ∂S β = − ρjk Gαβ (S) j dμg 2 M ∂u ∂uk 1 ∂S α ∂S β + g jk Gαβ (S) j (trg ρ) dμ. 4 M ∂u ∂uk

We would like to differentiate this a second time with respect to g; but this seems problematic because the variable g is restricted to lie in a manifold M−1 and not in an open set of all metrics. However, since we are evaluating this ˜G , this should not matter, derivative at a zero of XG and a critical point of E and it does not. To see this, note that if σ(t, r) is a surface in M−1 (with (t, r) in a neighbourhood of zero in R2 ) such that σ(0, 0) = g, ∂σ ∂t (0, 0) = h, ∂σ ∂2σ ∂r (0, 0) = h, ∂r∂t (0, 0) = k, then in differentiating the composition (t, r) → E˜G (σ(t, r)) twice at (0, 0), first with respect to t and then with respect to r, we would obtain the extra term DE˜G ([g])k. Now from formula (18) (20)

DE˜G ([g])k = −2

Re ξ, k T g

where k T is the trace-free part of k. Since Re ξ = −2∇E˜G ([g]) = 0, this extra term vanishes. With this in mind we differentiate (19) a second time with respect to g in the direction of a trace-free h. Using the fact that the derivative of g → g jk (cf. Tromba [24], p. 71) is −g jk (ρ · h), where ρ · h := ρjk hjk is the density of the L2 -metric, and that the derivative of g → trg h is −(h · ρ), we see that

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(21)

1 4

D2 E˜G ([g])(h, ρ) =

g jk (h · ρ)Gαβ (S)

− 1 − 2

ρjk Gαβ (S)

461

∂S α ∂S β dμg ∂uj ∂uk

∂Vhα ∂S β dμg ∂uj ∂uk

ρjk Gαβ, Vh


where Gαβ, denotes the derivative of G with respect to the th variable. Lemma 1. We have (22)

−Q(Vh , Vρ ) = − −

ρjk Gαβ (S) 1 2

∂Vhα ∂S β dμg ∂uj ∂uk

ρjk Gαβ, (S)Vh


where Q(Vh , Vρ ) denotes the second variation of the function g → E(g, X) (g being fixed) with respect to X at the harmonic conformal map S in the directions Vh , Vρ . Proof. Since S(g) is a critical point of X → E(g, X) we obtain ∂E (g, S(g)) = 0 for all g. ∂X

(23)

Differentiating this again with respect to g we have ∂2E ∂2E + = 0, ∂g∂X ∂X 2 that is

∂2E = −Q. ∂g∂X

Since the right-hand side of (22) is

∂2E ∂g∂X ,

the proof is complete.

Corollary 3. We have (24)

D2 E˜G ([g])(h, ρ) =

1 4


Proof. Immediate from (21) and Lemma 1.

∂S α ∂S β dμg − Q(Vρ , Vh ). ∂uj ∂uk

˜ G , a somewhat similar comWe must now find the derivative of G → DE putation. Again from (19)

462


(25)

DE˜G ([g])ρ = −

1 2

+

1 4

ρjk Gαβ (S)


g jk Gαβ (S)

∂S α ∂S β (trg ρ) dμg . ∂uj ∂uk

Differentiating this in the direction H, keeping in mind the S dependence on G and that trg ρ = 0, we get: ∂V α ∂S β ∂ (DE˜G ([g]ρ)H = − ρjk Gαβ (S) Hj (26) dμg ∂G ∂u ∂uk ∂S α ∂S β 1 − ρjk Gαβ, (S)VH

dμg 2 ∂uj ∂uk 1 ∂S α ∂S β ρjk Hαβ (S) j − dμg . 2 ∂u ∂uk As in Lemma 1 we may write this as 1 ∂S α ∂S β ρjk Hαβ (S) j dμg , (27) −Q(VH , Vρ ) − 2 ∂u ∂uk Q the second variation of X → E(g, X) at S. We can now write out a formula for the total derivative of our vector field X, DX(G, [g])(H, h). To this end we start from the formula defining X(G, [g]) = XG ([g]) by ˜G ([g]) =

XG ([g]), ρ. DE Differentiating this at a zero of XG ([g]) (i.e. at a critical point of EG ([g])), we obtain D2 EG ([g])(h, ρ) =

D[g] XG ([g])h, ρ. Applying formulas (23), (26), (27) we obtain Theorem 9. We have (28)

ρ, DX(G, [g])(H, h) ∂S α ∂S β 1 g jk (h · ρ)Gαβ (S) j dμg − Q(Vρ , Vh ) = 4 M ∂u ∂uk 1 ∂S α ∂S β − ρjk Hαβ (S) j dμg − Q(VH , Vρ ). 2 M ∂u ∂uk

Our principal goal is now to show that if the right-hand side of (28) vanishes for all (H, h), H a (0, 2)-symmetric tensor on N, h divergence-free and trace-free (with respect to g), symmetric (0, 2)-tensors on M , then ρ = 0. This then establishes the surjectivity of DX. We will need the following formula for Q(VH , Vρ ).

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Lemma 2. (29)

−Q(VH , Vρ ) =

g jk Hαβ (S) 1 + 2

463

∂Vρα ∂S β dμg ∂uj ∂uk

g jk Hαβ, (S)Vρ

∂S α ∂S β dμg . ∂uj ∂uk

Proof. Since S is a critical point of E(g, ·) we have the first variation equation ∂Vρα ∂S β 0= g jk Gαβ (S) j (30) dμg ∂u ∂uk ∂S α ∂S β dμg . + g jk Gαβ, (S)Vρ j ∂u ∂uk Now differentiate this with respect to G keeping in mind that S depends differentiably on G, and we obtain the equation ∂Vρα ∂S β 0 = Q(VH , Vρ ) + g jk Hαβ (S) j dμg ∂u ∂uk ∂S α ∂S β 1 g jk Hαβ, (S)Vρ j + dμg 2 ∂u ∂uk giving the result.

Let us, for the remainder of our calculations, assume that N ⊂ Rd is embedded isometrically. We will write the metric on Rd either as ·, · or Gαβ ≡ δαβ . Then the same calculation shows that (29) can be written as ∂Vρα ∂S β (31) g jk Hαβ (S) j dμg −Q(VH , Vρ ) = ∂u ∂uk ∂S α ∂S β 1 g jk DHαβ (S)[Vρ ] j dμg + 2 ∂u ∂uk where {Hαβ } now represents a (0, 2)-tensor on Rd and DHαβ (S)[Vρ ] denotes the derivative in the direction Vρ . Let H = {Hβγ } be the associated (1, 1)tensor on Rd , obtained by raising an index using {Gαβ }. From here on all tensors Hαβ will be variations of the metric tensor on Rd . Continuing, we obtain ∂S ∂Vρ jk H j , k dμg (32) g −Q(VH , Vρ ) = ∂u ∂u ∂S α ∂S β 1 g jk DHβγ (S)(Vρ ) j dμg , + 2 ∂u ∂uk S = (S 1 , S 2 , . . . , S d ), Vρ = (Vρ1 , Vρ2 , . . . , Vρd ).

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Let us now summarize what we know about ρ if ρ is orthogonal to the range of DX(G, [g]). From (32) we obtain that (33)

Q(Vρ , Vh ) =

1 4



for all h, and (34)

∂S α ∂S β dμg ∂uj ∂uk 1 ∂S ∂S = − ρjk H j , k dμg 2 ∂u ∂u

Q(VH , Vρ ) = −

1 2

ρjk Hαβ (S)

for all Vρ and VH . Thus we have from (32) (35)

1 2

∂S ∂S ρ H j , k dμg ∂u ∂u ∂S ∂Vρ = g jk H j , k dμg ∂u ∂u 1 ∂S α ∂S β g jk Gαγ DHβγ (S)[Vρ ] j + dμg , 2 ∂u ∂uk

jk

and this holds for all (1, 1)-tensors H. Suppose w0 ∈ M is not a branch point of S. Then locally, around S(w0 ) the image of a small neighbourhood of w0 is a submanifold, say S of N . For the rest of our calculation we will be further assuming that our tensors H have support in a neighbourhood of S(w0 ), are trace-free, and the (1, 1)ˆ obtained from H by raising an index preserves the tangent space tensor H ˆ of S, i.e. H(p) : Tp S → Tp S, and vanish on vectors normal to S. Moreover to ease our calculations we will introduce a conformal coordinate system about w0 for g (with gjk = λδjk ). We then have: Lemma 3. Under the above assumptions (36)

g jk Gαγ DHβγ (S)[Vρ ]

∂S α ∂S β dμg = 0. ∂uj ∂uk

Proof. For p ∈ Rd , let ei (p) be an orthonormal basis for Rd with e1 (p), e2 (p) an orthonormal basis for Tp S and with p → ei (p) smooth. Since H is trace-free on S we have (37)

β β α α 0 = Hβα (S)eα 1 (S)e1 (S) + Hβ (S)e2 (S)e2 (S).

Differentiating this with respect to S in the direction W we obtain

6.8 Scholia

0=

2

β DHβα (S)[W ]eα j (S)ej (S) + 2

j=1

2

465

β Hβα (S)Deα j (S)[W ]ej (S).

j=1

The second term is equal to (38)

2

2

HDej (S)[W ], ej (S) = 2

j=1

2

Dej (S)[W ], Hej (S) .

j=1

Write He1 = ae1 + be2 , then He2 = be1 − ae2 . Then the expression given by the right-hand side of (38) becomes 2a De1 (S)[W ], e1 − 2a De2 (S)[W ], e2 + 2b { De1 (S)[W ], e2 + De2 (S)[W ], e1 } . Since the ej (S), j = 1, 2 are an orthonormal frame all these terms vanish. ∂S Now the conformality of S yields the lemma (ej (S) = μ ∂u j ). We now know that for all (1, 1)-tensors as above ∂S ∂S ∂S ∂Vρ 1 1 jk jk ρ g H j , k dμg = H j , k dμg (39) 2 ∂u ∂u 2 ∂u ∂u and in local conformal coordinates this becomes 1 ∂S ∂S ∂S ∂Vρ ij ρ H i , j du1 du2 = (40) H j , j du1 du2 2 ∂u ∂u ∂u ∂u ∂S = H j , ∇ ∂ j Vρ du1 du2 ∂u ∂u #T ∂S " du1 du2 = H j , ∇ ∂ j Vρ ∂u ∂u where the superscript T denotes tangential components along S, and where (in conformal coordinates) ∇ ∂ j Vρ denotes the Gαβ covariant derivative of Vρ ∂u along S. Since (40) holds for all test tensors H which preserve that tangent space of S and since the Jacobian of S has maximal rank, we have the following Definition 1. We say that a pair of vector fields (Y, Z) on S are conjugate if Y (w) = a

∂S ∂S +b ∂u1 ∂u2

and

Z(w) = −b

∂S ∂S +a . ∂u1 ∂u2

For j = 1, 2 define the two vector field pairs (Yj , Zj ) on S by %T %T $ $ ∂S ∂S − ρ11 1 , Z1 = ∇ ∂ 2 Vρ − ρ12 1 , Y1 = ∇ ∂ 1 Vρ ∂u ∂u ∂u ∂u (41) %T % $ $ T ∂S ∂S Y2 = ∇ ∂ 1 Vρ − ρ21 2 , Z2 = ∇ ∂ 2 Vρ − ρ22 2 ∂u ∂u ∂u ∂u in suitable local conformal coordinates.

466


Theorem 10. If ρ is orthogonal to the range of the total derivative of the vector field X on Teichm¨ uller moduli space, then the local vector field pairs (Yi , Zi ) are always conjugate. Proof. This is a consequence of some elementary linear algebra and standard arguments in the calculus of variations. As a first step we prove: Lemma 4. Let Y and Z be vectors in R2 with v1 , v2 orthonormal vectors. Suppose that Y and Z are not conjugate to v1 and v2 in the sense that there do not exist scalars a and b such that Y = av1 + bv2

and

Z = −bv1 + av2 .

Then there exists a trace-free, symmetric, linear map T : R2 → R2 such that

T v1 , Y + T v2 , Z = |Y |2 + |Z|2 . Proof. We divide the proof into four cases. Case (i): Y and Z are linearly independent and Y is not orthogonal to Z. Set

Y

Y = av1 + bv2 ,

Z = cv1 + dv2 ,

⊥

Z ⊥ = dv1 − cv2 .

= bv1 − av2 ,

We would like to define T by T v1 := Y + αY ⊥ , T v2 := Z + βZ ⊥ , where α and β are to be chosen so that T is trace-free and symmetric. The condition that T is symmetric is

T v1 , v2 = T v2 , v1 , i.e.

' ' & &

Y, v2 + α Y ⊥ , v2 = Z, v1 + β Z ⊥ , v1 ,

or equivalently α(−a) − βd = c − b. The condition of being trace-free is 0 = T v1 , v1 + T v2 , v2 ' ' & & = Y, v1 + α Y ⊥ , v1 + Z, v2 + β Z ⊥ , v2 , that is, αb − βc = −a − d.

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467

Thus the system of equations for α and β can be written in matrix form −a −d α c−b = . b −c β −a − d If Y is not orthogonal to Z then ac + bd = 0, and so the linear equation above possesses a unique solution. Thus the assertion is proved in case (i). Case (ii): Y and Z are orthogonal and non-zero. Set Y = av1 + bv2 , Z = α(bv1 − av2 ),

α = 0.

Since Y and Z are not conjugate to (v1 , v2 ) it follows that α = −1. Define α 1 T v1 := Y + Z = (a + b)v1 + (b − a)v2 , 1+α α α 1 T v2 := Z − Y = (b − a)v1 − (a + b)v2 . 1+α α T is clearly trace-free and symmetric. Case (iii): Z = 0, Y = 0. Define T v1 := Y = av1 + bv2 , T v2 := W = bv2 − av1 . T is again trace-free and symmetric. Case (iv): Z = αY , α = 0, Y = 0. Define T v1 := (1 + α2 )Y = (1 + α2 )(av1 + bv2 ), T v2 := (1 + α2 )Y ⊥ = (1 + α2 )(bv1 − av2 ). This completes the proof of Lemma 4.

Now to finally complete the proof of Theorem 10 consider equation (39). ∂S ∂S We know that ∂u 1 and ∂u2 are orthogonal and of the same length, and are nonzero except at finitely many points which we shall ignore. Let U be an open coordinate neighbourhood of M and assume that (Yj , Zj ), j = 1 or j = 2, is not conjugate on U . Then U is the union of open dense sets of points such that either cases (i)–(iv) as described above hold. Let V ⊂ M be any open set on which one of these conditions hold. Then for each p ∈ S(V ) we can define a symmetric, trace-free, (1, 1)-tensor Tp : Tp S → Tp S as described above, with Yp , Zp vector fields on S, such that p → Tp is smooth. For any point w0 ∈ V let Ψ : V → R+ be a smooth

468


function with Ψ = 1 on a neighbourhood of w0 and having compact support in V . Define Hw := Ψ (w)TS(w) as “test tensor”. Going back now to (40), let Z1 and Y1 be the local tangent fields to S that were defined by $ Y1 = ∇

%T ∂ ∂u1

Vρ

− ρ11

∂S , ∂u1

$ Z1 = ∇

%T ∂ ∂u2

Vρ

− ρ12

∂S ∂u1

then using the symmetry of H and ρ and the fact that they are trace-free, (40) can be written as ∂S ∂S (42) H 1,Y + H 2,Z du1 du2 = 0 ∂u ∂u for all symmetric, trace-free H. It now follows from Lemma 4 that Z1 and Y1 must vanish on U , establishing half of Theorem 10. The result for (Y2 , Z2 ) follows in exactly the same way. Thus Yj and Zj are zero on U and hence conjugate. This contradiction shows global conjugacy, establishing Theorem 10. To obtain the final result that: If ρ is orthogonal to the total derivative of X then necessarily ρ = 0, we must now deal with the situation that the pairs (Yj , Zj ) are globally conjugate. Again let U be a local coordinate neighbourhood in M . We may assume that U contains no branch points of S. Let S be the image of U under S, and assume further that on U the metric g is in conformal coordinates gjk = λδjk . Then there exist local functions a, b, A, B such that " " (43)

" "

∇ ∇ ∇ ∇

#T ∂ ∂u1

Vρ #T

∂ ∂u2

Vρ #T

∂ ∂u1

Vρ #T

∂ ∂u2

Vρ

− ρ11

∂S ∂S ∂S = a 1 + b 2, 1 ∂u ∂u ∂u

− ρ12

∂S ∂S ∂S = −b 1 + a 2 , ∂u1 ∂u ∂u

− ρ21

∂S ∂S ∂S = A 1 + B 2, ∂u2 ∂u ∂u

− ρ22

∂S ∂S ∂S = −B 1 + A 2 . 2 ∂u ∂u ∂u

Notice that since S is conformal it follows that A = a + ρ11 ,

(44)

B = b − ρ12 .

Let us now denote the local variables by (u, v) instead of (u1 , u2 ). Since for any metric g the map S(g) : (M, g) → (N, G) is harmonic, it follows that (45)

γ(S) dw2 :=

2 2 ∂S − ∂S − 2i ∂S , ∂S dw2 ∂u ∂v ∂u ∂v

6.8 Scholia

469

is a holomorphic quadratic differential on M . Thus ∂ γ(S(g)) ≡ 0 for all g. ∂w

(46)

Differentiating (46) with respect to g in the direction ρ and using the fact that for S minimal γ(S) ≡ 0, we claim that on U ∂S ∂Vρ ∂S ∂Vρ ∂S ∂Vρ ∂S ∂Vρ (47) , − , − 2i , + , ∂u ∂u ∂v ∂v ∂u ∂v ∂v ∂u is holomorphic. In (47) we may replace the derivatives of Vρ by the tangential components of the covariant derivatives. To prove holomorphy we must write γ(S(g)) in terms of g. Let J be the associated almost complex structure. Then in a fixed coordinate system (u, v) with basis e1 , e2 (48)

γ(S(J)) := γ(S(g)) 2

2

= |DS(e1 )| − |DS(Je1 )| − 2i DS(e1 ), DS(Je1 ) where D denotes the derivative of S. If H = (ρjk ) is the linearization of J, then taking the derivative of (48) in the direction H we get that ∂S ∂S ∂ , ∇ ∂ Vρ − , ∇ ∂ Vρ 0 = 2 ∂u ∂v ∂w ∂u ∂v ∂S ∂S ∂ , ∇ ∂ Vρ + , ∇ ∂ Vρ − 2i ∂v ∂u ∂w ∂u ∂v +2

∂ ( DS(Je1 ), DS(He1 ) − i D(Se1 ), DS(He1 )) . ∂w

1 ∂S Since DS(He1 ) = ρ11 ∂S ∂u +ρ2 ∂v , it follows from (43) that on U this is equivalent to

2

∂S 2 1 ∂ ∂ 1 1 ∂S 1 (ρ1 − iρ2 ) + −i (ρ1 − iρ2 ) 0= ∂w ∂u ∂w ∂u

∂ = (1 − i) ∂w

(ρ11

−

2 1 ∂S iρ2 )

∂u

which implies that (47) is holomorphic. Thus 2 2 ∂S 1 ∂S (ρ11 − iρ12 ) = (ρ11 − iρ12 ) ∂u λ ∂u

470


is holomorphic. But ρ11 − iρ12 is holomorphic. If ρ11 − iρ12 is zero, we have 2 achieved our goal. If ρ11 − iρ12 is not identically zero it follows that λ1 | ∂S ∂u | is holomorphic and therefore a constant, say c, on the open set U . Since U 2 was arbitrary, it follows that λ1 | ∂S ∂u | is globally equal to c = 0. Thus S is an immersion. To get ρ = 0 we now use the fact that the scalar curvature of (M, g) is −1. Thus, if in local coordinates Δg denotes the Laplace–Beltrami operator, we have 1 Δg log λ = 1. 2

(49)

Lemma 5. If S is conformal, then away from a branch point, in local conformal coordinates, we have (50)

2 2 ∂S 1 ∂S 1 Δg log = − KN (S), 2 ∂u λ ∂u

KN (S) the planar sectional curvature of N at S determined by

∂S ∂u

and

∂S ∂v .

∂S 2 2 Proof. Let ϕ := | ∂S ∂u | = | ∂v | and Ψ := log ϕ. Then

ΔΨ =

ϕuu + ϕvv − ϕ

(ϕu )2 + (ϕv )2 ϕ2

.

Now, since S is harmonic, ϕu = 2 Du Su , Su = 2 Du Sv , Sv = −2 Dv Sv , Su , ϕv = 2 Dv Sv , Sv , where again D denotes covariant derivative, and we also use Su and Sv to denote partial derivatives. Differentiating again we have 1 ϕvv = Dv Dv Sv , Sv + Dv Sv , Dv Sv , 2 1 ϕuu = Du Du Sv , Sv + Du Sv , Du Sv . 2 Note also that, since Su , Sv = 0,

Du Su , Sv = − Du Sv , Su ,

Dv Su , Sv = − Dv Sv , Su .

6.8 Scholia

471

We claim that (51)

Dv Dv Sv , Sv + Du Du Sv , Sv = R(Su , Sv )Su , Sv .

R the curvature tensor on N . To see this, note that Du Du Sv = Du Dv Su = Dv Du Su + R(Su , Sv )Su . Since S is harmonic Du Su + Dv Sv = 0 and (51) follows. We may immediately conclude that % ϕuu + ϕvv 2 2$ 2 2 = R(Su , Sv )Su , Sv + |Dv Sv | + |Du Sv | . ϕ ϕ ϕ However Dv Sv = Dv Sv , Su

Su 2

+ Dv Sv , Sv

2

+ Du Sv , Sv

|Su |

Sv 2

|Sv |

and Du Sv = Du Sv , Su

Su |Su |

Sv |Sv |

2.

Note that

Du Sv , Sv = Du Su , Su = − Dv Sv , Su and

Du Sv , Su = − Sv , Du Su = Dv Sv , Sv . Consequently % 2 ϕuu + ϕvv 4 $ 2 2 = R(Su , Sv )Su , Sv + 2 Dv Sv , Su + Dv Sv , Sv . ϕ ϕ ϕ But % 4 $ (ϕu )2 + (ϕv )2 2 2

D = S , S +

D S , S v v v v v u ϕ2 ϕ2 and 1 ϕ

R(Su , Sv )Su , Sv λ ϕ2 ϕ KN (S). =− λ

1 2 Δg log |Su | = 2

From Lemma 5 and the observation that ρ = 0 implies λ−1 |Su |2 = const =: c we obtain

472


Lemma 6. Setting μ := (52)

1 2 cλ |Su | ,

then if ρ = 0, we have on all of M that

1 0 = − Δg log μ = cμKN (S) + 1, 2

c > 0.

Thus KN (S(x)) = −1/cμ on all of M . By rescaling the metric on N we may assume that c = 1 and so |Su |2 = λ globally. We therefore see that if ρ = 0 then S must be a conformal isometric immersion of (M, g) into (N, G). We claim that we can say more, namely that S is totally geodesic. Theorem 11. If ρ = 0 in the situation described above, the map S : (M, g) → (N, G) is a totally geodesic Riemannian immersion. Proof. We need to show that the “covariant derivative” ∇DS ≡ 0. Note that ∇DS is a tensor of type (1, 2) and is an element of Hom(T M ⊗ T M, S ∗ T N ). In local coordinates ∇DS = Aγjk duj ⊗ duk ⊗

∂ ∂uγ

where (53)

Aγjk = −

γ ∂S γ

∂S Γ + ∇ . ∂ ∂uj ∂uk ∂u jk

Let

∂S α ∂S β ∂uj ∂uk be the Dirichlet energy density, and note that e(S) := g jk Gαβ (S)

2 |Su |2 = 2. cλ Then since S is harmonic we have the Bochner formula (Tromba [24], p. 165) which in our situation reduces to (54)

1 2 Δg e(S) = KM (g)e(S) − KN (S)e(S) + |∇DS| . 2

By Lemma 6 we have KN (S) ≡ −1, and since e(S) = constant, KM = −1, Δg e(S) = 0, and so ∇DS ≡ 0, concluding the proof of the theorem. We are now in a position to prove the surjectivity of the derivative of the minimal surface vector field. Theorem 12. If ρ is a quadratic differential orthogonal to the range of the total derivative of the minimal surface vector field X, then ρ = 0.

6.8 Scholia

473

Proof. Consider the defining equation of the harmonic map S(g) : (M, g) → (N, G) as g varies in M−1 : 1 jk √ ∂S (55) g k =0 √ ∇ ∂j g g ∂u ∂u for all g. If ρ = 0, then by the last theorem we know that S is totally geodesic. Differentiating the above equation in the direction of a transverse trace-less tensor ρ we obtain " √ # ∂S ∂S 1 jk jk g∇ ∂k Vρ + g R Vρ , j √ ∇ ∂j g ∂u ∂u g ∂u ∂uk 1 √ ∂S = √ ∇ ∂ j ρjk g k g ∂u ∂u ∂S ∂S jk = ΔVρ + g R Vρ , j . ∂u ∂uk In conformal coordinates this can be written as ∂S 1 ∂S 1 1 ∂S (56) ΔVρ + R Vρ , j . = ∇ ∂ j ρjk λ ∂u ∂uj λ ∂u λ ∂uk Now we want to show that the right-hand side of (56) is zero. To this end we prove the following result: Lemma 7. In the case S that is totally geodesic and ρ is trace free as well as divergence free, the right-hand side of (56) vanishes identically. Proof. ∇

∂ ∂uj

ρjk

1 ∂S λ ∂uk

=

∂ρjk ∂uj

1 ∂S λ ∂uk

Since ρ is transverse traceless, for each k, ∂S ρjk ∇ ∂ j ( λ1 ∂u k ) which equals

+ ρjk ∇

∂ρjk ∂uj

∂ ∂uj

1 ∂S λ ∂uk

.

= 0. Now consider the term

∂u

−ρjk

1 ∂λ ∂S 1 ∂S + ρjk ∇ ∂ j . 2 j k λ ∂u ∂u λ ∂u ∂uk

Writing this out and using the fact that S is trace free this term becomes ) " ρ # ( 1 ∂λ ∂S ∂S ∂S 1 ∂λ ∂S ∂S 11 − (57) + + ∇ − ∇ ∂ ∂ ∂u2 ∂u2 λ λ ∂u1 ∂u1 λ ∂u2 ∂u2 ∂u2 ∂u1 ∂u1 ( ) 1 ∂λ ∂S 2ρ12 1 ∂λ ∂S ∂S − . + − + ∇ ∂ ∂u1 ∂u2 λ 2λ ∂u2 ∂u1 2λ ∂u1 ∂u2

474


Using the fact that S is harmonic and that the Christoffel symbols Γjk associated to g satisfy 2 2 1 1 = Γ21 = Γ11 = −Γ22 = Γ12

1 ∂λ , 2λ ∂u1

1 1 2 2 = Γ21 = Γ22 = −Γ11 = Γ12

1 ∂λ 2λ ∂u2

we can write the expression (57) as ( ) 2ρ11 ∂S 1 ∂S 2 ∂S −Γ11 1 − Γ11 2 + ∇ ∂ 1 ∂u ∂u1 λ ∂u ∂u ( ) 2ρ12 ∂S 1 ∂S 2 ∂S −Γ12 1 − Γ12 2 + ∇ ∂ 1 + . ∂u ∂u2 λ ∂u ∂u Comparing this with the local coordinate formula for ∇DS, we see that both terms above vanish identically. This proves Lemma 7. Returning to the proof of Theorem 12 we now conclude that ∂S ∂S jk ΔVρ + g R Vρ , j = 0. ∂u ∂uk Since N has strictly negative sectional curvature this implies that Vρ vanishes identically. Considering equation (28) and using the fact that the energy density is constant, it follows that h · ρ dμg = 0 for all h, or ρ = 0, which was our goal.

Then applying the transversality Theorem 8 we obtain an index theorem for closed minimal surfaces in (N, G) and a consequent generic nondegeneracy result: Theorem 13. Let Σ be a homotopy class of maps from a two-dimensional closed manifold M into a strictly negatively sectionally curved complete Riemannian manifold (N, G) with property (E) of Theorem 5, and assume that ˜G be the X ∈ Σ induces an injective map on the fundamental group. Let E uller’s moduli space whose critical C ∞ -smooth Dirichlet integral on Teichm¨ points are the branched minimal 2-surfaces in (N, G), contained in the homotopy class Σ, and with genus equal to that of M . Then for a Baire subset of ˜G are metrics on N with strictly negative curvature, the critical points of E nondegenerate. ∼ = If X ∈ Σ would induce an isomorphism X∗ : π1 (M ) → π1 (N ), then ˜G : T(M ) → R were proper and therefore classical Morse theory would E apply.

6.8 Scholia

475

We note that, if N = M and Σ is the component of the identity, then ˜G is pluri-subharmonic, and properness implies that for all G, the function E T(M ) is a Stein manifold (cf. Tromba [24]). With regards to Morse theory when X∗ is only injective on π1 (M ), we ˜G is equivariant on T(M ) with respect to the action of the surknow that E face modular group Γ = D/D0 (see Section 4.2). By Mumford compactness, ˜G induces a proper map on the Riemann moduli space R(M ) = T(M )/Γ . E Moreover Γ acts properly, but not freely on T(M ). The equivariant Morse ˜G with nondegenerate critical points, theory for equivariant functions like E where the action of Γ might not be free, but is proper, was worked out by N. Hingston [1]. The end result of her theory is that, in stating the Morse relations, equivariant cohomology replaces cohomology. Finally, a recent result by Kahn and Markovic shows that the condition of injectivity is satisfied by many examples. They have proved that the fundamental group of any closed hyperbolic 3-manifold contains a proper subgroup which is the fundamental group of a surface.

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The following references are selected from the bibliography of Vol. 1. Here we only cite authors whose results are relevant for the topics discussed in this volume. We also refer the reader to the following Lecture notes: MSG:

SDG: SMS: TVMA: GACG:

GTMS:

Minimal submanifolds and geodesics. Proceedings of the Japan–United States Seminar on Minimal Submanifolds, including Geodesics, Tokyo, 1977. Kagai Publications, Tokyo, 1978 Seminar on differential geometry, edited by S.T. Yau, Ann. Math. Studies 102, Princeton, 1982 Seminar on minimal submanifolds, edited by Enrico Bombieri. Ann. Math. Studies 103, Princeton, 1983 Th´ eorie des vari´ et´ es minimales et applications. S´ eminaire Palaiseau, Oct. 1983– June 1984. Ast´ erisque 154–155 (1987) Geometric analysis and computer graphics. Proceedings of the Conference on Differential Geometry, Calculus of Variations and Computer Graphics, edited by P. Concus, R. Finn, D.A. Hoffman. Math. Sci. Res. Inst. 17. Springer, New York, 1991 Global theory of minimal surfaces, edited by D. Hoffman. Proceedings of the Clay Mathematics Institute 2001 Summer School, MSRI, Berkeley, June 25–27, 2001. Clay Math. Proceedings 2, Am. Math. Soc., Providence, 2005

We also mention the following report by H. Rosenberg which appeared in May 1992: Some recent developments in the theory of properly embedded minimal surfaces in R3 . S´ eminaire Bourbaki 34, Exp. No. 759 (1992), 73 pp. Furthermore, we refer to: EMS: Encyclopaedia of Math. Sciences 90, Geometry V, Minimal surfaces (ed. R. Osserman), Springer, 1997. This volume contains the following reports: I.

D. Hoffman, H. Karcher, Complete embedded minimal surfaces of finite total curvature. II. H. Fujimoto, Nevanlinna theory and minimal surfaces. III. S. Hildebrandt, Boundary value problems for minimal surfaces. IV. L. Simon, The minimal surface equation.

U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften 341, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11706-0,

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Index

2M , 270 α-stationary, 154, 155 α-minimal cone, 164 α-stability of cones, 169 α-stable, 156, 196 α-stable hypersurface, 194 α-stationary submanifold, 154 t and ν, 40 A, 318, 258 A(M ), 256 A(Mrot ), 169 CS , 265 CS /D0 , 265, 270 C/D, 260 C/D0 , 260, 263 D0 , 256 D(M ), 256 D, 318, 332 Dν , 320, 332 Dν (s), 320 E-type, 74 E(Γ, S), 75 n (s), 316 H∗,ν n (t; s), 314 Hλν n , 316 H∗,ν Hνn , 314 Hν = Hνn=1 , 309 n , 307 Hλ0 n , 305, 315 Hλν n (t, s), 315 Hλν 1 , 307 Hλ0 Hn , 305, 314–316 n (s; t), 305 Hλν χ(V ), 404 χ(Wα ), 413

J(X), 329 J(X) = J(X), 329 M−1 := K −1 (−1), 259 M−1 /D, 260 M−1 /D0 , 260, 263 M/P, 258 MS −1 , 268 MS −1 /D0 , 270 Mλν , 303, 326 Mn λν , 304 N∗,ν := A × Dν = Pν , 320 Nα , 405 , 412 Nα ∗ , 412 Nα Nνλ , 321 Nνλ (t, s), 321 N := A × D, 318 N = A × D, 332 N(α)(τ ), 365 ND(X), 376 Og (D), 260 P, 258 (P), 277 (P0 ), 277 Pν = A × Dν , 332 QD(M ), 357 R(M ) := C(M )/D(M ), 256 R(M ) = T (M )/Γ , 256 S, 412 S := α∈A S(α), 345 S(α), 345 T (2M ), 265 T := C/D0 , 256 T = T (M ), 256 T -type, 74, 100

U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften 341, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11706-0,

531

532

Index

T (Γ, S), 75 C[w], 305 W, 325 Wα , 324 Wα (τ ), 366 Zn , 305 δEα (u, ϕ), 171 ΔM |A|2 , 162, 177, 182, 188, 219, 233 δ 2 Eα (M, X, Z), 152 δ 2 A(X, λ), 54 δ 2 A(X, Y ), 54 δ A(M, X), 147 δ Eα (M, X), 152 dim ND(X), 376 Γ := D/D0 , 256 Aˆ, 351 ˆ (α)(τ ), 365 N ˆ μν (τ ), 380 N ˆ (τ ), 380 N E-interval, 43 π : A × D → A, 324 π0 (GLc (H)), 402 Πμν = Π|Wμν , 385 πνλ := π|Wλν , 324 Π : N → Ar , 385 ΣX := ker DW(X), 331 Σ ⊂ MS −1 , 283 Γ := π(Γ ), 39 |∇M A|2 , 181, 187 |A|2 , 161 A A priori estimates, 413 An , 305 Abelian differential, 355, 356 Adjoint of Henneberg’s surface, 10 Ahlfors, L., 296 Almgren, F.J., 240 Almost complex structure J on M , 256 Annulus-type minimal surface spanned by two interlocking curves, 251 Area A = Hn (M ∩ K), 147 Area A of the symmetric graph Mrot , 169 Area element dA, 53 Area estimate, 112 Area formula, 147 Area functional A(X), 54 Associated vector field W to f , 423 Assumption A, 11 Asymptotic expansions, 21, 23, 76–78, 80, 82 Asymptotic growth condition, 184

B Banach, S., 399 Barbosa, J.L., 169 Bernatzki, F., 297 Bernstein, S., 135 Bernstein result for α-stable solutions of the symmetric m.s.e., 196 Bernstein theorem, 108, 236, 238 Bernstein theorem for α-stable hypersurfaces, 194 Bernstein theorem in a wedge, 108 Bernstein type result, 194 Bernstein’s theorem for minimal graphs in Rn+1 , n ≤ 5, 184 Bers, L., 236, 238 Birkhoff, G.D., 447 Bj¨ orling’s problem, 7 B¨ ohme, R., 299, 398, 399 Bombieri, E., 144, 167, 207, 238, 240–242, 245, 246 Bombieri–De Giorgi–Giusti examples, 240 Boundary configuration Γ, S, 11, 41 Brakke, K.A., 127 Branch points, 57 Branching type (λ, ν), 303 Bshouty, D., 132 Bundle N of branching type (μ, ν), 385 Bundle projection map πνλ = π|Wλ ν , 345 C Caffarelli, L.A., 241 Calabi, E., 245 Cao, H.D., 245 Chen, Y.-W., 127, 128 Cheng, S.Y., 245 Chern, S.S., 236, 242 Ciarlet, P.G., 35 Class M (Γ, S), 60 Class C(Γ, S), 41 Class M∗ (Γ, S), 43 Class M(Γ, S), 41 Class M+ ∗ (Γ, S), 97 Class M+ (Γ, S), 97 Coarea formula, 176 Codazzi equations, 160 n (s; t); Hn ), 305 codimR (Hλν n ; Hn ), 305 codimR (Hλν Codimension of Nμν (τ ) in N(τ ), 385 Codimension of Nνλ in N, 321 ˆ μν (τ ) in N ˆ (τ ), 385 Codimension of N

Index Cohn, H., 355 Collar theorem, 287 Complete stratification, 344 Complex structure, 255 Components of GLc (H), 402 Concus, P., 35 Condition (∗R ), 110 Condition (A), 73 Condition (B), 45, 46, 48 Condition (C), 48 Condition (C∗ ), 65 Condition (M1), 46, 48 Condition (M2), 46, 48 Condition of cohesion, 284 Condition (R), 72 Cone stability, 167 Configuration Γ, S, 37 Coordinate charts, 255 Corank of D W(X), 326 Corank of DV C(τ )(X), 376 Corank of the derivative DV K(τ )(X), 378 Courant, 128, 284, 285, 297 Courant–Shiffman theorem, 285 Courant’s method, 297 Covariant derivative, 159 Critical points of E, 283 Curvature of ∂G, 40 Cusp, 5, 26, 27 Cylinder surface S := Σ0 × R, 39 D DWα (X), 410, 411 De Giorgi, E., 144, 167, 238–240, 242, 245 Dierkes, U., 164, 169, 172, 226, 242, 246 Directrix Σ0 , 39 Dirichlet integral of the normal N , D(N ), 53 Dirichlet’s integral, 366, 430 Dirichlet’s integral Eα : Nα → R, 405 Dirichlet’s integral Eα (X, τ ), τ ∈ T (M ), 366 Divisor, 356 D(N ), 53, 54, 56 DoCarmo, M., 169 Domain G, 39 D¨ orfler, W., 35 Douglas condition, 129 Douglas’s sufficient condition, 255 Dziuk, G., 35

533

E E-type, 100 Eα : Nα → R, 430 Eα (M ), 152 Eα (u), 171 Eα (X), 405 Eα (X, τ ), 366 E(Γ, S), 86 Earle, C., 263 Ebin, D., 262 Ebin–Palais lemma, 262 Ecker, K., 218, 241, 245 Edge-creeping, 43, 88, 106, 127, 133 Eells, J., 263 Elworthy, K.D., 444 Embedded minimal surfaces of annulustype, 128 Equivalent nonparametric representation, 100 E(X, g), 278 Euler characteristic χ(Wα ) of Wα , 411, 413 Euler-characteristic of a Palais–Smale vector field, 404 Existence of unstable minimal surfaces, 447 Existence theorem for the Douglas problem, 293, 295, 296 Exponential map, 434 Exterior solutions, 141 F Federer, H., 146, 176, 245 Ferus, D., 240 Finn, R., 144, 236, 237, 238 First variation of A, 147 First variation of Eα , 152 Fischer, A.E., 259, 260, 262, 271, 297 Fischer-Colbrie, D., 244 Fleming, W., 236, 239, 245 Forced Jacobi field for a minimal surface X, 328, 329, 411 Formula for the index, 303 Fredholm, E.I., 399 Fredholm index, 299 Fredholm map, 299, 385 Fredholm vector field, 404 Free second variation of X, 69 2 Free second variation of X, δ A(X, λ), 69 Free trace, 5, 13 Freely stable, 69 Fr¨ ohlich, S., 244

534

Index

G Gauss curvature K(g), 259 Gauss equation, 158 Gauss map N , 78, 79, 83, 115 Gauss map N of a minimal surface X : B → R3 , 52 Gauss map N of X, 56 Gauss–Bonnet formula, 265 Gauss–Bonnet theorem, 323 General problem of Douglas, 250 General problem of Plateau, 250, 284 Generalization of the Kneser–Rad´ o lemma, 50 Generalized Dirichlet integral E(X, g), 278 Generalized graph, 39 Generic curve, 412 Generic finiteness, 303, 347, 398 Geodesic spray, 432, 434 Geodesic spray on a Hilbert manifold, 432 Giaquinta, M., 176, 244 Gilbarg, D., 140, 144, 145, 185, 238 Giusti, E., 128, 135, 144, 167, 239–242, 245 GLc (H), 402 Gradient of Eα , 406 Gradient vector field Wα : Nα → T Nα , 405 Gromoll, D., 156 Gr¨ otzsch, H., 296 Gulliver, R., 294 H h = hT T + LV g, 260, 269 Halpern, N., 287 Hardt, R., 146, 176 Hausdorff measure, 146 Height function γ, 40 Heinz, E., 135, 236, 237, 448 Heinz’s estimate, 217, 237 Hellinger, E., 399 Hengartner, W., 132 Henneberg’s surface, 7–9, 11 Hilbert, D., 399 Hildebrandt, S., 34, 126, 127, 244, 294 Hoffman, D.A., 246 Hohrein, J., 449, 456 Holomorphic quadratic differential, 261, 357 Hopf, E., 236, 237 Huisken, G., 218, 241, 245 Hutchinson, J.E., 35 Hwang, J.-F., 127

I I(λ), 303 I(λ, ν), 303 Imbusch, C., 448 Inclusion of f (B), 50 Inclusion of free boundary values, 48 Incompressibility condition, 285 ind Πμν , 385 index πνλ , 324, 345 Index of the Fredholm map πνλ , 324, 345 Index theorem, 299, 303, 412 Index theorem for genus zero surfaces, 411 Index theorem for minimal surfaces of higher genus, 353, 395 Index theorem for minimal surfaces without boundary branch points, 396 Index theorems, 399 index Πμν , 395 Integral curvature estimate, 180 Integral curvature estimate for α-stable hypersurfaces, 186 Integral curvature estimate for stable hypersurfaces, 181 Intruding vertices, 133 Isoperimetric inequality, 207 J J(X), 328, 329 Jacobi equation, 53, 55, 163 Jacobian Jf of the projected map f := π ◦ X, 56 Jenkins, H., 236, 238 J¨ orgens, K., 135, 137, 236, 237 J¨ orgens’s theorem, 137 Jost, J., 128, 156, 244, 245, 285, 294, 296, 297 K Karcher, H., 240 Kato, T., 399 Keen, L., 287 Keiper, J.B., 226 Klingenberg, W., 156 Kneser-type result, 132 Kra, I., 296 K¨ uhnel, W., 156 L Lc (H), 402 LV g, 260 Laplace–Beltrami-operator, 160, 161 Lawlor, G., 165, 167

Index Lawson, H.B., 240, 243 Lee, C.-C., 126 Lee’s theorem, 127 Lehto, O., 296 Lewy, H., 135, 237 Lie derivative of g, LV g, 260 Linear Fredholm maps, 399 Local representation of Fredholm maps, 300 Loop, 5, 26 Lower semicontinuity theorem for E(X, g), 287 M Martinazzi, L., 244 Massari, U., 207, 239 Massey, W., 292, 293 Mean curvature H, 146, 158 Mean curvature type equation, 164

Mean curvature vector H, 146 Mean value formula, 204, 205 Metric g, 259 Metric G of constant scalar curvature conformal to g, 265 Meyer, W., 156 Michael, F.H., 198, 246 Michael–Simon inequality, 206, 210, 217 Mickle, E.J., 237 Milnor, J., 419 Minimal cone, 164 Miranda, M., 144, 207, 236, 238, 239, 245 Mittelmann, M.D., 35 Modica, G., 176 Modular group Γ = D/D0 , 256 Moduli space, 296 Monotonicity formula, 203 Monotonicity inequality, 210 Monotonicity of free boundary values, 48 Morrey, C.B., 296 Morse lemma, 425, 428 Morse theory, 411 Moser, J., 140, 143, 238 Moser–Harnack inequality, 140 Moser’s Bernstein theorem, 144, 218, 241 Moser’s Bernstein theorem, generalization, 233 Multiplicities ν, 304 Multiplicities λ, 304 Multiply connected domains, 253 Mumford, D., 271 Mumford compactness theorem, 271

535

N Natural projection of the bundle N onto Ar , 385 Nehring, T., 127 Neil’s parabola, 8 Nielsen, J., 256 Nirenberg, L., 241 Nitsche, J.C.C., 34, 127, 135, 236–238, 241, 297 Nonconstant exterior solution, 141 Nondegenerate critical point, 411 Nondegenerate zero of a Palais–Smale vector field, 404 Nonexistence of stable cones, 172 Nonlinear Fredholm map, 399 Nonparametric area, 146 Nonparametric representation, 18, 39, 61 Nonparametric representation of X above G, 61, 72, 73, 107 Nonparametric representation on an annulus, 132 Nonsmooth supporting surfaces, 96 Normal form for Dirichlet’s integral, 436 Normal line L(p), 45 Notation (N), 115 Numerical approach to partially free problems, 35 O OG (X), 345 Observation of Sch¨ uffler, 352 ω : A × D → H s (S 1 , Rn ), 319 One-sided minimal surface, 253, 254 ˆ in A, 351 Open, dense set A Orbit Og (D) of D through g, 260 Oscillation of a nonconstant entire solution, 141 Osserman, R., 236–238, 242, 243 2 δ A(X, λ), 69 P Palais–Smale vector field, 404 Pape, A., 35 Partial conformality operator, 366 Partial conformality operator V C(τ ), 371 Phenomenon of degeneracy, 284 Phragm´ en–Lindel¨ of theorem, 127 Plateau’s problem, 428 Pointwise curvature estimates for α-stable minimal hypersurfaces, 225

536

Index

Pointwise curvature estimates for stable minimal hypersurfaces, 218 Polthier, K., 35 Projection π onto the plane Π, 38 Proper homotopy, 404 Proper map, 300 Protruding vertices, 133 Q Quadratic differential on M , 357 Quasiconformal mappings, 296 R Rad´ o, T., 236 range(DV C(τ )(X)), 375 Regular set, 43 Regular value, 300 Reparametrized graph, 114 Representation theorem, 18, 28 Ricci’s lemma, 160 Riemann curvature tensor, 158 Riemann mapping theorem, 423 Riemann moduli space R(M ), 256 Riemann space of moduli of M , 256 Riemann–Roch theorem, 261, 356 Riemannian structure, 432 Riemann’s period relations, 355 Riesz, F., 399 Rotational graph, 171 Rothe mapping, 443 S Sard theorem for functionals, 423 Sard–Brown theorem, 301 Sauvigny, F., 126, 127, 236 Scalar curvature R(g), 259 Schauder, J.P., 399 Schlicht domains, 253 Schoen, R., 163, 164, 180, 217, 241, 242, 285, 287 Schottky double, 264 Sch¨ uffler, K., 352, 399, 400, 438 Sch¨ uffler’s observation, 411 Sch¨ uffler’s theorem, 352 Schwarz’s formula, 7 Second fundamental form, 157 Second variation of A, 147 Second variation of area, δ 2 A(X, Y ), 54 Second variation of Eα , 152 Serrin, J., 236, 238

Set of classes of pointwise conformally equivalent metrics, 258 Set of minimal surfaces X in N of branching type (λ, ν), Mλν , 326 Shen, Y., 245 Shiffman, M., 284, 285, 297 Siebert, K.G., 35 Sign of DWα (Xj ), 413 Simon, L., 126, 145, 163, 164, 180, 198, 236, 238, 240–242, 246 Simons, J., 161, 164, 240, 241 Simons cone, 167, 240 Simons’s identity, 161, 177, 188 Singular minimal surface equation, 172, 197 Singular set, 43 Smale, S., 399 Smale transversality theorem, 301 Smale–Sard theorem, 300 Smoczyk, K., 244, 245 Sobolev inequalities, 245 Sobolev-type estimate, 217 Sou˘ cek, J., 176 Space of almost complex structures A = A(M ), 256 Space of forced Jacobi fields, J(X), 328 Space of holomorphic quadratic differentials, 261 Space of positive C ∞ -functions on M , P = P(M ), 258 Space C of complex structures, 256 Space M of Riemann C ∞ -metrics on M , 257 Spray of the weak metric, 431 Spruck, J., 241, 246 Stability, 347 ˆ under perturbations, 351 Stability of A Stability of minimal cones, 164, 167 Stable α-minimal cone, 164 Stable minimal cone, 164 Stable minimal surface, 55 Stationary minimal surface in C(Γ, S), 13 Stationary minimal surfaces, 11 Stationary submanifold, 150 Stationary surface in the configuration

Γ, S, 41 Steinmetz, G., 35 Stratification, 344 Stratification of harmonic surfaces by singularity type, 304, 318 Strict local minimizer, 428, 431 Strict minimizer, 425

Index Strong condition of cohesion, 129 Struwe, M., 411, 448 Sufficient condition for edge creeping, 86 Surface of lower type, 255 Symmetric complex structure, 265 Symmetric minimal surface equation, 171, 172, 195 Symmetric Riemann–Roch theorem, 358 Symmetry theorem, 19 T Th Hν , 309 Tid OG (X), 346 n , 316 TX H∗,ν n , 315 TX Hλν TX Pν + Jν (X) = TX N, 332 T (Γ, S), 86 Teichm¨ uller, O., 256 Teichm¨ uller space, 296, 359 Teichm¨ uller space for surfaces with k boundary components, 263 Teichm¨ uller space T (2M ) := CS /D0 , 265 Teichm¨ uller space T (M ), 256, 455 Teichm¨ uller theory, 255 Teichm¨ uller theory for unoriented surfaces M , 270 Theorem of Douglas, 284 Thiel, U., 344, 399, 400 Toeplitz, O., 399 Tomi, F., 271, 284, 285, 294–297, 299, 399 Tongue, 5, 26 Transition mappings, 274 Transversality, 301, 341 Transverse, 412 Tromba, A.J., 259, 260, 262, 263, 271, 284, 285, 295–297, 299, 398, 399, 423 Trudinger, N., 140, 144, 145, 185, 238, 245 Turowski, G., 127–133 U Uniformity condition (B) at infinity, 40 Uniqueness, 18 Uniqueness in M(Γ, S), 60 Uniqueness theorem, 18, 63, 65, 72, 107, 108, 422

537

Uniqueness up to conformal equivalence, 60 V V Cα (τ ), 366, 369 V C(τ ), 369 Vector bundle structure, 359 Vertex of E-type, 43 Vertex of T -type, 43 Vertex of edge-creeping type, 43 Vertex of transversal type, 43 Vertices gj := σ(tj ), 40 Vitali covering theorem, 206 V Kα (τ ), 378 V Kα (τ )(X), 378 W Wα : Nα → T Nα , 405 Wagner, H.J., 35 Wang, M.T., 244, 245 Weak Riemannian metric, 406 Weak Riemannian structure on Nα , 434 Wedge, 74 Weighted “α-perimeter”, 195 Weil–Petersson metric, 270 Weingarten equations, 53 Weingarten map, 60, 157 Weingarten operator, 66 Widman, K.O., 244 Winklmann, S., 242, 244 Wohlrab, O., 35 X Xin, Y.L., 244, 245 Y Yang, L., 245 Yau, S.T., 163, 164, 180, 217, 241, 242, 245, 285, 287 Z Zeros of Wα on Nα , 409 Zero(V ), 404 Zhu, S., 245 Zieschang, H., 292

Global analysis of minimal surfaces

Minimal surfaces

Minimal surfaces

Regularity of minimal surfaces

Minimal surfaces of codimension one

A Survey of Minimal Surfaces

A survey of minimal surfaces

A Survey of Minimal Surfaces

A Survey of Minimal Surfaces

Minimal Surfaces of Codimension One

Crescent-Shaped Minimal Surfaces

Minimal Surfaces in R3

The Global Theory of Minimal Surfaces in Flat Spaces

Minimal Surfaces and Functions of Bounded Variation

Minimal surfaces and functions of bounded variation

A theory of branched minimal surfaces

A Theory of Branched Minimal Surfaces

A Theory of Branched Minimal Surfaces