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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Contents

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 B¨ohme & 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,

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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

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1 Minimal Surfaces with Supporting Half-Planes

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 Bj¨orling’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

1 Minimal Surfaces with Supporting Half-Planes

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

1 Minimal Surfaces with Supporting Half-Planes

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

1 Minimal Surfaces with Supporting Half-Planes

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

1 Minimal Surfaces with Supporting Half-Planes

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.

1.4 Classiﬁcation of the Contact Sets

−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

1 Minimal Surfaces with Supporting Half-Planes

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 .

1.4 Classiﬁcation of the Contact Sets

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 Minimal Surfaces with Supporting Half-Planes

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

1 Minimal Surfaces with Supporting Half-Planes

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.

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1 Minimal Surfaces with Supporting Half-Planes

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 <

24

1 Minimal Surfaces with Supporting Half-Planes

u1 < u2 < 1. (It follows from formula (2) of Section 1.5 that u2 = −u1 > 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

1 Minimal Surfaces with Supporting Half-Planes

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

1 Minimal Surfaces with Supporting Half-Planes

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 Rad´o’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

1 Minimal Surfaces with Supporting Half-Planes

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.

1.9 Proof of the Representation Theorem

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 Rad´o’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

1 Minimal Surfaces with Supporting Half-Planes

{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

1.9 Proof of the Representation Theorem

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

1 Minimal Surfaces with Supporting Half-Planes

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 Rad´o’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–Rad´o 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

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 2,

37

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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

2 Embedded Minimal Surfaces with Partially Free Boundaries

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 of X, i.e. for every oneparameter family of surfaces X ∈ C(Γ, S), − 0 < < 0 , which is diﬀerentiable with respect to and satisﬁes X0 = X. We know that any stationary surface in Γ, S is a minimal surface which meets S orthogonally away from the edges and maps C homeomorphically onto Γ . Deﬁnition 1. Let M(Γ, S) be the class of minimal surfaces X ∈ C(Γ, S) which are stationary in Γ, S, i.e. the class of stationary disk-type minimal surfaces in the boundary conﬁguration Γ, S. Since C(Γ, S) is nonempty, there is at least one minimizer X of Dirichlet’s integral in C(Γ, S), and it follows that X ∈ M(Γ, S); see Section 4.6 of Vol. 1. However, it is by no means clear that a surface X ∈ M(Γ, S) lies completely on the same side of S as Γ ; it is quite conceivable that X passes through S several times. Our mathematical solution X ∈ M(Γ, S) might therefore be very diﬀerent from a soap ﬁlm spanning Γ, S which, clearly, cannot penetrate a material support surface S and, therefore, has to lie on the same side of S as Γ . Hence, if the mathematical solution X ∈ M(Γ, S) is to model reality, we have to verify that it does not pass S. This is one of the main problems that we have to solve in the sequel; see Sections 2.5, 2.6, and 2.9. Moreover, the property of X to lie on one side of S will also be crucial for proving that X lies as a graph above G. For the convenience of the reader we summarize some of the properties of surfaces in M(Γ, S) which were already proved in Vols. 1 and 2. Proposition 1. Suppose that σ ∈ C 3 (R, R2 ), and let X be a minimal surface of class M(Γ, S). Then we have: (i) X ∈ C 2,β (B ∪ I, R3 ) for all β ∈ (0, 1), and there is a function λ ∈ C 0 (I) such that (1) Xv (u, 0) = λ(u)n X(u, 0) on I. (ii) There is a constant c(σ) such that (2) |dX| ≤ L(Γ ) + c(σ)D(X) I

42

2 Embedded Minimal Surfaces with Partially Free Boundaries

and therefore |dX| ≤ 2L(Γ ) + c(σ)D(X).

(3) ∂B

(iii) X|∂B ∈ AC(∂B, R3 ), and in particular X|∂B ∈ C 0 (∂B, R3 ) whence X ∈ C 0 (B, R3 ). ˚ R3 ). Moreover, (iv) If also Γ ∈ C 2,α for some α ∈ (0, 1) then X ∈ C 2,α (B ∪ C, the number of branch points of X on B \ {1, −1} is ﬁnite, and the total curvature of X satisﬁes |K| dA < ∞. B

Proof. The regularity properties stated in (i) and (iv) can be found in Chapter 2 of Vol. 2. Equation (1) expresses the fact that X intersects S perpendicularly along its free trace on S; to prove it, one uses the equation Xu · Xv = 0 and the fact that Xu (u, 0) is tangent to S at the point X(u, 0). The estimate (2) can be derived from Theorem 2 of Section 4.6 in Vol. 2, and (3) is an immediate consequence of (2) since L(Γ ) = C |dX|. Statement (iii) follows from (ii) on account of Tsuji’s theorem (see Vol. 1, Section 4.7, Theorem 1). The statements in (iv) about the number of branch points and the total curvature of X follow from the Gauss–Bonnet theorem and the asymptotic expansions derived in Section 3.2 of Vol. 1, and in Sections 2.10 and 3.4 of Vol. 2, using also property (ii); see Section 2.11 of Vol. 2, formulas (23) and (24). If σ is merely piecewise smooth, we still obtain that X 3 |I is continuous. In fact, the following is true: Proposition 2. For any X ∈ M(Γ, S) we have X 3 ∈ C ∞ (B ∪ I) as well as Xv3 (u, 0) = 0 for u ∈ I. Therefore X 3 can be extended to a function on the closure of the unit disk D := (u, v) ∈ R2 : u2 + v 2 < 1 , which is harmonic in D and continuous on D. In particular, X 3 is continuous on B. Proof. Let ϕ ∈ Cc1 (B ∪ I) and set Φ := (0, 0, ϕ). Then X := X + Φ ∈ C(Γ, S) for any ∈ R, and so we obtain d D(X ) =0 d =0 whence

∇X 3 · ∇ϕ du dv = 0 for any ϕ ∈ Cc1 (B ∪ I). B

2.1 The Geometric Setup

43

By linear regularity theory it follows that X 3 ∈ C ∞ (B ∪ I) and Xv3 = 0 on I. Set H(u, v) := X 3 (u, v) for (u, v) ∈ B and H(u, v) := X 3 (u, −v) for (u, −v) ∈ B. Since X 3 is continuous on C, the extension H is continuous on ∂D, and X 3 ∈ H21 (B) implies H ∈ H21 (D). On account of Schwarz’s reﬂection principle H is harmonic in D and therefore also continuous on D. Next we introduce an important subclass of M(Γ, S), the class M∗ (Γ, S). Deﬁnition 2. M∗ (Γ, S) is the class of those surfaces X ∈ M(Γ, S) for which f := π ◦ X furnishes a weakly monotonic mapping of I into Σ. Proposition 3. If X ∈ M∗ (Γ, S), then X ∈ C 0 (B, R3 ), and f := π ◦ X yields a continuous, weakly monotonic mapping of ∂B onto ∂G such that f (C) = Γ and f (I) = Σ. Proof. Since X maps C homeomorphically onto Γ which lies as a generalized graph above Γ , we infer that f |C furnishes a continuous, weakly monotonic mapping of C onto Γ . Because f |I is a weakly monotonic mapping of I into Σ, the one-sided limits f (−1 + 0) and f (1 − 0) exist and are elements of Σ. It follows that f |∂B yields a monotonic map into ∂G, and since the Dirichlet integral of f satisﬁes D(f ) ≤ D(X) < ∞, we infer by a “bridge-argument” that f |∂B is continuous. As a consequence we obtain f (I) = Σ and f (∂B) = ∂G. Taking Proposition 2 into account, we ﬁnally see that X = (f, X 3 ) is continuous on B. On account of the preceding result we conclude that the pre-image of any vertex qj of Σ with respect to the mapping f˜ := f |I = π ◦ X|I associated with a surface X ∈ M∗ (Γ, S) is either a single point uj ∈ I or a closed interval Ij = [uj , uj ] ⊂ I. In the ﬁrst case we call qj a vertex of transversal type for X, as the minimal surface X intersects the edge Ej := {qj } × R of S in a single point, Xj (u, 0). In the second case we have X(u, 0) ∈ Ej for all u ∈ Ij , i.e. the surface X adheres to Ej along a full interval X(Ij ). We call this phenomenon edge creeping, and the corresponding vertex qj is said to be a vertex of edge-creeping type for X. Let us ﬁx this notation as Deﬁnition 3. Let X ∈ M∗ (Γ, S), f := π ◦ X, and f˜ := f |I . We call qj a vertex of T-type for X if f˜−1 (qj ) = {uj }. If f˜−1 (qj ) = Ij = [uj , uj ] with uj < uj , then qj is said to be a vertex of E-type for X. With X we associate the open subset I ∗ = I ∗ (X) := I \ f˜−1 (Q), Q := {q1 , q2 , . . . , ql }, consisting of all points of I which are not mapped by X into one of the edges Ej := {qj }×R of S, j = 1, 2, . . . , l. We call I ∗ the regular set of X, whereas f˜−1 (Q) is said to be the singular set of X, and the interval Ij = [uj , uj ] associated with an E-type vertex qj for X is called an E-interval of X.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

Proposition 4. Suppose that σ is piecewise C 3 -smooth. (i) If X ∈ M∗ (Γ, S) has the regular set I ∗ , then X ∈ C 2,β (B ∪ I ∗ , R3 ) for any β ∈ (0, 1), and there is a function λ ∈ C 0 (I ∗ ) such that (4) Xv (u, 0) = λ(u)n X(u, 0) for all u ∈ I ∗ . Moreover, for any E-interval Ij = [uj , uj ] of X there is an > 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 .

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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,

2.2 Inclusion and Monotonicity of the Free Boundary Values

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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).

2.2 Inclusion and Monotonicity of the Free Boundary Values

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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–Rad´o 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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 ).

2.4 Properties of the Gauss Map, and Stable Surfaces

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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∗ ,

2.4 Properties of the Gauss Map, and Stable Surfaces

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

|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 .

2.4 Properties of the Gauss Map, and Stable Surfaces

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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:

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2 Embedded Minimal Surfaces with Partially Free Boundaries

(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).

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

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

(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.

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

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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|<ε0 of admissible variations (1) where Z satisﬁes (3). Now we want to compute (d2 /dε2 )A(X ε )|ε=0 for variations X ε of the form (1). Then we have 1 Xuε = Xu + ελu N + ελNu + ε2 Zu + o(ε2 ), 2 1 2 ε Xv = Xv + ελv N + ελNv + ε Zv + o(ε2 ), 2 whence Xuε ∧ Xvε = EN + ε[λ(Xu ∧ Nv + Nu ∧ Xv ) + (λv Xu ∧ N + λu N ∧ Xv )] ε2 + {(Xu ∧ Zv + Zu ∧ Xv ) + 2(λλu N ∧ Nv + λλv Nu ∧ N ) 2 + 2λ2 Nu ∧ Nv } + o(ε2 ). On account of the formulae (2), (5), (8), and (9) of Section 2.4 it follows that Xuε ∧ Xvε = EN − ε(λu Xu + λv Xv ) + ε2 λ2 EKN + ε2 (λλu Nu + λλv Nv ) 1 + (Xu ∧ Zv − Xv ∧ Zu ) + o(ε2 ) 2 with E = |Xu |2 whence |Xuε ∧ Xvε |2 = E2 + 2ε2 λ2 E2 K + ε2 E|∇λ|2 + ε2 EN · (Xu ∧ Zv − Xv ∧ Zu ) + o(ε2 ) by virtue of N ⊥ Xu , Xv , Nu , Nv , and of the conformality relations for X. By (7) of Section 2.4 and ΔX = 0 we obtain N · (Xu ∧ Zv − Xv ∧ Zu ) = (N ∧ Xu ) · Zv − (N ∧ Xv ) · Zu = Xv · Zv + Xu · Zu = (Xu · Z)u + (Xv · Z)v .

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2 Embedded Minimal Surfaces with Partially Free Boundaries

Thus the function W(ε) deﬁned by W(ε) := |Xuε ∧ Xvε | satisﬁes W(ε)2 = E2 + ε2 {E|∇λ|2 + 2E2 Kλ2 + E[(Xu · Z)u + (Xv · Z)v ] + o(ε2 )}. It follows that 0=

∂ ˙ W(ε)2 = 2W(0) · W(0) ∂ε ε=0

whence we obtain ˙ W(0) =0 away from the branch points. Furthermore we have 1 1 ∂2 ∂2 ∂2 2 2 W(ε) = W(ε) = W(ε) , 2 2 2 ∂ε2 ∂ε 2E ∂ε 2 W(ε) ε=0 ε=0 ε=0 and so, away from the branch points, it follows that ¨ W(0) = |∇λ|2 + 2EKλ2 + (Xu · Z)u + (Xv · Z)v . Thus we obtain for A(X ε ) = B W(ε) du dv that d ε A(X ) (5) = 0, dε ε=0 as expected, and d2 ε (6) A(X ) = [|∇λ|2 + 2EKλ2 + (Xu · Z)u + (Xv · Z)v ] du dv, dε2 B ε=0 if D(N ) = X |K| dA < ∞. By the Gauss–Bonnet formula we have K dA < ∞ if Γ ∈ C 2,α and σ ∈ C 3 . X

Note that the right-hand side of (6) diﬀers from the standard second variation δ 2 A(X, λ) only by the divergence term [(Xu · Z)u + (Xv · Z)v ] du dv, B

which can be transformed into the boundary integral (Xu · Z) dv − (Xv · Z) du. ∂B

If X is a local minimizer of A we have (7)

d2 ε A(X ) ≥ 0. 2 dε ε=0

This motivates the following deﬁnition, taking Lemma 1 into account:

2.6 Uniqueness of Freely Stable Minimal Surfaces

69

Deﬁnition 1. Suppose that X ∈ M(Γ, S) and D(N ) < ∞. (i) The expression 2

δ A(X, λ) := δ 2 A(X, λ) + E(X, λ)

(8)

with δ 2 A(X, λ) =

(|∇λ|2 + 2EKλ2 ) du dv B

and (Xu · Z)u + (Xv · Z)v du dv,

E(X, λ) := (9)

B

Z := −λ2 κ(X)[N · τ (X)]2 n(X) is called free second variation of X. (ii) The surface X is said to be freely stable if 2

δ A(X, λ) ≥ 0

(10)

for every λ ∈ C 1 (B) with λ = 0 on C.

The preceding discussion yields: Proposition 1. Any X ∈ M(Γ, S) is freely stable if it is a minimizer of area in C(Γ, S). Now we can establish an important result which is basic for proving uniqueness of freely stable surfaces in M(Γ, S). Proposition 2. Suppose that Γ ∈ C 2,α and σ ∈ C 3 satisfy Condition (B) (or (M1 ), or (M2 ) respectively), and let X be a freely stable surface in M(Γ, S) with the Gauss map N = (N 1 , N 2 , N 3 ). Then we have N 3 (w) > 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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 + ,

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

⎛

⎛ ⎞ ⎞ 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 .

2.7 Asymptotic Expansions

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:

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2 Embedded Minimal Surfaces with Partially Free Boundaries

(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

2.7 Asymptotic Expansions

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).

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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.

2.7 Asymptotic Expansions

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) + · · · ,

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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”.

2.7 Asymptotic Expansions

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

2.7 Asymptotic Expansions

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 Rad´o’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∗ .

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 .

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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}.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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}.

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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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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}.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

(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]

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces

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).

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 Poincar´e’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→∞

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces

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 Arzel`a– ∗ 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 Fr´echet 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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

˚ 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.

2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces

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)

2 Embedded Minimal Surfaces with Partially Free Boundaries

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

2.9 Embedded Minimizers for Nonsmooth Supporting 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

2 Embedded Minimal Surfaces with Partially Free Boundaries

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

2 Embedded Minimal Surfaces with Partially Free Boundaries

∗ 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

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

∗ 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

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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.

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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.

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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.

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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)

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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

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2 Embedded Minimal Surfaces with Partially Free Boundaries

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 .

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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

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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 ∗ .

2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge

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.

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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,

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(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 Phragm´en–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.

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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 .

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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

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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

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3 Bernstein Theorems and Related Results

to invoke a deep connection between the minimal surface equation on the one hand and the Monge–Amp`ere 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. J¨orgens [1] implies the preceding Theorem 1. Theorem 1. Suppose u ∈ C 2 (R2 ) solves the Monge–Amp`ere 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

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3 Bernstein Theorems and Related Results

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

3.1 Entire and Exterior Minimal Graphs of Controlled Growth

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.

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3 Bernstein Theorems and Related Results

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

3.1 Entire and Exterior Minimal Graphs of Controlled Growth

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.

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3 Bernstein Theorems and Related Results

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).

3.1 Entire and Exterior Minimal Graphs of Controlled Growth

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.

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3 Bernstein Theorems and Related Results

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.

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3 Bernstein Theorems and Related Results

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

3.2 First and Second Variation Formulae

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 .

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3 Bernstein Theorems and Related Results

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 ).

3.2 First and Second Variation Formulae

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

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3 Bernstein Theorems and Related Results

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.

3.2 First and Second Variation Formulae

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).

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3 Bernstein Theorems and Related Results

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

3.2 First and Second Variation Formulae

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.

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3 Bernstein Theorems and Related Results

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 .

3.2 First and Second Variation Formulae

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

3 Bernstein Theorems and Related Results

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

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3 Bernstein Theorems and Related Results

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 ).

3.3 Some Geometric Identities

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

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3 Bernstein Theorems and Related Results

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.

3.3 Some Geometric Identities

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

3 Bernstein Theorems and Related Results

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

3 Bernstein Theorems and Related Results

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

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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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3 Bernstein Theorems and Related Results

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

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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 Sim˜oes [1]. A complete classiﬁcation of cones over products of (more than two) spheres was given by Lawlor [1]. 8

168

3 Bernstein Theorems and Related Results

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).

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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 ,

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3 Bernstein Theorems and Related Results

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}.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

3 Bernstein Theorems and Related Results

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.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

3 Bernstein Theorems and Related Results

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

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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|)

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3 Bernstein Theorems and Related Results

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,

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

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3 Bernstein Theorems and Related Results

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

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

3 Bernstein Theorems and Related Results

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.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

3 Bernstein Theorems and Related Results

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 .

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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 <

|∇M ξ|4+2q dHn M

2 n.

Theorem 3 is proved.

184

3 Bernstein Theorems and Related Results

Corollary 1. Let M be a complete, stable minimal hypersurface in Rn+1 and suppose that the asymptotic growth condition lim (R−4−2q Hn (BR (x0 ))) = 0

(31)

R→∞

holds for some q ∈ [0, geodesic ball

2 n)

and some x0 ∈ M , where BR (x0 ) denotes the

{x ∈ M : r(x) := distM (x0 , x) < R}

in M.

Then M must be a plane. Proof. By assumption and the&Theorem of Hopf and Rinow we have that BR (x0 ) is compact and M = R>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.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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|<2R or, since a.e. Dϕ = Dη · uR + ηDuR ,

|Du|2 |Du|2 dx ≤ η dx 1 + |Du|2 1 + |Du|2 {|x|
(DuDη)uR = − 1 + |Du|2 {|x|<2R}

≤ R |Dη| dx. {|x|<2R}

Consequently we get

H (BR (X0 ) ∩ M ) = n

n

dH = {|x|2 +u2
BR (X0 )∩M

≤

{|x|

1 + |Du|2 dx

1 + |Du|2 dx ≤ c(n)Rn ,

186

3 Bernstein Theorems and Related Results

and (i) follows. Ad (ii). We have to show that

n 2 2 |A| ξ dH ≤ M

|∇M ξ|2 dHn

M

holds for all ξ ∈ Cc1 (M, R) where M = graph u ⊂ Rn+1 . To this end we recall Jacobi’s ﬁeld equation, cp. Proposition 2 of Section 3.3: ΔM ν + |A|2 ν = 0,

(34)

where ν = √(−Du,1)2 is the upward unit normal on M = graph u. We put 1+|Du|

w := − log ν n+1 = log(1 + |Du|2 )1/2 and ﬁnd successively ∇M w = −(ν n+1 )−1 ∇M ν n+1 and

ΔM w = (−ν n+1 ΔM ν n+1 + |∇M ν n+1 |2 )(ν n+1 )−2 .

Using (34) it follows that ΔM w = |A|2 + |∇M w|2 , which–upon multiplying with ξ 2 and integrating by parts–yields

|A|2 ξ 2 dHn + |∇M w|2 ξ 2 dHn M

M

=−

∇M w · ∇M ξ 2 dHn ≤ M

|∇M w|2 ξ 2 dHn + M

|∇M ξ|2 dHn . M

The stability of M = graph u follows from this inequality. Theorem 4 is proved. We prove now curvature estimates for α-stable minimal surfaces. Recall that, for any α > 0, a C 2 -submanifold M ⊂ Rn+1 is called α-stable (cp. Deﬁnition 4, Section 3.2) in U ⊂ Rn+1 if either M ∩ U ⊂ Rn × {0} or the mean curvature is given by H(x) = α(xn+1 )−1 ν n+1 for all x ∈ M ∩ U and the stability inequality

|xn+1 |α {α−1 H2 + |A|2 }ξ 2 dHn ≤ |xn+1 |α |∇M ξ|2 dHn M

M

holds for all ξ ∈ Cc1 (M ∩ U, R). Theorem 5. Let M ⊂Rn+1 be an α-stable hypersurface of class C 4 and 2 ) and each nonnegative function ξ ∈ Cc1 (M ) we α > 0. For each q ∈ [0, n+α have the estimate

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

187

2+q √ 1 2 2 |x | α|A| + √ H ξ 4+2q dHn α M

≤ c(n, α, q) |xn+1 |α |∇M ξ|4+2q dHn

n+1 α

M

for some constant c depending only on n, α and q. A crucial step in the proof of Theorem 5 is the following sharp estimate from below for the term |∇M A|2 , cp. Lemma 3 for the case α = 0. Lemma 4. Let M ⊂ Rn+1 be an α-stationary smooth hypersurface, i.e. H(x) = α(xn+1 )−1 ν n+1 . Then the estimate |∇M A|2 ≥ (1 + 2[n + −1 (n − 1)]−1 )|∇M |A||2 − 2(1 + )[n + −1 (n − 1)]−1 |∇M H|2 holds a.e. on M for any > 0. Proof. Pick a point x ∈ M with |A(x)| = 0 and introduce normal coordinates near x, so that at x, gij (x) = δij , Γijk (x) = 0 and hij (x) = λi δij for all i, j, k = 1, . . . , n. As in the proof of Lemma 3 we ﬁnd (cp. relation (24)) h2jj,i . (35) |∇M A|2 − |∇M |A||2 ≥ 2 i =j

By Schwarz’s inequality and since hij is diagonal (cp. proof of Lemma 1), (36)

(hii,j )2 = h2ii,j + h2ii,i .

|∇M |A||2 ≤

i,j

Furthermore, since H =

j

i

i =j

hjj we get

hii,i = H,i −

hjj,i ,

i =j

and (36) gives for any > 0 |∇M |A||2 ≤

h2ii,j +

≤

i =j

H,i −

i

i =j

h2ii,j

+

2 hjj,i

i =j

(1 +

2 )H,i

+ (n − 1)(1 +

−1

i

= (n + −1 (n − 1))

)

i =j

i =j

h2ii,j + (1 + )|∇M H|2 .

h2jj,i

188

3 Bernstein Theorems and Related Results

Finally (35) implies the estimate |∇M A|2 − |∇M |A||2 ≥ 2(n + −1 (n − 1))−1 |∇M |A||2 − 2(1 + )(n + −1 (n − 1))−1 |∇M H|2

which proves Lemma 4. We combine Simons’s identity (9) with Lemma 4 and obtain Lemma 5. Let M ⊂ Rn+1 be α-stationary. Then we have the inequality (37)

ΔM |A|2 ≥ (2 + 4(n + −1 (n − 1))−1 )|∇M |A||2 − 4(n + −1 (n − 1))−1 (1 + )|∇M H|2 − 2|A|4 + 2hij H,ij + 2Hhij him hmj .

Now we have to carefully analyze the term hij H,ij using the fact that H = α(xn+1 )−1 ν n+1 . Lemma 6. Let M ⊂ Rn+1 be an α-stationary hypersurface of class C 4 . Then the following estimate holds true (38) HΔM H + α|A|ΔM |A| + ≥

√ 1 √ H2 + α|A|2 α

2

2 2α |∇M H|2 + |∇M |A||2 n+α n+α − α(xn+1 )−1 H∇M H · ∇M xn+1 − α2 (xn+1 )−1 |A|∇M |A|∇M xn+1

for all x ∈ M with |A(x)| = 0, xn+1 = 0. Proof. At a given point x ∈ M , |A(x)| = 0 we introduce normal coordinates and an orthonormal frame τ1 , . . . , τn at x ∈ M . Since H = α(xn+1 )−1 ν n+1 we compute at the ﬁxed point x, using ∇i ∇j = Di Dj , H,ij = 2H(xn+1 )−2 Di xn+1 Dj xn+1 + (xn+1 )−1 HDi Dj xn+1 − α(xn+1 )−2 Dj xn+1 Di ν n+1 − α(xn+1 )−2 Di xn+1 Dj ν n+1 + α(xn+1 )−1 Di Dj ν n+1 . By virtue of the Gauss and Weingarten equations, which at x take the form Di ν = −hi τ and Di τj = hij ν,

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

189

we infer αH,ij = 2αH(xn+1 )−2 Di xn+1 Dj xn+1 − α(xn+1 )−1 Hhij ν n+1 + α2 (xn+1 )−2 hi D xn+1 Dj xn+1 + α2 (xn+1 )−2 hj D xn+1 Di xn+1 − α2 (xn+1 )−1 Di [hj D xn+1 ]. In view of Codazzi’s equations hij,k = hik,j we have Di [hj D xn+1 ] = hj,i D xn+1 + hj Di D xn+1 = hij, D xn+1 + hj hi ν n+1 and therefore (39)

αhij H,ij = 2αH(xn+1 )−2 hij Di xn+1 Dj xn+1 − H2 |A|2 + 2α2 (xn+1 )−2 hij hi D xn+1 Dj xn+1 − α2 (xn+1 )−1 hij hij, D xn+1 − αHhij hj hi .

Since ΔM |A|2 = 2|A|ΔM |A| + 2|∇M |A||2 we infer from (37) and (39) for any > 0 the estimate (40)

α|A|ΔM |A| + α|A|4 + H2 |A|2 ≥ 2αH(xn+1 )−2 hij Di xn+1 Dj xn+1 + 2α2 (xn+1 )−2 hij hi D xn+1 Dj xn+1 −

α2 n+1 −1 (x ) (∇M |A|2 · ∇M xn+1 ) 2

+ 2α(n + −1 (n − 1))−1 |∇M |A||2 − 2α(n + −1 (n − 1))−1 (1 + )|∇M H|2 . In a similar way we infer from (39) the identity (41)

HΔM H + α−1 H4 + H2 |A|2 = 2(xn+1 )−2 H2 |∇M xn+1 |2 + 2αH(xn+1 )−2 hij Di xn+1 Dj xn+1 − α(xn+1 )−1 H∇M H∇M xn+1 .

190

3 Bernstein Theorems and Related Results

Now we add relations (40) and (41) and get HΔM H + α|A|ΔM |A| + α−1 H4 + α|A|4 + 2H2 |A|2 ≥ 2(xn+1 )−2 H2 |∇M xn+1 |2 + 4αH(xn+1 )−2 hij ∇i xn+1 ∇j xn+1 + 2α2 (xn+1 )−2 hij hi ∇j xn+1 ∇ xn+1 − α(xn+1 )−1 H∇M H∇M xn+1 −

α2 n+1 −1 (x ) ∇M |A|2 ∇M xn+1 + 2α[n + −1 (n − 1)]−1 |∇M |A||2 2

− 2α(1 + )[n + −1 (n − 1)]−1 |∇M H|2 . On the other hand, we infer from H = α(xn+1 )−1 ν n+1 H,i = −α(xn+1 )−2 ν n+1 ∇i xn+1 − α(xn+1 )−1 hi ∇ xn+1 = −(xn+1 )−1 H∇i xn+1 − α(xn+1 )−1 hi ∇ xn+1 and therefore |∇M H|2 = (xn+1 )−2 H2 |∇M xn+1 |2 + 2α(xn+1 )−2 Hhi ∇i xn+1 ∇ xn+1 + α2 (xn+1 )−2 hi hij ∇ xn+1 ∇j xn+1 . Inserting this identity into the above estimate and rearranging terms yields HΔM H + α|A|ΔM |A| + α−1 H4 + α|A|4 + 2H2 |A|2 ≥ 2α[n + −1 (n − 1)]−1 |∇M |A||2 + (2 − 2α(1 + )[n + −1 (n − 1)]−1 )|∇M H|2 − α(xn+1 )−1 H∇M H · ∇M xn+1 −

α2 n+1 −1 (x ) ∇M |A|2 ∇M xn+1 , 2

which holds for all > 0 and all points x with |A(x)| = 0, xn+1 = 0. Choosing = n−1 α we conclude HΔM H + α|A|ΔM |A| + ≥

√ 1 √ H2 + α|A|2 α

2

2 2α |∇M H|2 + |∇M |A||2 n+α n+α − α(xn+1 )−1 H∇M H · ∇M xn+1 − α2 (xn+1 )−1 |A|∇M |A| · ∇M xn+1 ,

which coincides with (38). Lemma 6 is proved.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

191

Proof of Theorem 5. We multiply inequality (38) by |xn+1 |α and obtain |xn+1 |α HΔM H + α|xn+1 |α |A|ΔM |A| 2 √ 1 n+1 α 2 2 √ H + α|A| | + |x α

(42)

≥

2 2α |xn+1 |α |∇M H|2 + |xn+1 |α |∇M |A||2 n+α n+α − H∇M H · ∇M |xn+1 |α − α|A|∇M |A| · ∇M |xn+1 |α .

Clearly, this inequality is satisﬁed for all x ∈ M with |A(x)| = 0 provided that α > 1. Furthermore it follows from an integration by parts that (42) holds true in a distributional sense locally on M and for all α > 0. In fact we have |A| ∈ Lip(M ) and ∇M |xn+1 |α is integrable on M because ν n+1 = 0 whenever xn+1 = 0, i.e. M must locally be a graph of a regular function ˆi , . . . xn+1 ) for some i ∈ {1, . . . , n} in a neighbourhood of any xi = f (x1 , . . . , x point x0 ∈ M ∩ {xn+1 = 0} and also

|xn+1 |α−1 dHn M ∩B (x0 )

|xn+1 |α−1

=

+i . . . dxn+1 < ∞ 1 + |Df |2 dx1 . . . dx

Kδ

for a suitable set Kδ ⊂ Rn . Only the distributional version of (42) will be used in the sequel. Before we continue we may still put (42) in a more convenient form by abbreviating √ 1 1 T := √ H2 + α|A|2 = √ (H2 + α|A|2 ). α α Diﬀerentiation yields 2 ∇M T = √ [H∇M H + α|A|∇M |A|] α and

√

αΔM T = 2|∇M H|2 + 2HΔM H + 2α|∇M |A||2 + 2α|A|ΔM |A|, √ 1 1/2 −1/2 √ H∇M H + α|A|∇M |A| , ∇M (T ) = T α

|∇M T 1/2 |2 = T −1 [α−1 H2 |∇M H|2 + 2H|A|∇M H · ∇M |A| + α|A|2 |∇M |A||2 ] √ √ 1 1 −1 2 2 2 2 √ H + α|A| √ |∇M H| + α|∇M |A|| ≤ T α α √ 1 = √ |∇M H|2 + α|∇M |A||2 . α

192

3 Bernstein Theorems and Related Results

Using these expressions in (42) gives √

α n+1 α |x | ΔM T + |xn+1 |α T 2 2 2 2 n+1 α 2 |x |xn+1 |α |∇M |A||2 ≥ 1+ | |∇M H| + α 1 + n+α n+α √ α ∇M T · ∇M |xn+1 |α , − 2

which may be simpliﬁed to 1 n+1 α |x | ΔM T + α−1/2 |xn+1 |α T 2 2 2 1 |xn+1 |α |∇M T 1/2 |2 − ∇M T · ∇M |xn+1 |α . ≥ 1+ n+α 2

(43)

We shall use the distributional form of (43), i.e. we multiply by ξ 2 T q , where q ≥ 0 and ξ ∈ Cc1 (M ), ξ ≥ 0 and integrate over M : 1 2

|xn+1 |α ΔM T ξ 2 T q dHn M

≥

1+

2 n+α

− α−1/2

|xn+1 |α |∇M T 1/2 |2 ξ 2 T q dHn M

|xn+1 |α T 2+q ξ 2 dHn − M

1 2

(∇M T · ∇M |xn+1 |α )ξ 2 T q dHn . M

Integration by parts gives

|xn+1 |α (∇M T · ∇M ξ)ξT q dHn + M

≤ α−1/2

− 1+

1 2

|xn+1 |α (∇M T · ∇M T q )ξ 2 dHn M

|xn+1 |α T 2+q ξ 2 dHn M

2 n+α

which is equivalent to

|xn+1 |α |∇M T 1/2 |2 ξ 2 T q dHn , M

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

193

2 |xn+1 |α |∇M T 1/2 |2 T q ξ 2 dHn n+α M

+ |xn+1 |α (∇M T · ∇M ξ)ξT q dHn

1+

M

+

1 2

|xn+1 |α (∇M T · ∇M T q )ξ 2 dHn M

≤ α−1/2

|xn+1 |α T 2+q ξ 2 dHn . M

By virtue of 1 q (∇M T · ∇M T q ) = |∇M T |2 T q−1 = 2q|∇M T 1/2 |2 T q 2 2 this implies 1+

2 + 2q |xn+1 |α |∇M T 1/2 |2 T q ξ 2 dHn n+α M

|xn+1 |α (∇M T · ∇M ξ)ξT q dHn +

(44)

M

≤ α−1/2

|xn+1 |α T 2+q ξ 2 dHn . M

From here on we may argue as in the proof of Theorem 3, i.e. we replace ξ in q+1 the stability inequality (2) by ξT 2 and ∇M ξ by ∇M ξT

q+1 2

+

q + 1 q−1 T 2 ∇M T ξ. 2

This gives (45)

α

−1/2

|xn+1 |α T 2+q ξ 2 dHn M

2 q+1 |xn+1 |α T 1+q |∇M ξ|2 + |∇M T |2 T q−1 ξ 2 2 M q + (q + 1)T (∇M T · ∇M ξ)ξ dHn .

≤

Using that |∇M T 1/2 |2 T q = and qT q ξ(∇M T · ∇M ξ) ≤

1 |∇M T |2 T q−1 4

2 2 −1 q+1 q ξ |∇M T |2 T q−1 + T |∇M ξ|2 2 2

194

3 Bernstein Theorems and Related Results

we add both inequalities (44) and (45); thus 2 1 − q 2 − 2q 2 |xn+1 |α T q−1 |∇M T |2 ξ 2 dHn 4 n+α M −1 ≤ 1+ |xn+1 |α T q+1 |∇M ξ|2 dHn . 2 M 2 2 If q ∈ [0, n+α ) it is possible to choose > 0 such that ( n+α −q 2 (1+2)) > 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}.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

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

3 Bernstein Theorems and Related Results

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.

3.4 Nonexistence of Stable Cones and Integral Curvature Estimates

197

Multiplying equation (48) with ϕ and integrating by parts one obtains

|Du|2 (1 + |Du|2 )−1/2 dx {|x|
αϕu−1 (1 + |Du|2 )−1/2 dx + |uR |0

≤ B2R (0)

This implies

M ∩BR (X0 )

|Dη| dx. B2R

dHn ≤

{|x|
≤ (z0 + R) B2R (0)

1 + |Du|2 dx

Du

div 1 + |Du|2

dx + c(z0 + R)Rn−1

≤ c(n)(z0 + R)Rn−1 for some constant c = c(n), whence

n n+1 α α |x | dH ≤ (z0 + R) M ∩BR (X0 )

dHn

M ∩BR (X0 )

≤ c(n)(z0 + R)α+1 Rn−1 ≤ c(n, α)[z0α+1 + Rα+1 ]Rn−1 and (49) follows for suﬃciently large R. By Corollary 2 we conclude that M = graph u must be a vertical hyperplane or identical to the coordinate plane {xn+1 = 0}, both of which is impossible. Theorem 6 is proved. Finally we consider C 2 -solutions of the singular minimal surface equation |u|α Du = α|u|α u−1 1 + |Du|2 (50) div 1 + |Du|2 or equivalently, assuming α > 1, Du α −1 2 −1/2 div − αu (1 + |Du| ) = 0, (51) |u| 1 + |Du|2 which is the Euler equation of the (singular elliptic) integral

Eα (u) = |u|α 1 + |Du|2 dx if α > 1. Then we have

198

3 Bernstein Theorems and Related Results

Corollary 3. Suppose that n + α < 4 + 2

2 n+α

and α > 1. Furthermore

assume that u ∈ C (R ) is an entire α-stable solution of the singular minimal surface equation (51) or (50). Then we have u ≡ 0 on Rn . 2

n

Proof. In view of Remark 3 in Section 3.4 we have either u ≡ 0 on Rn or u(x) = 0 for every x ∈ Rn , in which case u satisﬁes the symmetric minimal surface equation Du = αu−1 (1 + |Du|2 )−1/2 . div 1 + |Du|2 Now Theorem 6 is applicable and the result follows.

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities In this section we assume throughout that M ⊂ Rn+k is a smooth (i.e. C 2 ) properly embedded submanifold in Rn+k . It is the aim to prove important generalizations of the classical Sobolev inequality holding for functions f deﬁned on minimal submanifolds and more generally on submanifolds with arbitrary mean curvature. In our exposition we will closely follow the method of Michael and Simon [1], and in particular the lecture notes by L. Simon [8]. Their inequality and its proof will be presented in Theorem 3. The importance of this estimate for the analysis “upon the surface” is evidenced in the next section where it is a crucial step in the pointwise curvature estimate for minimal hypersurfaces. Furthermore we present a version of the Michael–Simon inequality for α-minimal hypersurfaces in Theorem 4. These Sobolev type results are derived through a consideration of suitable monotonicity formulae which are presented in Theorem 1 and Lemma 4 respectively. A further consequence of these monotonicity results are useful mean value formulae, which generalize the well-known mean value inequality for subharmonic functions in Rn , cp. Theorem 2. We proceed now to derive the important monotonicity formula, Theorem 1. The divergence theorem (Chapter 4, Vol. 2) states that for any M ⊂ Rn+k as above with smooth boundary ∂M and any C 1 -vectorﬁeld X : M → Rn+k we have the identity

divM X dHn = − X · H dHn + X · n dHn−1 , (1) M

M

∂M

where n denotes the exterior unit normal ﬁeld to ∂M which is tangent to M along ∂M , and k H=− (divM Ni )Ni i=1

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

199

denotes the mean curvature vector. Recall that the mean curvature vector H = nH diﬀers slightly from the notation introduced in Chapter 4, Vol. 2. Equation (1) is obtained from the standard divergence theorem

divM X dHn = X · n dHn−1 M

∂M

for tangential vector ﬁelds by decomposing an arbitrary vector ﬁeld X into its tangential and normal parts, X = XT + X⊥

with X ⊥ =

k (X · Ni )Ni , i=1

and observing that divM X ⊥ =

k

(X · Ni ) divM Ni = −X · H.

i=1

Consider an open set U ⊂ Rn+k such that U ∩ M = ∅, U ∩ ∂M = ∅ and some nonnegative function h ∈ C 1 (U ). Fix some point ξ ∈ U and some number ρ > 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

3 Bernstein Theorems and Related Results

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

3 Bernstein Theorems and Related Results

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

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

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

3 Bernstein Theorems and Related Results

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

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

(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

3 Bernstein Theorems and Related Results

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<σ<ρ0 0 ρ0

5ρ0 2 −n −n −n ≤ 2 + ρ f (ρ) dρ + ρ f (ρ) dρ 5ρ0 0 ρ0 ≤ 2−n +

2 2 sup (ρ−n f (ρ)) + ρ−n f (∞), 5 0<ρ<ρ0 5(n − 1) 0

whence we had 3 7 1 2 1 3 ≤ sup (ρ−n f (ρ)) ≤ + · = 5 5 (0,ρ0 ) 4 5 4 20

which is a contradiction.

The following covering lemma is an important tool in the proof of the Michael–Simon inequality, Theorem 3. We state it here without proof and refer to the pertinent literature on measure theory. Lemma 3 (Vitali Covering Theorem). Let F be a family of closed balls B in Rn with uniformly bounded radii, i.e. sup{diamB : B ∈ F} < ∞. Then there exists a countable disjoint subfamily F ⊂ F, F = {B ρj (ξj )}∞ j=1 , such that ∞ , , B⊂ B 5ρj (ξj ). B∈F

j=1

Theorem 3 (Michael–Simon inequality). Let M ⊂ Rn+k be an ndimensional C 2 -submanifold and U ⊂ Rn+k be open such that U ∩ M = ∅, U ∩ ∂M = ∅. Suppose h ∈ Cc1 (U ) is a non-negative function with compact support in U . Then we have the inequality h

(16)

n n−1

dH

n

n−1 n

≤c

M

(|∇M h| + h|H|) dHn M

for some constant c depending only on n. An immediate consequence of Theorem 3 is Proposition 1. For M and h as in Theorem 3 and 1 ≤ p < n we have the inequality h

(17) M

np n−p

dH

n

n−p np

≤ c(n, p)

(|∇M h|p + hp |H|p ) dHn M

p1 .

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

Proof. Apply inequality (16) to the function hγ where γ := use H¨ older’s inequality to conclude h

γn n−1

M

dH

n

p(n−1) n−p

207

> 1 and

n−1 n

≤ c(n)

(hγ |H| + γhγ−1 |∇M h|) dHn M

≤ c(n)

(γ−1)p p−1

h

dH

n

p−1 p

M

+ c(n)γ

h

(γ−1)p p−1

dH

n

p−1 p

M γn n−1

=

np n−p

h

np n−p

and dH

M

n

p

n

p1

M

Since

h |H| dH p

(γ−1)p p−1

|∇M h| dH p

n

p1 .

M

=

np n−p

we obtain

n−1 n

≤ c(n, p)

h

np n−p

dH

n

p−1 p

M

(|∇M h| + h |H| ) dH p

p

p

n

p1

M

which proves (17).

It is interesting to observe that (16) implies an isoperimetric inequality on the surface M ; in fact, (16) is equivalent to such an inequality, cp. Bombieri [2,4]. To see that an isoperimetric inequality follows from estimate (16) we take a suitable sequence of nonnegative Cc1 -functions on M which approximate the characteristic function of an open set K ⊂ M with ∂K suﬃciently smooth, e.g. of Lipschitz-class C 0,1 . Relation (16) immediately implies Proposition 2 (Isoperimetric Inequality). Let K ⊂ M be open with Lipschitz boundary ∂K. Then we have the inequality

n−1 n n−1 n n ≤ c(n) H (∂K) + |H| dH (18) [H (K)] K

where c(n) denotes the constant in (16). Remark. The same argument as above can be used to prove (18) for arbitrary Hn -measurable subsets K ⊂ M if we replace the Hausdorﬀ-measure of the boundary Hn−1 (∂K) by the “perimeter of K”, see e.g. Massari and Miranda [1]. Now we proceed with the proof of Theorem 3. To this end let us recall inequality (11) which states that for 0 < σ ≤ ρ < R

208

3 Bernstein Theorems and Related Results

σ

−n

M ∩Bσ

−n

h dH ≤ ρ n

h dHn

M ∩Bρ ρ

+

τ −n

M ∩Bτ

σ

[h|H| + |∇M h|] dHn dτ

where R is such that B R = BR (ξ) ⊂ U . Since h has compact support in U we can let R → ∞ and apply Lemma 2 to the functions deﬁned by

1 h dHn f (ρ) := ωn M ∩Bρ (ξ) and

1 ωn

g(ρ) :=

M ∩Bρ (ξ)

[h|H| + |∇M h|] dHn .

Relation (14) and the condition sup(0,ρ0 ) (ρ−n f (ρ)) ≥ 1 are satisﬁed whenever ξ ∈ {x ∈ M : h(x) ≥ 1}. For each such ξ ∈ M ∩ {h ≥ 1} Lemma 2 guarantees the existence of a ρ = ρ(ξ) ∈ (0, ρ0 ), where ρ0 = 2[f (∞)]1/n such that

(19) h dHn M ∩B5ρ (ξ)

−1 ≤ 5 ωn n

h dH

n

n1

M ∩Bρ (ξ)

M

[h|H| + |∇M h|] dHn .

By Vitali’s covering theorem, Lemma 3, applied to the covering {Bρ(ξ) (ξ)}ξ∈M ∩{h≥1} , there exist countably many disjoint balls Bρj (ξj ), j ∈ N, ξj ∈ {x ∈ M : h(x) ≥ 1} and such that {x ∈ M : h(x) ≥ 1} ⊂

∞ ,

B5ρj (ξj ).

j=1

Applying (19) for each ξj , j ∈ N, and summing over j we obtain

(20) h dHn {ξ∈M :h(ξ)≥1}

≤ 5n ωn−1

h dHn

1/n ∞

M

≤ 5n ωn−1

j=1

h dHn M

n1

M ∩Bρj (ξj )

[h|H| + |∇M h|] dHn

[h|H| + |∇M h|] dHn . M

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

209

Consider now a nondecreasing function γ ∈ C 1 (R) with 1 for t > > 0, γ(t) := 0 for t < 0 and apply (20) with γ(h − t), t ≥ 0, in place of h. We then have, writing {h > 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

3 Bernstein Theorems and Related Results

∞

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

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

(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

3 Bernstein Theorems and Related Results

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).

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

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<σ<ρ0

where ρ0 := 5(f (∞))1/n and n ≥ 2. Furthermore let the inequality

ρ (26) σ −n f (σ) ≤ ρ−n f (ρ) + τ −n [g(τ ) + Θf (τ )] dτ σ

be satisﬁed for all 0 < −σ < ρ < ∞ with some constant Θ ∈ R such that Θ · ρ0 ≤ 12 . Then there exists a number ρ ∈ (0, ρ0 ) with the property 1 n 5 ρ0 g(ρ). 2 Proof. Suppose that (27) were false for every ρ ∈ (0, ρ0 ). Then (25) and (26) would imply the relation f (5ρ) ≤

(27)

1≤

sup (σ −n f (σ)) 0<σ<ρ0

ρ0 2 · 5−n ρ0 −n + ρ f (5ρ) dρ + Θ ρ−n f (ρ) dρ ≤ ρ0 0 0 ρ0

5ρ0 2 −n −n −n ρ f (ρ) dρ + ρ f (ρ) dρ ≤ ρ0 f (∞) + 5ρ0 0 ρ0

ρ0 +Θ ρ−n f (ρ) dρ ρ−n 0 f (ρ0 )

0

≤ 5−n +

2 2 1 sup (ρ−n f (ρ)) + ρ−n f (∞) + sup (ρ−n f (ρ)) 5 0<ρ<ρ0 5(n − 1) 0 2 0<ρ<ρ0

= 5−n +

2 9 · 5−n + sup (ρ−n f (ρ)) 5(n − 1) 10 0<ρ<ρ0

≤

2 9 1 + + sup (ρ−n f (ρ)) 25 125 10 0<ρ<ρ0

since n ≥ 2.

214

3 Bernstein Theorems and Related Results

Thus we would have 1 1 7 ≤ sup (ρ−n f (ρ)) ≤ 10 10 0<ρ<ρ0 125

which is a contradiction. Proof of Theorem 4. Deﬁne the functions f and g by

f (ρ) = ωn−1 (ξ n+1 )−α |xn+1 |α h dHn M ∩Bρ (ξ)

and g(ρ) =

ωn−1 (ξ n+1 )−α

M ∩Bρ (ξ)

|xn+1 |α |∇M h| dHn ,

where ξ = (ξ 1 , . . . , ξ n+1 ) ∈ M ∩ supp h. The monotonicity inequality (22) can now be rephrased as

ρ α σ −n f (σ) ≤ ρ−n f (ρ) + f (τ ) dτ. τ −n g(τ ) + minx∈supp h (xn+1 ) σ If the inequality (28)

5α minx∈supp h (xn+1 )

ωn−1 (ξ n+1 )−α

|x

| h dH

n+1 α

n

n1

M

≤

1 2

is fulﬁlled we infer from Lemma 5 for each ξ ∈ {x ∈ M ∩ supp h : h(x) ≥ 1} the existence of a number n1

|xn+1 |α h dHn ρ < 5 ωn−1 (ξ n+1 )−α M

such that

(29) M ∩B5ρ (ξ)

≤

|xn+1 |α h dHn

n1

5 n+1 −1 n+1 −α ωn (ξ ) |xn+1 |α h dHn 2 M

· |xn+1 |α |∇M h| dHn M ∩Bρ (ξ)

holds true. By Vitali’s covering Theorem, Lemma 3, we can select a sequence of disjoint balls Bρ1 (ξ1 ), . . . , Bρn (ξn ), . . . which are centered at ξi ∈ {x ∈ supp h ∩ M : h(x) ≥ 1}

3.5 Monotonicity and Mean Value Formulae. Michael–Simon Inequalities

and that {x ∈ M : h(x) ≥ 1} ⊂

∞ ,

215

B5ρj (ξj ).

j=1

Note that for each ξj we have ξjn+1 ≥ . Then (29) applied to each ξ = ξj and summing over j yields the estimate

(30) |xn+1 |α h dHn M ∩{h≥1}

≤

5 n+1 −1 −α 1 [ωn ] n 2

|xn+1 |α h dHn

n1

M

|xn+1 |α |∇M h| dHn .

M

We conclude the proof similarly as the one of Theorem 3, i.e. we choose a non-decreasing C 1 (R) function γ with γ(t) = 1 for t > δ > 0 and γ(t) = 0 for t < 0. By virtue of (30) applied to the function γ(h − t) in place of h we deduce the inequality

(31) |xn+1 |α dHn M ∩{h>t+δ}

n1 5 n+1 −1/n −α/n n n+1 α ωn |x | dH ≤ 2 M ∩{h>t}

· |xn+1 |α γ (h − t)|∇M h| dHn . M

Using the obvious estimate

(t + δ)

1 n−1

M ∩{h>t}

≤

n1

M ∩{h>t}

we can then conclude that

1 (t + δ) n−1 M ∩{h>t+δ}

5 n+1 − n1 − α ≤ ωn n 2

|x

|x

| dH

n+1 α

n

n1 | (h + δ)

n+1 α

n n−1

dH

n

|xn+1 |α dHn

n1 M ∩{h>t}

|x

| (h + δ)

n+1 α

n n−1

d · − |xn+1 |α γ(h − t)|∇M h| dHn . dt M

dH

n

216

3 Bernstein Theorems and Related Results

Finally we apply Lemma 1 with μ := |xn+1 |α Hn M , α = T0 = ∞ and obtain the estimate

- n n . n−1 |xn+1 |α h n−1 − δ n−1 dHn n M ∩{h≥δ}

n n−1 ,

t0 = δ and

n1 n 5 n+1 − n1 − α n n+1 α ≤ ωn n |x | (h + δ) n−1 dH 2 M ∩supp h

· |xn+1 |α |∇M h| dHn M

from which Theorem 4 follows by letting δ tend to zero. Using H¨ older’s inequality we obtain from Theorem 4

Proposition 3. Let M and h be as in Theorem 4. Then we have the estimate n−p p1 np np α |xn+1 |α h n−p dHn ≤ c(n)− n |xn+1 |α |∇M h|p dHn M

M

for all p ∈ [1, n) provided that

−α−n −1 ωn

M ∩supp h

|x

| dH

n+1 α

n

n1

≤

1 . 10α

Of particular interest for later applications is the case p = 2, i.e. n−2

n 2n 2α n n+1 α n−2 |x | h dH ≤ c(n)− n |xn+1 |α |∇M h|2 dHn . (32) M

M

We apply H¨older’s inequality with exponents p=

n+α−2 n · n−2 n+α

and obtain

and

q=

n(n + α − 2) 2α

n+α

|xn+1 |α h n+α−2 dHn M

n

≤

|xn+1 |α h n−2 dHn

p1

M

M ∩supp h

|xn+1 |α dHn

q1

2

whence, after replacing h by h , n+α−2 n+α 2(n+α) n n+1 α n+α−2 |x | h dH M

≤

|x M

| h

n+1 α

2n n−2

dH

n

n−2 n M ∩supp h

|xn+1 |α dHn

2α n(n+α)

.

3.6 Pointwise Curvature Estimates

217

By virtue of (32) we get |x

| h

n+1 α

2(n+α) n+α−2

dH

n

n+α−2 n+α

M

≤ c(n)

− 2α n

M ∩supp h

|x

| dH

n+1 α

n

2α n(n+α)

|xn+1 |α |∇M h|2 dHn .

M

Thus we arrive at Proposition 4. Suppose M and h ∈ Cc1 (U ) are as in Theorem 4. Then we have the estimate |x

(33)

| h

n+1 α

2(n+α) n+α−2

dH

n

n+α−2 n+α

≤ C(n, α)

M

|xn+1 |α |∇M h|2 dHn M

provided that supp h is small in the sense that

−n−α ωn−1

Here C(n, α) = c(n)(

M ∩supp h

|xn+1 |α dHn

n1

≤

1 . 10α

1/n 2α ωn n+α . 10α )

Proposition 4 provides the Sobolev-type estimate which will be needed in the next section.

3.6 Pointwise Curvature Estimates In this section we study pointwise curvature estimates for minimal and αminimal hypersurfaces. In particular we derive a higher-dimensional analogue of Heinz’s estimate (Theorem 2, Section 2.4, Vol. 1) due to Schoen, Simon, and Yau [1], and Dierkes [13]. The proofs will make extensive use of the results obtained so far in this chapter, especially we make use of Simons’s identity (Theorem 1 of Section 3.3) and consequences thereof, in particular Lemmas 3 and 5 in Section 3.4. Further essential tools are the Michael–Simon inequalities which are derived in Section 3.5, Theorems 3 and 4, and the integral curvature estimates Theorems 3 and 5 in Section 3.4. The interior pointwise curvature bounds are then established by a Moser-type iteration argument “on the surface”. This yields a local L∞ -bound of the length |A| of the second fundamental form which is derived in Theorems 1 and 2. A somewhat diﬀerent pointwise estimate for the product |A| · v where 1 + |Du|2 , formulated in Theorem 3, is obtained by an application v = of the mean value inequality for subharmonic functions on a minimal graph in combination with Simons’s identity and Jacobi’s ﬁeld equation. This esti-

218

3 Bernstein Theorems and Related Results

mate implies a generalization of Moser’s Bernstein Theorem (Corollary 3 in Section 3.1) due to Ecker and Huisken [1]. We ﬁrst proceed to describe the pointwise estimates for minimal hypersurfaces M ⊂ Rn+1 . As in Section 3.4 it is convenient to distinguish between geodesic balls BR (x0 ) in M and the intersection of M with Euclidean balls BR (x0 ), i.e. we deﬁne for x0 ∈ M BR (x0 ) := {x ∈ M : distM (x, x0 ) < R} and BR (x0 ) ∩ M := {x ∈ M : |x − x0 | < R}. Both sets are given by a Lipschitz continuous function r := rx0 , where rx0 (x) = distM (x, x0 ) or rx0 (x) := |x − x0 | respectively. Note that we have |∇M r| ≤ 1 a.e. on M . We put KR (x0 ) := {x ∈ M : rx0 (x) < R} to denote either the geodesic ball BR (x0 ) or the intersection BR (x0 ) ∩ M . The following result establishes local curvature estimates. Theorem 1. Let M ⊂ Rn+1 be a stable minimal hypersurface of class C 2 and suppose KR (x0 ) M with Hn (KR (x0 )) ≤ KRn . If n ≤ 5, the estimate (1)

sup KR/4 (x0 )

|A|2 ≤ C(n, K)R−2

holds true for some constant C depending only on n and K. Remarks. 1. If M ⊂ Rn+1 , n ≤ 5, is complete and the area-growth assumption Hn (BR (x0 )) ≤ KRn holds for all R > 0, we can let R → ∞ in formula (1) and obtain that |A(x0 )|2 = 0 for every x0 ∈ M . From this we conclude that M must be a hyperplane. 2. Similarly, if M = graph u where u ∈ C 2 (Rn ) is an entire solution of the minimal surface equation, we can apply Theorem 1 with KR (x0 ) = BR (x0 ) ∩ M , R > 0 arbitrary, since in this case the growth assumption Hn (BR (x0 ) ∩ M ) ≤ KRn is always fulﬁlled and also every minimal graph is stable, cp. the proof of Theorem 4 in Section 3.4. Thus we have for all R > 0 and every x0 ∈ M the estimate sup M ∩BR/4 (x0 )

|A|2 ≤

C(n, K) R2

provided that n ≤ 5. Evidently this implies Bernstein’s Theorem up to the dimension n = 5. 3. A simple modiﬁcation of the proof of Theorem 1 shows that for all θ ∈ (0, 1) there is a constant C = C(n, K, θ) such that

3.6 Pointwise Curvature Estimates

219

sup |A|2 ≤ C(n, K, θ) · R−2 . KθR (x0 )

We start to prove Lp -estimates for the curvature function |A|; these inequalities will later be iterated. Select arbitrary numbers 0 < σ < ρ < R/2 and let η = η(x) be a non negative cut-oﬀ function deﬁned by 1 on Kρ−σ (x0 ) = {x ∈ M : r(x) < ρ − σ}, η(x) = 0 on M \ Kρ (x0 ) such that |∇M η| ≤ cσ −1 with some constant c not depending on ρ or σ. (It is clear that for each σ < ρ as above such an η exists.) Lemma 1. Put f := |A|p where p > 1 and suppose σ, ρ, R and η are as above. Then we have the estimate 1/λ −2 n f 2 dHn for λ = n−2 , 3 ≤ n ≤ 5, Cp σ Kρ n 2λ (2) f dH ≤ 2 Cp σ −1 Kρ f 2 dH for λ = n = 2, Kρ−σ where C = C(n, K) and = (n) > 1 are constants depending only on n, K or n respectively. Proof. We infer from Simons’s identity (10) and Lemma 3 in Section 3.4 2 |∇M |A||2 − 2|A|4 . ΔM |A|2 = 2|∇M A|2 − 2|A|4 ≥ 2 1 + n Let δ > 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

3 Bernstein Theorems and Related Results

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 H¨older’s inequality,

222

3 Bernstein Theorems and Related Results

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

3.6 Pointwise Curvature Estimates

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

3 Bernstein Theorems and Related Results

(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.

3.6 Pointwise Curvature Estimates

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

3 Bernstein Theorems and Related Results

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

3.6 Pointwise Curvature Estimates

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

3 Bernstein Theorems and Related Results

(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 H¨older’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

3.6 Pointwise Curvature Estimates

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

|xn+1 |α f 2 η 2 dHn

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 η 2 dHn

|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),

<

1 . 10α

230

3 Bernstein Theorems and Related Results

such that |∇M η| ≤ 2/σ, where 0 < σ < ρ < R/2. Relation (22) then yields the estimate 1/λ (24) Kρ−σ

|xn+1 |α f 2λ dHn

≤ c 2 κτ ≤ c 3 κτ

−1

−1

p1+τ p1+τ

−1

−1

R−2 σ −2

Kρ

Kρ

|xn+1 |α f 2 dHn + c2 σ −2

Kρ

|xn+1 |α f 2 dHn

|xn+1 |α f 2 dHn ,

which holds for 0 < σ < ρ < R/2 and R > 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

3.6 Pointwise Curvature Estimates

(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

3 Bernstein Theorems and Related Results

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

3.6 Pointwise Curvature Estimates

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

3 Bernstein Theorems and Related Results

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

3.6 Pointwise Curvature Estimates

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

3 Bernstein Theorems and Related Results

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. Rad´o [10], E. Hopf [1,4], L. Bers [2,3], E. Heinz [1], R. Finn [4,5], K. J¨orgens [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 J¨orgens’s result [1] (cp. Theorem 1, Section 3.1) concerning entire solutions of the Monge-Amp`ere 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. J¨orgens [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)

3 Bernstein Theorems and Related Results

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

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3 Bernstein Theorems and Related Results

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

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3 Bernstein Theorems and Related Results

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).

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3 Bernstein Theorems and Related Results

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 (Fr¨ohlich–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<2

then u1 , . . . , uk are linear functions on Rn . In their paper Xin and Yang [1] the authors also generalize Theorem 3 in Section 3.6 due to Ecker and Huisken [1] to the codimension greater one case. Theorem 12 (Xin and Yang [1]). Let M = {(x, u(x)) ∈ Rn × Rk } be an √ √ entire minimal graph with k ≥ 2. If g < 2 and (2 − g)−1 = o(R4/3 ) where 2 2 2 1 k R = |x| + |u| , then u , . . . , u are linear functions and hence M is an n-dimensional plane. Xin and Yang [2] also proved curvature estimates in the spirit of Schoen, Simon, and Yau [1] for minimal submanifolds of higher codimension. In their argument the stability condition is replaced by a suitable assumption on the Gauss map of the submanifold; for details we refer to that paper. There are many further interesting results on the Bernstein problem which we so far did not mention or discuss, e.g. the papers by Calabi [3], Cheng and Yau [1], Cao, Shen, and Zhu [2], Jost and Xin [2], Jost, Xin, and Yang [1] and Trudinger and Wang [1], Xin [2] to mention just a few of them. 3.7.4 Sobolev Inequalities Sobolev inequalities are an important tool in the theory of elliptic partial differential equations. The ﬁrst inequality of this type for functions deﬁned on a minimal graph was obtained by M. Miranda [2]. An essential ingredient in his argument is an isoperimetric inequality on minimal surfaces proved earlier by Federer and Fleming [1]. Diﬀerent and reﬁned versions of the Sobolev inequality on minimal surfaces were later given by Bombieri, De Giorgi, and Miranda [1], Bombieri [2] and Bombieri and Giusti [1], and it was proved that an isoperimetric inequality on a minimal hypersurface is equivalent to a

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3 Bernstein Theorems and Related Results

Sobolev inequality, see Bombieri [1]. The general Sobolev inequality as presented in Theorem 3 of Section 3.5 and its proof are due to Michael and Simon [1], cp. also Simon [8]. Their argument is ﬂexible enough to be extendable to diﬀerent situations like submanifolds of Riemannian manifolds (Hoﬀman and Spruck [1]) and also to α-minimal surfaces, cp. Dierkes [13].

Part II

Global Analysis of Minimal Surfaces

Chapter 4 The General Problem of Plateau: Another Approach

In this chapter we shall present another solution of the general problem of Plateau that, justiﬁedly, is often called the Douglas problem. This is the question whether a conﬁguration Γ1 , Γ2 , . . . , Γk of several closed curves Γj may bound multiply connected minimal surfaces that could be of higher genus and even nonorientable. After a brief introduction to the problem we shall exhibit several examples of boundary conﬁgurations and of minimal surfaces illustrating the general Plateau problem. In Section 4.2 we shall outline an approach to Teichm¨ uller theory of compact oriented Riemann surfaces which, in Section 4.3, is extended to oriented Riemann surfaces with boundary. These results are basic for the variational treatment of the general Plateau problem as we have to deal with variations of the complex structure of two-manifolds. The second basic tool is a compactness theorem of Mumford which will be combined with Courant’s condition of cohesion to obtain nondegenerate solutions of the variational problem connected with the general Plateau problem. This variational problem will be discussed in Section 4.5. We shall outline the crucial ideas for solving this problem and state some of the principal existence results due to Douglas, Courant and Shiﬀman. Finally, in Section 4.6, we shall tackle the general Plateau problem for orientable surfaces in combination with an obstacle problem in order to obtain a suﬃcient condition for the solvability of the general Douglas–Plateau problem that can be formulated in geometric-topological terms.

4.1 The General Problem of Plateau. Formulation and Examples In Chapter 4 of Vol. 1 we discussed the classical problem of Plateau as it was solved by Douglas and Rad´o, and we presented the solution found by Courant and, independently, by Tonelli. In the restricted sense formulated in Deﬁnition 1 of Vol. 1, Section 4.2, Plateau’s problem consists in ﬁnding 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 4,

249

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4 The General Problem of Plateau: Another Approach

a disk-type minimal surface spanning a prescribed closed Jordan curve. This is to say, for a given closed Jordan curve Γ we are to ﬁnd a harmonic and conformal mapping of the disk B = {w ∈ C : |w| < 1} into R3 which maps the boundary of B topologically onto Γ . As we already pointed out, this is neither the most general nor the most natural way to formulate Plateau’s problem, but merely the simplest and most convenient one as we do not run into the problem that parameter domains of the same topological type may be of diﬀerent conformal type. By Riemann’s mapping theorem all simply connected bounded domains in the plane are conformally equivalent. Nevertheless, boundary conﬁgurations consisting of several closed curves may bound multiply connected minimal surfaces as we see in the classical case of minimal surfaces of revolution, the catenoids. Moreover, soap ﬁlm experiments show that certain conﬁgurations may bound minimal surfaces of higher topological structure, say, nonorientable ones, or minimal surfaces that are not of genus zero. Figures 1–9 depict several such contours as well as minimal surfaces spanning them. In certain cases it is not diﬃcult to see that topologically more complicated minimal surfaces may have smaller area than disk-type surfaces bounded by the same contour. The ﬁrst to study Plateau’s problem in a general form was Jesse Douglas; he attacked this question in a series of profound and pioneering papers. Hence many authors speak of the general problem of Douglas (or of the Douglas problem) instead of what Douglas himself called the General problem of Plateau. Given in R3 a conﬁguration Γ = Γ1 , Γ2 , . . . , Γk consisting of k closed and mutually disjoint Jordan curves Γj , ﬁnd a minimal surface of prescribed Euler characteristic, orientable or not, that span the contour Γ . This problem also has many generalizations, We might replace R3 by RN , N ≥ 2, or by any N -dimensional Riemannian manifold, say, by the N -sphere S N or by the N -dimensional hyperbolic space H N . Moreover, instead of minimal surfaces (i.e. surfaces of vanishing mean curvature) we might look for surfaces of prescribed mean curvature, and instead of ﬁxed boundary conﬁgurations we could investigate other boundary arrays leading to free or partially free boundary problems for minimal surfaces as were considered in Chapters 1, 2, and in Chapter 1 of Vol. 2. In all such cases, solutions of higher topological type may occur. In order to tackle solutions of this kind it is necessary to have some hold on their parameter domains which are Riemann surfaces with boundaries (non-oriented surfaces can be treated by passing to the double cover). For this purpose, some results from the theory of Riemann surfaces will be needed, particularly results concerning the so-called Teichm¨ uller theory. In Sections 4.2 and 4.3 we shall outline an approach to Teichm¨ uller theory due to Fischer and Tromba which avoids functions-theoretic tools and yields a very natural geometric method to obtain the results needed for treating the general Plateau problem.

4.1 The General Problem of Plateau. Formulation and Examples

251

Fig. 1. A soap ﬁlm experiment: Catenoids held by two coaxial circles. Courtesy of Institut f¨ ur Leichte Fl¨ achentragwerke, Stuttgart–Archive

Fig. 2. Minimal surfaces bounded by two closed curves. Courtesy of K. Polthier (Figs. 2a, c); J. Hahn and K. Polthier (Fig. 2b)

Fig. 3. An annulus-type minimal surface spanned by two interlocking curves

Fig. 4. Two views of a minimal surface of genus zero bounded by three closed curves. Courtesy of K. Polthier

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4 The General Problem of Plateau: Another Approach

Fig. 5. The catenoid (below), and four of its relatives bounded by 4, 5, and 6 closed curves respectively, constructed by Wohlgemuth. Courtesy of E. Boix, J. Hoﬀman, and M. Wohlgemuth

4.1 The General Problem of Plateau. Formulation and Examples

253

Fig. 6. A closed curve bounding a one-sided minimal surface. This curve also spans a disk-type minimal surface

Fig. 7. (a) A knotted closed curve Γ (cloverleaf). (b) A one-sided minimal surface of genus zero bounded by Γ

Fig. 8. A one-sided minimal surface of genus 2 in a closed curve

For an introduction to the Douglas problem, dealing with surfaces parametrized over multiply connected domains in the complex plane (“schlicht domains”), we refer to the last chapter in Vol. 1. We do not plan to give an exhaustive description of all results obtained for the general Plateau problem. Rather the reader should take this chapter as another exposition of the problem and as a way to become acquainted with some of the more sophisticated methods used to treat boundary value problems for minimal surfaces of higher topological type and of more compli-

254

4 The General Problem of Plateau: Another Approach

Fig. 9. A one-sided minimal surface of genus 2 in a closed curve

Fig. 10. Two-sided minimal surface of higher genus bounded by a single closed curve. (a) A Chen-Gackstatter surface of genus one. (b) A Chen-Gackstatter surface of genus two. (c) An Enneper-type surface of genus two with a Y -handle. Courtesy of J. Hahn and K. Polthier (Fig. 10a); K. Polthier and Wohlgemuth (Figs. 10b, c)

Fig. 11. A rectiﬁable Jordan curve bounding an area-minimizing surface of inﬁnite genus

cated boundaries. As an example of what can be done, we shall discuss a few existence results of Tomi and Tromba in Section 4.6. Some of the existence results of Douglas, Courant and Shiﬀman will be formulated without proof in Section 4.5. References to the literature and to more complete discussions of

4.2 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces

255

the general Plateau problem will be provided in Section 4.5 as well as in the Scholia (Section 4.7). In Douglas’s approach to the general Plateau problem the main feature is a suﬃcient (but not necessary) condition ensuring the existence of a solution. This condition requires that the greatest lower bound of area (or of Dirichlet’s integral) for nondegenerate admissible surfaces is less than the greatest lower bound for surfaces of lower type. Here a surface is called of lower type if at least one of the following two possibilities occurs: (i) It has a smaller Euler characteristic than the prescribed one; (ii) It is degenerate in the sense that it consists of two or more disconnected surfaces whose total Euler characteristic is less than or equal to the prescribed characteristic and which are bounded by subconﬁgurations of Γ = Γ1 , . . . , Γk whose union is the prescribed conﬁguration Γ . Some of the examples pictured in Figs. 1–9 illustrate Douglas’s suﬃcient condition. We shall see why for the general Plateau problem extra conditions have to be posed in order to guarantee the existence of solutions. On account of the enclosure theorems presented in Chapter 4 of Vol. 2 it is in any case clear that the general Plateau problem is not always solvable. However, there is still a big discrepancy between necessary and suﬃcient conditions for solvability. In general, no conditions are known that are both necessary and suﬃcient.

4.2 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces In this section we shall outline a geometric approach to Teichm¨ uller theory for compact oriented surfaces without boundary and of genus greater than one. Here a surface M is meant to be two-dimensional manifold of class C ∞ . It is well-known that there is a collection of coordinate charts {Gj , ϕj } consisting of open sets Gj ⊂ M satisfying Gj = M , and of bijective mapis holomorphic. Such a collection of pings ϕj : Gj → C such that ϕj ◦ ϕ−1 k charts {Gj , ϕj } is called a complex structure; we denote a complex structure on M by c. When we think of M having a ﬁxed complex structure c we shall designate this by writing the pair (M, c). However, a given M may have many complex structures. For example, let f : M → M be a C ∞ diﬀeomorphism. Then we can construct a new complex structure f ∗ c, the pull-back of c by f , considering the coordinate pairs {f −1 (Gj ), ϕj ◦ f }. The maps (ϕj ◦ f ) ◦ (ϕk ◦ f )−1 are also holomorphic, and hence f ∗ c is indeed a complex structure, and f : (M, f ∗ c) → (M, c) is a holomorphic mapping. It seems natural to identify (M, c) with (M, f ∗ c) and to consider equivalence classes of complex structures on a ﬁxed manifold M under this equivalence

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4 The General Problem of Plateau: Another Approach

relations. To be precise, let C = C(M ) be the space of all complex structures on M , and denote by D = D(M ) the space of all orientation preserving C ∞ -diﬀeomorphisms of M to itself. Then consider the space R(M ) := C(M )/D(M ) of equivalence classes of complex structure on M as described above. The space R(M ) is called the Riemann space of moduli of M . Riemann conjectured that R(M ) is a (6g − 6)-dimensional space in case that the genus g of M is greater than one, as we have assumed. To investigate the structure of R, Teichm¨ uller introduced a somewhat larger space T = T(M ) deﬁned by T := C/D0 where C/D0 is the quotient space of C with respect to the subgroup D0 of diﬀeomorphisms f ∈ D which are homotopic to the identity map on M . D0 can also be characterized as the set of diﬀeomorphisms which are homotopic to the identity through diﬀeomorphisms. One calls T(M ) the Teichm¨ uller space of M . Then we have R(M ) = T(M )/Γ where Γ := D/D0 is the modular group of M . Nielsen [1,2] proved that Γ is a discrete group. (Actually, Γ turned out to be isomorphic to the group of outer automorphisms of M .) One may attempt to understand the moduli space R(M ) by ﬁrst investigating T(M ) and then studying the action of Γ on T(M ). In the following we shall not employ any of Teichm¨ uller’s ideas. Rather we shall outline a description of Teichm¨ uller spaces based on work by Fischer and Tromba [1]. It is somewhat diﬃcult to investigate the space C of complex structures on M directly. We shall come to understand C and the action of D on it through a somewhat circuitous route. Deﬁnition 1. An almost complex structure J on M is a C ∞ -section of the (1, 1)-tensor bundle T11 (M ) over M such that J 2 = −I, I the identity map. More colloquially, for each p ∈ M, J(p) is a linear map of the tangent space Tp M into itself such that J 2 (p) = −I(p), where I(p) is the identity on Tp M and where the mapping p → J(p) is of class C ∞ . We say that J is orientation preserving if, for each p and for non-zero Xp ∈ Tp M , the pair (Xp , J(p)Xp ) is a positively oriented basis for Tp M . We shall denote by A(M ), or simply A, the space of all almost complex structures. It is not diﬃcult to see that with any given structure c ∈ C we can associate a unique J ∈ A. This can be done as follows. Let (ϕj , Gj ) be a coordinate chart ∈ c. Deﬁne Ji (p) : Tp M → Tp M by

4.2 A Geometric Approach to Teichm¨ uller Theory of Oriented Surfaces

Jj (p) = dϕ−1 j

0 1

−1 0

257

dϕj

is the canonical almost complex structure on R2 . where 01 −1 0 We have Jj (p) = Jk (p) if dϕ−1 j

−1 0

0 1

or if dψ

dϕj = dϕ−1 k

ψ := ϕj ◦ ϕ−1 k , −1 0 dψ −1 = 0 1

0 1

−1 0

0 1

−1 0

dϕk

,

where ψ is a complex analytic mapping. Write ψ = x + iy, x = x(u, v), y = y(u, v). Then dψ is represented by the matrix xyuu xyvv which, by the Cauchy– Riemann equations, is of the form a −b xu −yu = . b a yu xu Also,

a b

−b a

−1

1 = 2 a + b2

a −b

b a

.

Thus we have 1 2 a + b2

a b

−b a

0 −1 1 0

a −b

b a

=

0 1

−1 0

and consequently Jj (p) = Jk (p). By this computation we see that each c ∈ C induces a J ∈ A if we deﬁne J(p) = Jj (p) for any coordinate chart (Gj , ϕj ) ∈ c. We shall see that the converse is also true, i.e., given any J ∈ A, there is a unique c ∈ C which induces J in the above described manner. This converse is a much deeper fact which is only true in the two-dimensional case. The diﬀeomorphism group D acts on A in a natural way. For f ∈ D deﬁne (f ∗ J)(p) := dfp−1 Jf (p) dfp . It is clear that f ∗ J ∈ A if J ∈ A, and one easily sees that for the correspondence c → J we have f ∗ c → f ∗ J. Thus the map which sends c to J is D-equivariant. We shall now introduce another space of objects into the picture, the space of all C ∞ -Riemannian metrics M on M . Deﬁnition 2. The space M of Riemann C ∞ -metrics on M is the space of positive C ∞ -sections of the (0, 2) tensor bundle T02 (M ).

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4 The General Problem of Plateau: Another Approach

This is to say, if g ∈ M, then for each p ∈ M the metric g(p) is a positive deﬁnite symmetric bilinear form on Tp M, g(p) : Tp M × Tp M → R, so that p → g(p) is of class C ∞ . It is easy to see that M is a Fr´echet manifold since M is open in the linear space S2 of all symmetric two-tensors on M (S2 is deﬁned in the same way as M except that positive deﬁniteness is not required). The diﬀeomorphism group D acts on M as follows: (f ∗ g)(p)(V, W ) = g(f ((p))(df p V, df p W )), where df p : Tp M → Tf (p) M denotes the tangent mapping deﬁned by f . Let P be the space of all positive C ∞ -functions on M , and let M/P denote the quotient space of M with respect to P; that is, we identify two metrics g1 and g2 if g1 (p) = ρ(p)g2 (p) for all p, where ρ is a strictly positive C ∞ -function on M . In brief, M/P is the set of classes of pointwise conformally equivalent metrics g. Theorem 1. There is a bijective equivalence (in fact, a diﬀeomorphism) between A and M/P. Proof. Given g ∈ M, there is a standard way to construct a new and unique nondegenerate, alternating, antisymmetric bilinear form μg (p) : Tp M × Tp M → R such that, if Vp , Wp is an oriented basis for Tp M , then μg (p)(Vp , Wp ) > 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:

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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)

4 The General Problem of Plateau: Another Approach

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

4 The General Problem of Plateau: Another Approach

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 .

4.3 Symmetric Riemann Surfaces and Their Teichm¨ uller Spaces

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 Ga