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).

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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