Minimal Surfaces II: Boundary Regularity (Grundlehren Der Mathematischen Wissenschaften)

Ulrich Dierkes Stefan Hildebrandt

Albrecht Mister Ortwin Wohlrab

Minimal Surfaces II Boundary Regularity

With 59 Figures and 4 Colour Plates

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Ulrich Dierkes

Stefan Hildebrandt Albrecht Kuster

Universitat Bonn, Mathematisches Institut Wegelerstral3e 10, D-5300 Bonn, Federal Republic of Germany Ortwin Wohlrab Mauerseglerweg 3, D-5300 Bonn, Federal Republic of Germany

Mathematics Subject Classification (1991): 53A 10, 35J60

ISBN 3-540-53170-X Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-53170-X Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data Minimal surfaces/Ulrich Dierkes... [etal.] v. cm. - (Grundlehren der mathematischen Wissenschaften; 295-296) Includes bibliographical references and indexes. Contents: 1. Boundary value problems - 2. Boundary regularity. ISBN 3-540-53169-6 (v.1: Berlin). - ISBN 0-387-53169-6 (v. 1: NewYork). ISBN 3-540.53170-X (v.2: Berlin). -ISBN 0-387-53170-X (v.2: New York) 1. Surfaces- Minimal. 2. Boundary value problems. I. Dierkes, Ulrich. II. Series.

QA644.M56 1992 516.362 - dc2O 90-27155 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in 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 a copyright fee must always be obtained from Springer-Verlag. Violations fall under the prosecution act of the German Copyright Law. c; Springer-Verlag Berlin Heidelberg 1992 Printed in the United States of America

Typesetting: Asco Trade Typesetting Ltd., Hong Kong 41/3140-543210 Printed on acid-free paper

Preface

This second volume is mainly devoted to the study of the boundary behaviour of minimal surfaces satisfying various kinds of fixed and free boundary conditions. In addition we treat the so-called thread problem, and we also provide an introduction to the general Plateau problem. Volume 2 can be read independently of Volume 1 although we use some terminology and results from the preceding material. We would like to mention that a substantial part of this volume originated in joint work of the second author with M. Griiter, E. Heinz, J.C.C. Nitsche, and F. Sauvigny. Other parts are drawn from work by H.W. Alt, R. Courant, J. Douglas, G. Dziuk, R. Gulliver, E. Heinz, W. Jager, H. Lewy, C.B. Morrey J.C.C. Nitsche, F. Tomi and A. Tromba. The material of Chapter 11 is essentially based on the

joint paper Tomi/Tromba [4], and we are very much indebted to Anthony Tromba who wrote for us the main part of this chapter. We are grateful to Klaus Bach, Frei Otto, and Eric Pitts for providing us with photographs of various soap film experiments, and to David Hoffman, Hermann Karcher, Konrad Polthier, and Meinhard Wohlgemuth for allowing us to use some of their computer drawings of minimal surfaces. The support of our work by the Computer Graphics Laboratory of the Institute of Applied Mathematics at Bonn University and of the Sonderforschungsbereich 256 was invaluable. We are especially grateful to Eva Kuster who polished both style and grammar of these notes, and to Anke Thiedemann who professionally and with untiring patience typed many versions of our manuscript. We should also like to thank Katrin Rhode and Gudrun Turowski for checking parts of the computations in Chapter 7. Without the generous of support of SFB 72 and SFB 256 at Bonn University this book could not have been written. Last but not least we should like to thank the patient publisher and his collaborators, in particular Joachim Heinze and K. Koch, for their encouragement and help. Bonn, April 30, 1991

Ulrich Dierkes, Stefan Hildebrandt, Albrecht Kuster, Ortwin Wohlrab

Contents of Minimal Surfaces II

Introduction

..................................................

1

Part III. Boundary Behaviour of Minimal Surfaces

............ Potential-Theoretic Preparations ............................ Solutions of Differential Inequalities ..........................

Chapter 7. The Boundary Regularity of Minimal Surfaces 7.1

7.2 7.3

7.5 7.6 7.7 7.8

...................................

33

The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs Holder Continuity for Minima Holder Continuity for Stationary Surfaces

43 48 60

C1,112-Regularity

83

.............................. ..................... ..........................................

Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary 7.9 A Different Approach to Boundary Regularity 7.10 Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences 7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces 7.12 Scholia

.....

................. ......................... ..................................................

Chapter 8. Singular Boundary Points of Minimal Surfaces 8.1

7 21

The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

7.4

6

............

102 109 117 121 128

141

The Method of Hartman and Wintner, and Asymptotic Expansions at Boundary Branch Points

.................................

142

8.2 A Gradient Estimate at Singularities Corresponding

to Corners of the Boundary .................................

163

...................

173

8.4

Minimal Surfaces with Piecewise Smooth Boundary Curves and Their Asymptotic Behaviour at Corners An Asymptotic Expansion for Solutions of the Partially

8.5

Free Boundary Problem .................................... Scholia ..................................................

186 196

8.3

VIII

Contents of Minimal Surfaces II

Chapter 9. Minimal Surfaces with Supporting Half-Planes ............

198

An Experiment ............................................

199

9.2

Examples of Minimal Surfaces with Cusps on the Supporting Surface

202

9.3 9.4 9.5

Set-up of the Problem. Properties of Stationary Solutions ........ 206

9.1

.................................. Classification of the Contact Sets ............................

208

Nonparametric Representation, Uniqueness, and Symmetry of Solutions

..............................................

9.6

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

9.7 9.8

Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II Final Results on the Shape of the Trace. Absence of Cusps.

9.9

Proof of the Representation Theorem .........................

Minima of Dirichlet's Integral ............................... .

Optimal Boundary Regularity ...............................

9.10 Scholia ..................................................

213

216 218 221 223 229

1. Remarks about Chapter 9. 2. Numerical Solutions. 3. Another Uniqueness Theorem for Minimal Surfaces with a Semifree Boundary.

Part IV. Ramifications: The Thread Problem. The General Plateau Problem

Chapter 10. The Thread Problem .................................

250

10.1 Experiments and Examples. Mathematical Formulation

of the Simplest Thread Problem .............................

..................................................

250 255 271 291

Chapter 11. The General Problem of Plateau .......................

293

10.2 Existence of Solutions to the Thread Problem ..................

10.3 Analyticity of the Movable Boundary ......................... 10.4 Scholia

11.1 The General Problem of Plateau. Formulation and Examples .... 293 11.2 A Geometric Approach to Teichmuller Theory of Oriented Surfaces 299 11.3 Symmetric Riemann Surfaces and Their Teichmuller Spaces ...... 307

11.4 The Mumford Compactness Theorem ........................

..................................................

315 319 328 339

..................................................

341

11.5 The Variational Problem ................................... 11.6 Existence Results for the General Problem of Plateau in l3 11.7 Scholia Bibliography

......

Index of Names ................................................ Subject Index

.................................................

Index of Illustrations Minimal Surfaces II Minimal Surfaces I

397

400

.......................................... ...........................................

415 417

Sources of Illustrations of Minimal Surfaces II ......................

422

Contents of Minimal Surfaces I

Introduction

..................................................

I

Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces Chapter 1. Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

............................

6

1.1

Surfaces in Euclidean Space .................................

7

1.2

Gauss Map, Weingarten Map. First, Second, and Third Fundamental Form. Mean Curvature and Gauss Curvature Gauss's Representation Formula, Christoffel Symbols, Gauss-Codazzi Equations, Theorema Egregium, Minding's Formula for the Geodesic Curvature Conformal Parameters. Gauss-Bonnet Theorem Covariant Differentiation. The Beltrami Operator

1.3

1.4 1.5 1.6

Scholia

......

......................... ................ .............. ..................................................

11

25 34

40 48

1. Textbooks. 2. Annotations to the History of the Theory of Surfaces. 3. References to the Sources of Differential Geometry and to the Literature on Its History.

..................................... First Variation of Area. Minimal Surfaces ..................... Nonparametric Minimal Surfaces ............................ Conformal Representation and Analyticity of Nonparametric Minimal Surfaces .......................................... Bernstein's Theorem ....................................... Two Characterizations of Minimal Surfaces ...................

Chapter 2. Minimal Surfaces

53

2.1 2.2 2.3

54

2.4 2.5 2.6

2.7 2.8 2.9

Parametric Surfaces in Conformal Parameters. Conformal Representation of Minimal Surfaces. General Definition of Minimal Surfaces A Formula for the Mean Curvature Absolute and Relative Minima of Area Scholia

....................................... .......................... ....................... ..................................................

1. References to the Literature on Nonparametric Minimal Surfaces. 2. Bernstein's Theorem. 3. Stable Minimal Surfaces. 4. Foliations by Minimal Surfaces.

58 61

65 71

74 77 80 85

X

Contents of Minimal Surfaces I

Chapter 3. Representation Formulas and Examples

of Minimal Surfaces ............................................

89

3.1

The Adjoint Surface. Minimal Surfaces as Isotropic Curves in C3.

3.2 3.3 3.4

Behaviour of Minimal Surfaces Near Branch Points Representation Formulas for Minimal Surfaces Bjorling's Problem. Straight Lines and Planar Lines of Curvature

101

on Minimal Surfaces. Schwarzian Chains ......................

120 135

3.5

Associate Minimal Surfaces .................................

............ ................

Examples of Minimal Surfaces ...............................

90 107

1. Catenoid and Helicoid. 2. Scherk's Second Surface: The General Minimal Surface of Helicoidal Type. 3. The Enneper Surface. 4. Bour Surfaces. 5. Thomsen Surfaces. 6. Scherk's First Surface. 7. The Henneberg Surface. 8. Catalan's Surface. 9. Schwarz's Surface.

3.6 3.7 3.8

Complete Minimal Surfaces ................................. Omissions of the Gauss Map of Complete Minimal Surfaces

.....

Scholia ..................................................

175 181 192

1. Historical Remarks and References to the Literature. 2. Complete Minimal Surfaces of Finite Total Curvature and of Finite Topology. 3. Complete Properly Immersed Minimal Surfaces. 4. Construction of Minimal Surfaces. 5. Triply Periodic Minimal Surfaces.

Part II. Plateau's Problem and Free Boundary Problems

Chapter 4. The Plateau Problem and the Partially Free Boundary Problem for Minimal Surfaces

..........................

221

4.1

Area Functional Versus Dirichlet Integral .....................

226

4.2

Rigorous Formulation of Plateau's Problem and of the

Minimization Process ......................................

231

4.3 4.4 4.5

Existence Proof, Part I: Solution of the Variational Problem

234 239

The Courant-Lebesgue Lemma .............................. Existence Proof, Part II: Conformality of Minimizers of the Dirichlet Integral

4.6

.....

..........................................

Boundary Problem

........................................

Boundary Behaviour of Minimal Surfaces with Rectifiable Boundaries 4.8 Reflection Principles 4.9 Uniqueness and Nonuniqueness Questions 4.10 Scholia 4,7

242

Variant of the Existence Proof. The Partially Free

..................................... ....................................... .................... ..................................................

1. Historical Remarks and References to the Literature. 2. Branch Points. 3. Embedded Solutions of Plateau's Problem. 4. More on Uniqueness and Nonuniqueness. 5. Index Theorems, Generic Finiteness, and Morse-Theory Results. 6. Obstacle Problems. 7. Systems of Minimal Surfaces.

253

259 267 270 276

XI

Contents of Minimal Surfaces I

Chapter 5. Minimal Surfaces with Free Boundaries 5.1

5.2 5.3 5.4 5.5 5.6 5.7 5.8

5.9

..................

Surfaces of Class Hz and Homotopy Classes of Their Boundary Curves. Nonsolvability of the Free Boundary Problem with Fixed Homotopy Type of the Boundary Traces Classes of Admissible Functions. Linking Condition Existence of Minimizers for the Free Boundary Problem Stationary Minimal Surfaces with Free or Partially Free Boundaries and the Transversality Condition Necessary Conditions for Stationary Minimal Surfaces Existence of Stationary Minimal Surfaces in a Simplex Stationary Minimal Surfaces of Disk-Type in a Sphere Report on the Existence of Stationary Minimal Surfaces in

...................... ............ .........

............................

328 335 339 341

Convex Bodies ............................................

343

Nonuniqueness of Solutions to a Free Boundary Problem. Families of Solutions

......................................

345

..................................................

365

Chapter 6. Enclosure Theorems and Isoperimetric Inequalities for Minimal Surfaces with Fixed or Free Boundaries

................

6.2 6.3 6.4 6.5

305 318 321

.......... .......... ..........

5.10 Scholia

6.1

303

Applications of the Maximum Principle and Nonexistence of Multiply Connected Minimal Surfaces with Prescribed Boundaries Touching H-Surfaces and Enclosure Theorems. Further Nonexistence Results Isoperimetric Inequalities Estimates for the Length of the Free Trace Scholia

367

.

368

............................... ................................... .................... ..................................................

372 382 396 420

1. The Isoperimetric Problem. Historical Remarks and References to the Literature. 2. Experimental Proof of the Isoperimetric Inequality. 3. Estimates for the Length of the Free Trace. 4. Enclosure Theorems and Nonexistence.

..................................................

427

Index of Names ................................................

483

.................................................

486

Bibliography

Subject Index

Index of Illustrations Minimal Surfaces I Minimal Surfaces II

........................................... ..........................................

...................... ................................. after page

501

506

Sources of Illustrations of Minimal Surfaces I

508

Colour Plates I-VIII

218

Introduction

The present volume 2 consists of parts III (Boundary behaviour of minimal

surfaces) and IV (Ramifications: The thread problem. The general Plateau problem) of our text. Part III is less geometric and more analytic in nature and can be considered as a special topic in regularity theory for nonlinear boundary value problems of elliptic systems. Nevertheless regularity results are not only an interesting exercise in generalizing classical results on conformal mappings to minimal surfaces and to H-surfaces, but they also have interesting applications in geometry, for instance in establishing compactness results, index theorems, or geometric inequalities such as estimates on the length of the free trace (see Section 6.4) or generalized Gauss-Bonnet formulas (cf. Section 7.11). Actually, the notions of regular curve, regular surface, regularity are used in an ambiguous way. On the one hand, regularity of a map X : 6 R3 can mean that X is smooth and belongs to a class C', C a, ... , C,._ , CO, C", or to a Holder class C, or to a Sobolev class

H. The regularity results obtained in Chapter 7 are to be understood in this sense. On the other hand, a map X : b -+ R3 viewed as a parameter representation of a surface in R3 is called regular or a regular surface if its Jacobi matrix (X., Xp) has rank 2, i.e. if at all of its points the surface has a well defined tangent space. Nowadays regular surfaces are usually denoted as immersions or immersed surfaces in order to avoid ambiguities of the kind mentioned above.

In Chapter 7 we investigate the boundary behaviour of minimal surfaces subject to Plateau boundary conditions or to free boundary conditions. Roughly speaking we show that a minimal surface is as smooth at the boundary as the data of the boundary conditions to which it is subject. There is a basic difference between grappling with the regularity problem for area minimizing surfaces or for merely stationary solutions of boundary value problems. For minimizers it is always possible to derive a priori estimates while examples show that it is generally impossible to establish a priori estimates for stationary solutions of free boundary problems. Thus it becomes necessary to apply indirect methods if one wants to prove boundary regularity of minimal surfaces subject to free boundary conditions. For a more complete understanding of the boundary behaviour of minimal surfaces one not only has to investigate their class of smoothness at the boundary, but it is also necessary to find out whether singular points occur at the boundary and if so, how a minimal surface behaves in the neighbourhood of such points.

2

Introduction

This question is tackled in Chapter 8. If a minimal surface X (w), w = u + iv, is given in conformal parameters, then its singular (= nonregular) points are exactly its branch points wo, which are characterized by the relation X,,(wo) = 0,

In Chapter 8 we shall derive asymptotic expansions of minimal surfaces at boundary branch points which can be seen as a generalization of Taylor's formula to the nonanalytic case. Moreover, we shall also derive expansions of minimal surfaces with nonsmooth boundaries (e.g. polygons) at boundary points which are mapped onto vertices of the nonsmooth boundary frame. Asymptotic expansions are very useful if one wants to treat subtle geometric and analytic problems. A nonstandard problem of this kind is studied in Chapter 9. Suppose that a minimal surface X : 0 -+ R3 solves a free or partially free boundary problem where the supporting surface S has a nonempty boundary F. Then new problems arise if the free trace E = X I an attaches to T ; in particular

one may ask what shape E assumes and how it touches T. The regularity questions of this problem are answered by the results of Chapters 7 and 8, but a detailed investigation of E has only been carried out for the special case considered in Chapter 9. A uniqueness result for solutions of special semifree boundary problems completes Part III of our notes. There are many other types of boundary problems. For instance, systems of minimal surfaces form intriguing free boundaries. However, the basic existence and regularity results appear to be inaccessible by means of the methods devel-

oped in our notes. Thus we confine ourselves to the brief survey given in the Scholia 5.10 of volume 1 and to references to the literature. Instead we consider two other problems in the fourth and final part of this treatise, the thread problem and the general Plateau problem. The thread problem is a remarkable generalization of the isoperimetric problem to minimal surfaces and leads to many beautiful experiments. In Chapter 10 we shall treat both existence and regularity results for minimal surfaces with movable boundaries consisting of threads. The final chapter, essentially written by A. Tromba, is based on the joint paper [5] by F. Tomi and A. Tromba and deals with the general Plateau problem. This is the question of how to span a minimal surface of prescribed topological type in a boundary configuration consisting of finitely many disjoint Jordan curves. Our presentation of this topic will be less complete than that of the other chapters as we shall use results which will not be proved in these notes. The main purpose of this chapter is to provide a geometric approach to Teichmuller theory and to give an idea of how it can be used for solving general Plateau problems. The presentation of volume 2 is somewhat more advanced than that of the first volume although we have tried to give an essentially self-contained exposition of the basic facts from potential theory used in Chapter 7. Only a few results of regularity theory will be borrowed from other sources, usually only information needed for more refined statements. Concerning asymptotic expansions at corners we rely on some results taken from Vekua's treatise [1, 2]. Chapter 10 is more or less on the same level as Chapters 7-9 while Chapter 11 might offer more difficulties; further study is recommended as Chapter 11 only provides an introduction to the general Plateau problem.

Introduction

3

The Scholia serve as sources of additional information. In particular we try to give credit to the authorship of the results presented in the main text, and we sketch some of the main lines of the historical development. References to the literature and brief surveys of relevant topics not treated in our text complete the picture. Our notation is essentially the same as in the treatises of Morrey [8] and of Gilbarg-Trudinger [1]. Sobolev spaces are denoted by HP instead of W',a; the definition of the classes C°, Ck, C°°, and C'`,' is the same as in Gilbarg-Trudinger [1]; C`° denotes the class of real analytic functions; CC(Q) stands for the set of C°°-functions with compact support in Q. For greater precision we write C'(Q, Ip3) for the class of Ck-mappings X : 0 - R3, whereas the corresponding class of scalar functions is denoted by C"(Q), and similarly for the other classes of differentiability. Another standard symbol is B,(wo) for the disk {w = u + iv e C: Iw - w0I < r} in the complex plane. If formulas become too cumbersome to read, we occasionally write B(wo, r) instead of B,(wo). In general we shall deal with minimal surfaces defined on simply connected bounded parameter domains Q which, by Riemann's mapping theorem, are all conformally equivalent to each

other. Hence we can pick a standard representation B for 0: we take it to be either the unit disk {w: IwI < 1} or the semidisk {w: Iwi < 1, Im w > 0}. In the first case we write C for 8B, in the second C denotes the semicircle {w: IwI = 1, Im w > 0} while I stands for the interval {u c E8: Jul < 1}. On some occasions it is convenient to switch several times from one meaning of B to the

other. Moreover, some definitions based on one meaning of B have to be transformed mutatis mutandis to the other one. This may sometimes require slight changes but we have refrained from pedantic adjustments which the reader can easily supply himself.

Part III

Boundary Behaviour of Minimal Surfaces

Chapter 7. The Boundary Regularity of Minimal Surfaces

In this chapter we deal with the boundary behaviour of minimal surfaces with particular emphasis on the behaviour of stationary surfaces at their free boundaries. This and the following chapter will be the most technical and least geometric parts of our lectures. They can be viewed as a section of the regularity theory for nonlinear elliptic systems of partial differential equations. Yet these results are crucial for a rigorous treatment of many geometrical questions, and thus they will illustrate what role the study of partial differential equations plays in differential geometry.

The first part of this chapter, comprising Sections 7.1-7.3, deals with the boundary behaviour of minimal surfaces at a fixed boundary. Consider for example a minimal surface X : B R 3 which is continuous on B and maps aB onto some closed Jordan curve F. Then we shall prove that X is as smooth on B as r, more precisely, that X is of class C°(B, l3) (or X E C'(B, R3), or X E Cm,u(B, R3)) if r is of class C° (or T e C', or T E C",", respectively). These results are worked out in Section 7.3. In Section 7.1 we shall supply some results

from potential theory that will be needed, and in Section 7.2 we shall derive various regularity results and estimates for vector-valued solutions X of differential inequalities of the kind

I4XI - 2h. The function w e BR,

HH(w, )q(C)d2,,

Yh(w) := J

(16)

B,

is of class C2 and, setting a := IgIo,BR, we obtain J

IY(w)-Yh(w)I 0

whence we infer that V y, tends uniformly to z on every 0 c c BR. Together with the uniform convergence of yh to yon 0 c c BR as h - 0 we infer that y e C' (BR) and Vy(w) = z(w) for any w c- BR. Consequently {IH(w, 0)1 + IVwH(w, 0)I}

IY(w)I + IVWY(w)I BR

0} and {t(e"' - w) : t > 0}, we obtain O(w) = 1w - e'll" cos(pa(w))

where la(w)l < 2. Thus we infer from (29) that (30)

Hi/i(w) /Dr

< h*(w) - h*(e"°)
2(1 - aM)IVXI2

in 0. In particular, if aM < 1, then IXI2 is subharmonic in Q.

Proof. Because of I <X, AX>I < IX IIAX I < aMIVXI2,

the inequality (2) is an immediate consequence of the identity (3)

AIX12 = 21 VX12 + 2<X, AX>

which holds for every mapping X of class C2.

ER2 which

?. The Boundary Regularity of Minimal Surfaces

11

R") nC2(BR(wo), U8") satisfies (1) in Lemma 2. Suppose that X E C°(BR(wo), BR(wo) and aM < I are satisfied. BR(w(,). Assume also that IX(w)l < M for w e Then for any p e (0, R) we have 27rM

I

z


)(cp):= I-'(9)I2 of x(e`') satisfy rl" _ 21-T'12 + 2d(p wo

Hence we obtain

2nk

I <E, .

for all qp e CO, 2n].

Since E can be chosen as an arbitrary vector of I1BN, we conclude that I g-'((p) I

_< 2nk

for all q e R,

and therefore

I'"I l defined on the unit disk B = { w e C : I w I < 11. The same results can be carried over without

any problem to minimal surfaces X : B - R', N > 2. At the end of this section we shall sketch analogous results for minimal surfaces X : B -+ . t in an n-dimensional Riemannian manifold ./1. The main theorem of this section is the following result.

Theorem 1. Consider a minimal surface X : B l3 of class C°(B u y, R3) n C2(B, R3) which maps an open subarc y of aB onto an open Jordan arc T of ll which is a regular curve of class for some integer m >_ 1 and some u e (0, 1). Then X is of class Cm, "(B u y, R3). Moreover, if F is a regular real analytic Jordan arc, then X can be extended as a minimal surface across T.

In fact, we shall only prove a slightly weaker result. We want to show that the statement of the theorem holds under the assumption T e C" " with m > 2 and 0 < µ < 1. It remains to verify that the assumption r e implies x e C1'"(B u y, 1183). This can be carried out by employing the method of Section 7.8 where the behaviour of minimal surfaces at free boundaries is investigated. Other methods to prove this initial step can be found in Nitsche [16, 20] and [28] (see

Kapitel V, 2.1), Kinderlehrer [1], and Warschawski [5]. It will turn out that the method to be described also covers the boundary behaviour of surfaces of prescribed mean curvature at a smooth arc. Thus we shall deal with this more general result. Theorem 2. Let X e C°(B u y, R3) n C2(B, 1l83) be a solution of the equations

AX = 2H(X)X A X

(1)

IXu12=1X12,

(2)

<X.,X>=0

in B which maps an open subarc y = {e`°: 01 < 0 < 02} of aB into some closed regular Jordan arc r of R3, i.e., X (w) e r for all w e y. Then the following holds :

(i) If .*'(w) := H(X(w)) is of class L,,(B), and if r e C2, then we obtain that X e C1."(B u y, 183) for any /c a (0, 1).

(ii) If H is of class class C2 "(Buy, If83).

C C2.", 0 < µ < 1, then X(w) is of

7. The Boundary Regularity of Minimal Surfaces

34

Proof. (i) It suffices to show that for any wo e y there is some 6 > 0 such that X E C' "(Sb(wo ), R'), 0 < p < 1, provided that .mo(w):= H(X (w)) is of class L,,(B) and that T E C2. Here SS(wo) denotes as usual the two-gon B n B6(wo)

Thus we fix some wo e y. Without loss of generality we may assume that X(wo) = 0. For sufficiently small p > 0 we can represent Fn .7Y-p(O) in the form x2 = 92(t), x3 = t, Itl < 2to, x' = 91(t), where the functions g'(t) and g2(t) are of class C2, and by a suitable motion in (3)

383 we can arrange that 9k(0) = 0,

(4)

k = 1, 2,

9k(0) = 0,

choosing the parameter t appropriately. We may also assume that wo = 1 and that y = {w E 8B : I w - 11 < Ro } for some Ro E (0, 1). Choosing to > 0 and R E (0, Ro] sufficiently small we can achieve that (5)

I91(t)I2 + I92(t)I2 0. In addition, we have (9)

xW

= YW + 9k(x3)xw,

k=1,2,

and therefore (10)

IxWI2 + 1412 < 2IYwI2 + 2Iy2

2

+ 2IxwI2 L i9k(x3)I2 k=1

< 2

since (5) and (6) imply k=1

I9k(x3)I2 < s

Now we write the conformality relations (2) as (11)

0 = <Xw, X.> = (4W)2 + (x,2v)2 + (x42

7 3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

35

whence 1412 < IxwI2 + Ixw12

therefore 2IXwi2 G 1412 + 1412.

(12) (12)

From (10) and (11) we infer 4IXwI2 < 21 Y.I2

whence

IVXI2 < 8IVYI2.

(13)

From (8*) and (13) we derive the differential inequality (14)

I AYI < 8aJVYI2

on SR(1),

and we know already that

Y=O on

(15)

aBnOSR(l).

R2, and we obtain

Thus we can apply Theorem 2 of Section 7.2 to Y: SR(1) Y c C1'"(S(1), Q82) for any a E (0, R) and any U E (0, 1).

Combining (9) and (11) it follows that 2

2

0=

(16)

2

k=1

(Y)2 +2 k=1

9k(x3)Ywxw + { 1 + Y, I9k(x3)I2 j(xw)2 . k=1 )

If we introduce Pk(t) :=

(17)

2

9k(t) q(t)

4(t) := I + I I9k(t)I2 k=1

this relation can be rewritten as 2

2

3

Ixw + ki

(18)

=1

k

Pk(x3)Y]2 = Y Pk(x3)yw

Q(x3) k=1

(Yw)2.

As the right-hand side of (18) is continuous in SE(1), it follows that [...] and therefore also x,3 are continuous. Thus we arrive at X e C1(Se(l), R3) for any a e (0, R).

Multiplying (18) by -w2, we obtain (19)

[iwx,3y +

Y_

k=1

pk(x3)iwyw]2 =

13

I (wyw)2 - I{

Q(x ) k=1

Pk(x3)Wyk Y_

k=1

2

W

)}

Introducing polar coordinates r, (p with w = ret', we find that wxw = Z(rx; - ixV), (1= 1, 2,3)

wyk,=i(ry:-iy,

)

(k = 1, 2).

For W E y' := aB n aSE(1), we infer from (15) that the right-hand side of (19) is

36

7. The Boundary Regularity of Minimal Surfaces

equal to z

z" lq(x3)I-2

4

l2

k 9(x3)JYr - U 9k(x3)Yr

and this expression is real and nonnegative on account of (5) and (6). The left-hand side of (19) is of the form

(a + ib)2 = (a2 - b2) + 2iab with z

2

a := 2I x' + I

21 xr + Y Pk(x3)Y

b

Pk(x3)Y'

,

From the relations

a2 - b2 2- 0 and ab = 0 we infer that b = 0, that is, 2

Pk(x3)Y: = 0 on y'. ; + k=1

(20)

X3

Thus we have found: (21)

Idx31 + IVx31 < const

on Se(1),

arx3 E C°""(y')

Now we choose a cut-off function ri E Q° which is rotationally symmetric with respect to the pole w = 1 and satisfies 7l(w) = 1 for w e Ba(1), 6 :=

2e,

1(w) = 0 for

1w-11 >4s. Set W E Se(1).

Y(w) := Yj(W)x3(W),

(22)

We have yr = 11x, + 777x3, and therefore Yr E C°,'`(Y')

(23)

From the identity Ay = 774X3 + x3471 + 2071' 0x3

and from (21) we infer that

14y1 < const

(24)

on Se(1).

Finally we have (25)

y(w)=0 forallweSe(1)with

4e = <Xu, f'>

=<X.,X.+iX.>=1Xu12-r<Xu,X0>-IXu12 =

(u) = flu, x3(u)),

and (32) is verified. Thus C(u) = x3(u) is a solution of the integral equation (35)

F(u, C(u,)du.

C(u) = fo

It can easily be shown that there is some constant M > 0 such that

IF(w,C)-F(w,(')I < -MIC - C'I

40

7. The Boundary Regularity of Minimal Surfaces

e BRo. Then it follows from a standard fixed point holds for all w e SR and argument that there is a number S e (0, R) such that the integral equation (36)

z(w) = f

F(_w, z(M))dw,

w e Sb,

0o

has exactly one solution z(w), w e Sa, in the Banach space 4(Sa) of functions z: Sa -+ C which are holomorphic in S,, and continuous on S. (As usual, the proof of this fact can easily be carried out by Picard's iteration method.') Similarly one sees that the real integral equation (35) has (for u e la) exactly one solution 1;(u), u e IIi whence we obtain 4(u) = x3(u) and 1'(u) = z(u) for Jul < S, that is, (37)

z(u) = x3(u) for u e la.

Consequently z(w) is real-valued on la, and by Schwarz's reflection principle we can extend z(w) to a holomorphic function B8. Now we consider the mapping 0: Sa -+ C3, defined by (38)

O(w) = f(w) - g(z(w')),

which is continuous on Ss, holomorphic in Sa, and purely imaginary on Ia, since we have (39)

0(u) = f(u) - g(z(u)) = X(u) + iX*(u) - X(u) = iX*(u)

on account of (33). Applying the reflection principle once again, we can extend q(w) to a holomorphic function on B0, and therefore also (40)

f(w) = 0(w) + g(z(w))

is extended to a holomorphic mapping on B8.

We conclude this section by sketching the proof of a generalization of Theorem 1, employing the method of the proof of Theorem 2.

Theorem 4. Let .# be a Riemannian manifold of class C2, and let F be a regular Jordan arc in .' which is of class Cz. Moreover let X E C2(B, .,#), B = {w e C : I w I < l} be a minimal surface in . Finally we assume that y is an open subarc of 8B such that X e C°(B u y, A&) and that X(y) c F. Then we have:

(i) XeCi'"(Buy,A')forany It e(0, 1). (ii) If f and Fare of class m > 2, 0 < p < 1, then X e C`."(Buy, .11). (iii) If .# and rare real analytic, then X is real analytic in B u y and can be extended as a minimal surface across y. Proof. We shall sketch a proof of (i). The results of (ii) can be derived from (i) by employing a bootstrap reasoning together with potential-theoretic estimates, as

described in the proof of Theorem 2 and in Remark 1. The proof of (iii) now 3 The integral in (36) is a complex line integral independent of the path from 0 to w within S8.

7 3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

41

follows from a general theorem by Morrey [8] (cf Theorem 6.8.2, pp. 278-279). The reasoning of H. Lewy described in the proof of Theorem 3 can apparently not be carried over to the present case of a minimal surface in a real analytic

Riemannian manifold. We refer the reader to Hildebrandt [3], p. 80, for an indication how Morrey's result can be used to prove (iii). Let us now turn to step (i). We fix some point wo e T. Then there is some R > 0 such that X maps SR(wo) := B n BR(wo) into some coordinate patch on the manifold . # since X is continuous on B u y. Introducing local coordinates (x', x2, ..., x") on this patch, we can represent X in the form X(w) = (x1(w), x2(w), ..., x"(w))

for w E SR(wo)

with X E C0(9R(wo) u y, R') n C'(SR(wo), R").

Suppose that the line element ds of

on the patch is given by

ds2 = gkl(x)dxkdx',

(41)

where repeated Latin indices are to be summed from 1 to n, and let (42)

Tk = 2grJ(gjr,k + grk,j - gjk,r)

be the Christoffel symbols corresponding to gkl, where (g'') = (gjk)-t. Then we have the equations

dx' + Ijk(X) {x,',xu + xjx' } = 0,

(43)

1 < 1 < n,

and (44)

gk1(X)xuxV = O.

gkl(X)xuxa = gkl(X)xvxv,

(Equations (43) replace the equations 4x' = 0 holding in the Euclidean case, and equations (44) are the Riemannian substitute of the conformality relations (2).)

Without loss of generality we may assume that wo = 1, and we set SR := SR(I), 0 < R < 1, and y' = 8B n OSR. We can also assume that the coordinate patch containing X(SR) is described by {x e R': xj < 1} and that X(1) = 0. Furthermore, we can assume that Tin { I x I < 11 is described by x' = x2 xn-1 = 0, and that gkt E C1, g,,(0) = bkl. Thus we have IX(w)I < 1

for w c- SR

and

and w E y' .

x'(w) = 0 for a = 1, ..., n - 1 We write (44) as k

wESR,

I

gkl(X)xwxw=O,

which can be transformed into (45)

an(X) xw2

g(X) / - gnn(X) (xn+g_(g)

xx

2

g(X) - Z(X)

7. The Boundary Regularity of Minimal Surfaces

42

(summation with respect to repeated Greek indices

is supposed to run from 1 to

n - 1).

The definiteness of the matrix (gk,) implies ml < g,,,,(x)

for Jxl < 1

and Igk,(x)I < m2

obtain from (45) that where m, and m2 denote two positive constants. Then we there is some constant m3 > 0 such that n-1

Ivxni2
0 - \gannnWxw

on y" := y' n 0Se (cf. the computations leading to (20)). Hence we have (47)1

X

_

ga,(O, ..., 0, x"(e'')) gn,(0,..., 0, x"(e`'))

xa(e`m) r

on y".

Setting

p = -rki(X){xkxt, + xkxt,},

(48)

it follows that (49)

dx"=p in Se,

x,=f ony",

where x" is of class C' on Se, of class C2 on Se, p e L.(Se), f e C°-µ(y"). Then potential-theoretic reasoning yields x" E C""(Se) for 0 < E < a and therefore XE

Alternating between (49) and (50)

d Y = Q in Se,

Y=0 ony" ,

7.4 The Boundary Behaviour of Minimal Surfaces at Their Free Boundary

43

where

Q':= -raX)ljXukxI kl1 u +

XkX1) u

v

we obtain higher regularity of X at the boundary part T. This completes the sketch of the proof.

7.4 The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs The boundary behaviour of minimal surfaces with free boundaries is somewhat more difficult to treat than that of solutions of Plateau's problem. In fact, Courant [9, 15] has exhibited a number of examples indicating that the trace of a minimal

surface with a free boundary on a continuous support surface S need not be continuous. One of his examples even shows that the trace curve can be unbounded although S is smooth (but not compact). Unfortunately Courant's examples are not rigorous as their construction is based on a heuristic principle, the bridge theorem, which has not yet been established for solutions of free boundary problems, and therefore we shall describe Courant's idea only in the Scholia. However, one of Courant's constructions is not based on the bridge theorem and has recently been made perfectly rigorous by Cheung [1].

F_

E_

Fig. 1. A noncompact, Lipschitz continuous, nonclosed supporting surface S which satisfies no chord-arc condition. The configuration bounds an unbounded minimal surface of the type of the disk.

44

7. The Boundary Regularity of Minimal Surfaces

We consider here a modification of Cheung's example. The supporting surface S (see Fig. 1) in our example will be defined as follows. Let us define sets B,, B2, C, E+, G and curves y,,, /3+ by Bt B2

x=0,-1 1 and S > 0, if it is closed and if any two points P and Q of S whose distance IP - QJ is less than or equal to S can be connected in S by a rectifiable arc F* whose length L(F*) satisfies

L(F*) < MI P - QI. For example, every compact regular C1-surface S without boundary satisfies a chord-arc condition, and the same holds true if the boundary 8S is nonempty but smooth.

Let us first deal with the semifree problem. We now denote by B the parameter domain

B={w=u+ iv: JwJ0} the boundary of which consists of the circular are

C={w=u+iv: JwJ=1,v>-0} and of the interval

I={uel:Iul consisting of a closed set Sin D satisfying a chord-arc condition and of a Jordan curve Fin l3 whose endpoints P1 and P2 lie on S, Pt 0 P2. As in Section 4.6 we define the class of admissible surfaces for the semifree problem as the set le(T, S) of mappings X e HZ(B, 1183) satisfying

(i) X(w)eS .'1-a.e.onI; (ii) X : C -+ ]-is a continuous, weakly monotonic mapping of C onto F such

that X(1)=PI,X(-1)=P2. Let us also introduce the sets

Zd:={weB:Jwld}, (0 0.

Proof. Let X be a minimizer of the Dirichlet integral in e(T, S). Then X is harmonic in B, satisfies the conformality relations IXul = IXvI,

(4)

0,

<X,,,

and

D(X) = e. For any point wo e B we define IVXI2dudv.

O(r, wo) :=

(5)

We begin by proving that for any d e (0, 1) and for any wo c- I with I wo I < 1 - d the inequality O(r, wo) = 0,

hence (9)

O(r) = 2 fo p-t fo I X9(p, 0)12dOdp.

There is a set A" c [0, d] of 1-dimensional measure zero such that (10)

I X0 (r, 0) 12 d9 < oo

for r e (0, d) - N

Jo,

and that the absolutely continuous function O(r) is differentiable at r e (0, d) - .N' and satisfies (11)

W'(r) = 2r-1

o

fo

I Xe(r, 0)I2 de.

7.5 Holder Continuity for Minima

51

We can therefore assume that, for r e (0, d) - X, the function X (r, 0) is an absolutely continuous function of 0 E [0, rc]; in particular, the limits

Q1(r) := lim X(r, 0),

Q2(r) := lim X(r, 0) 0-+o

exist for r e (0, d) - K.

Consider now any r E (0, d) - ,' for which JnIX0(r,0)IzdO 0. Then X is of class P') for some it n (0, 1).

Fixing a third point P3 e T and requiring X(i) = P3 we can even derive an a priori estimate for [X]",e analogous to (3). In particular, the chord-arc condition for S u T implies the Holder continuity of any minimizer X in the corners w = ± 1 which are mapped by X on the points Pt and P2 where the arc T is attached to S.

If we consider minimal surfaces bounded by a preassigned closed Jordan curve F of finite length, we can even drop the minimizing property of X since we then can avoid the detour via the comparison surface Z = Y o r obtained from X and H. Instead we derive an inequality of the type (21) directly by applying the isoperimetric inequality to the part X 1 of the minimal surfaces. Leaving a detailed discussion to the reader we just formulate the final result:

Theorem 3. Let F be a closed rectifiable Jordan arc in l' of the length L(F) satisfying a chord-arc condition with constants M and S. Denote by a(T) a family of minimal surfaces Y e '(T) bounded by F (see Section 4.2 for the definition) which maps three fixed points on C = 3B onto three fixed points on T. Then there exists a number R > 0 such that for all X e j5(F) we have (29)

J

r

I VX l2 dude < (r/R)2"D(X) for all r > 0

S (wo)

with the exponent y = (1 + M)-2, and (30)

[X]o+",s

7r-1#2.

B, (r)

Then we obtain the trivial inequality 02(r)

(37)

IX9(r, 0)I2 d0 = iti-2'D(X)rO'(r)

O(r) < 2TL/3-2D(X) B,(r) 02(r)

I X8(r, 0)12 dO < 7r-'#'.

Case 2. B, (r)

Then for any two points P := X(r, 0) and P' := X(r, 0') on X(Cr(wo)) we have lz IXe(r,0)I2d0} IP, - PI 2, (or of class C',

with 0 < #< 1) is a two-dimensional manifold of class C' (or of class C','9) embedded in 683, with or without boundary, which has the following two properties: (i) The boundary 8S of the manifold S is a regular one-dimensional submanifold of class C"' (or C"'.") which can be empty. (ii) Assumption (B) is fulfilled.

Assumption (B), a uniformity condition at infinity, is defined next. We write x = (xt, x2, x3), y = (Yt, y2, Y3), ... for points x, y, ... in R'. Definition 2. A support surface S is said to fulfil Assumption (B) if the following

holds true: For each xo e S there exist a neighbourhood all of xo in R3 and a C2-diffeomorphism h of R3 onto itself such that It and its inverse g = h-t satisfy: (i) The inverse g maps all onto some open ball BR(O) = {y e 683 : yI < R} such

that g(xo) = 0; 0 < R < 1. (ii) If 3S is empty, then

g(Snall)_ {yeBR(0): y3 =0}. If OS is nonvoid, then there exists some number a = a(xo) e [-1, 0] such that

7. The Boundary Regularity of Minimal Surfaces

62

g(SnO1l)={yEBR(0): y3=0,yl>c} g(3Snah)= {yeB.(0): y3 = 0, y' =v} holds true. If xo e 0S, then o = 0, and o < -R if aS n a1I is empty. (iii) There are numbers m i and m2 with 0 < ml < m2 such that the components gik(Y) = hy,(Y)hyk(Y)

of the fundamental tensor of R3 with respect to the curvilinear coordinates y satisfies mt1b12 Q for all w e 1r(wo).

For any Z = (zt, z2, z3) E H2 '(Q, 93), £ c C, we define the transformed Dirichlet integral (or: energy functional) E0(Z) by 1

E0(Z) := 2

(3)

zVzV] dudv

and we set E(Z) := EB(Z).

(3')

We note that E0(Z) = Dn(h o Z) for all Z E H2(Q, R3),

(4)

whence, by (1'), (5)

For every

EQ(Y) = D0(X),

E(Y) = D(X)-

_ (91, cp2, (p3) a HZ n L,.(B, I3) and for XE := h(Y + so) we have lim {E(Y + so) - E(Y)} = lim {D(Xe) - D(X)). -+0 E e-+0 E

The left-hand side is equal to the first variation 6E(Y, 0) of E at Yin direction of 0, and a straightforward computation yields (6)

+ yco}dudv

6E(Y, 0) = SB

+ Jgik,z(Yyyuk + y'yk}cp'dudv while the right-hand side tends to (7)

V 70)dudv,

6D(X, W0) =

s

fB

Po := hv(Y)¢

7. The Boundary Regularity of Minimal Surfaces

64

because of

X = h(Y), with

q'0:=

8-0

Xe-X} =hy(Y)0.

Thus we have 6E(Y, 0) = 6D(X, WO).

(8)

Now we can reformulate the conditions which define stationary points X of the Dirichlet integral in terms of the transformed surfaces Y = g(X). Recall Definition 2 in Section 5.4:

If X is a stationary point of Dirichlet's integral in t(F, S) and if Xe = X+

E) is an outer variation (type II) of X with Xe E le(1, S) for 0 < e, < E0,

we have

lint {D(Xe) - D(X)} _ I 0 e +0 e B

for V. := Vf(-, 0), and this is equivalent to SE(Y, 0) >- 0.

(9)

This holds true in particular for every 0 E CC(B, R3) and thus we have both

8E(Y,0)>:0 and 6E(Y, -¢)>0 whence

5E(Y,¢)=0 forallOECI(B,683).

(10)

A partial integration yields

-J

+YVi (P., } dude B

=

J

Cgi1(Y)4Y`rP' + gii,x(Y)(YUYu + Y'Y')p'] dude

for any 0 E CC(B, 683), and we infer from (6) and (10) that (11)

[gu(Y)dY` + {gu,x(Y) - igix,i(Y)}(yiy. + y y )]ggldudv = 0 .(,B

for all

e Q' (B, 683).

Then the fundamental lemma of the calculus of variations yields (12)

gu(Y)zY` +

{9ii,x(Y)

-

Yvyv) = 0.

7.6 Holder Continuity for Stationary Surfaces

65

Introducing the Christoffel symbols of the first kind, rtk = 2191k,i - 9ik,t + 9il,k}

we can rewrite (12) in the form 91k(Y)dY` + litk(Y)(YUiYu + Y,Yk) = 0

(13)

using the symmetry relation I;tk = T'kti, and this implies (14)

4Y' + I;k(Y)(YUYU + Yvyv) = 0,

1 = 1, 2, 3,

if, as usual, rk = 9'j.k and (g`'") = (g'. As one can reverse the previous computations, we have found: The equation AX = 0 is equivalent to the system (14).

Moreover, we infer by a straight-forward computation from (1') and from 9ik-h`h` - y+ yk. The conformality relations

IX.IZ = IX"IZ,

<X., X,> = 0

are equivalent to 9,k(Y)YuYu = 9jk(Y)YvYv,

(15)

9;k(Y)Y',Yv = 0.

The advantage of the new coordinate representation Y(w) over the old representation X(w) is that we have transformed the nonlinear boundary condition X(I) c S into linear conditions as described in (2'). We pay, however, by having to replace the linear Euler equation AX = 0 by the nonlinear system (14). The variational inequality (9) will be the key to all regularity results. Together with the conformality relations (15) it expresses the fact that X = h o Y is a stationary point of the Dirichlet integral in the class '(T', S). (Here r can even be empty if X is a stationary point for a completely free boundary configuration; however, to have a clear-cut situation, we restrict our attention to partially free problems.)

The two main steps of this section are: (i) First we prove continuity in B u I, that is, up to the free boundary 1, using an indirect reasoning. The corresponding result will be formulated as Theorem 1. (ii) In the second step we establish Holder continuity on B u I employing the hole-filling technique. The corresponding result is stated as Theorem 2. Let us begin with the first step by formulating Theorem 1. Let S be an admissible support surface of class C2, and suppose that X(w) is a stationary point of Dirichlet's integral in the class 16(17, S). Then X(w) is continuous on B u 1.

The proof of this result will be based on four lemmata which we are now going to discuss.

7. The Boundary Regularity of Minimal Surfaces

66

Lemma 1. Let X : B -+ ll be a minimal surface. For any point w* e B we introduce x* := X(w*) and the set

K,,(x*):= {weB:IX(w)-x*I 2. lim sup 12 J P-++o nP OnK,(x*)

Proof. Fix some w* e B and some Q in B with w* e 0. We can assume that x* = X(w*) = 0. Then we introduce the set

ali1:={w:w=w*+teie,t>_0,0e ,IX(w*+re`B)I = 0

,

we have

{...}

21VXI21X-x*I2,

and the factor 1/2 will be essential for the following reasoning. It follows that

IVX12,.(P - IX -x*1)dudv

(18) 1

2

A'(p- IX -x*I)IVXI21X-x*Idudv t s,

0

Fig. 3. The graph of A(t).

7.6 Holder Continuity for Stationary Surfaces

71

where e denotes some positive number (in other words, we consider a family {, } of cut-off functions AE(t) with the parameter e). Then we obtain 1

SOXI2dudv 0.

Moreover, the reader readily checks that I V V(E, .)IL2(B) 0 such that

2'°-' < r/R < 2-`'. Since 6 = 2

and co(r) is nondecreasing, we see that c0(r) 1,0 0 such that fs'(WO) IVYl2dudv S f (36)

2IDYI2dudv

S2.(wo)

0. Combining (38) in a suitable way with interior estimates for X, we arrive at

(28). We can omit this reasoning since it would be a mere repetition of the arguments used in the second part of the proof of Theorem 1 in Section 7.5. Finally, Morrey's Dirichlet growth theorem yields X e Co8'(B v 1, R3). Thus we have proved Theorem 2 in the case that OS is empty.

82

7. The Boundary Regularity of Minimal Surfaces

The general case where aS is not necessarily empty can be settled by a slight modification of our previous reasoning. First we note that the test function

Wk=(yk - wkK 2, k=2,3 is admissible in (34), where co' and are chosen as before. Then we obtain an 0=(0,Q22,(P3),

inequality which coincides with (35) except for the term ml

J.

VYI2 2dudv,

which is to be replaced by mt SB ( I22

+ Vy32)2dudv.

However, this expression can be estimated from below by

Y22dudv 1 + K* SB V since there is a constant K* > 0 such that Ivy' I2 < K*(IVy212 + IVy312)

holds true, and this inequality is an immediate consequence of the conformality relations (15), written in the complex form

>=0, where we have set _ 1, m > 1, do e (0, 1), d e (0, do ), I t < do - d. Then Y, A, Y e HZ (Zd, IF'"), and

I

JI

JZd

t-O

Zap

The operators 0 and d, commute; more precisely, (d,V Y) (w) = (V4, Y)(w) for w E Zd ,

and similarly (DY),(w) = (V Y)(w)

for w E Zd.

Moreover, we have the product rule

dr(tPY)=(A,9)Y+cpd,Y=(A,q,)Y+cp,d,Y on Zd for scalar functions (p, and

lB

(pd_,tydudv = - f

for 0 < tj 0. In fact, we have 0 E HZ n L., (B, R'), and

Y(w) + eo(w) = Y(w) + eA_,{rl2d,Y}(w)

= 2 Y(w) +'2Y_,(w) + (1 - Al - 22)Y(w) where 22 := et-21]?r(w),

0 < Itl 2r+ItI,weB. Therefore we have Y(w) + EO(w) = Y(w)

for I w - wo l >_ 2r + I t 1.

On the other hand, if I w - woI < 2r + I t I, W E B, then we have

Iw±t-wol 0 such that I dr9Jk(Y) + df9;k,1(Y) < K4 , YI .

(6)

On account of (3)-(6), there is a number c = c(m2, K, K*) independent of t such that mt J rI2I0diYl2dudv B

< c{ f

(IOd,YI + (DY I4,YI)dudv

B

+ f"?72 IOdtYJ IDYj Id, Yldudv

+

Jn24tYAtYIvYI2

+ IodtYIIVYj + IV4YIIVYI)dud}

By means of the elementary inequality 2ab < Ea2 + b2

88

7. The Boundary Regularity of Minimal Surfaces

for any e > 0, we obtain the estimate 11 21VA,Yl2dudv SB

c* [r-2 J

eJ

e

B

+

IA,Yl2dudv S2r(WO)

2IAYI2VY12dudv+ SB ri2IdYI2DY2dudv] SB

Choosing e:= m1/2, we can absorb the first integral on the right-hand side by the positive term on the left-hand side, and secondly, we have

<J IVyl2dudv

Id,Yl2dudv <J

J

B

S2.(wo)

B

< ml1

IIVYII2dudv = ml

IVX12dudv = 2m'D(X). fB

SB

Thus we arrive at (7)

r S2 (wo)

f

< c**Lr-2D(X) +

I

r121d,YI2(IVyI2 + IVYI2)dudvJ.

S2.(wo)

Moreover, we claim that the estimate (28) in Theorem 2 of Section 7.6 implies the existence of some number co independent of r and t such that v2Id'YI2(IVYI2

+ IVYI2)dudv

SS2(w0) (8)

< cor2a

ff,

n2I0d'Yl2 dude + r-2D(X) } . J

2r (w0)

Let us defer the proof of the inequality (8) until we have finished the derivation of the L2-estimates of V2 X. Then we can proceed as follows: We choose r e (0, p) so small that c**cor2a < 1/2. Then we infer from (7) and (8) the existence of a number c1 independent of t such that 12I0d,YI2dudv : o-wl. Consequently, r Iq(w)I IC -

wl-2"dew c far

Jn

Iq(w)I IKo -

wl-2"dew,

and, by case 1, r Iq(w)I ICo -

wl-2vd2w

< c(a,

v)Qr2a-2"

Jn

Thus we have found that lq(w)l IC - WI-2vd2w < c(a, v)Qr2a`-2v

for all C E 0.

for

Consequently, (20)

fa

for

Iq(w)l Iw - CI-2"l0O(C)I2d2wd2 2. Then X is of class C"'-I,,(B u 1, R3) for any a e (0, 1). Moreover, if S is of class Cm,,s for some m > 2 and some /3 E (0, 1), then X is an element of u I, 683).

Proof. Recall that, according to Definition 1 in Section 5.4, a stationary minimal surface in W(r, S) is an element of c(r, S) n C'(B u I, 683) n C2(B, R3) which is harmonic in B, satisfies the conformality relations <Xu, Xv> = 0,

IXU12 = IXv12,

and intersects S perpendicularly along its free trace I given by the curve X : I --+ 683.

Pick some wo e I, and set xo := X(wo). Without loss of generality we can assume that xo = 0, and that for some cylinder (1)

C(R) := {(xt, x2, x3) :

Ixt 12

+ Ix212 0 such that S,(wo) is mapped by X into the cylinder C(R). Since X is perpendicular to S, the vectors X(w) are collinear for any w e 1,(wo) := I n B,(wo). Consequently we have X, = <Xe, n(X)>n(X)

on 1,(wo),

that is, x j = xkV n'`(X)n'(X) on I,(wo)

for j = 1, 2, 3.

104

7. The Boundary Regularity of Minimal Surfaces

If we set

K = 1, 2,

,K:=fK/w it follows that

- fL(x1, x2)xL } on 4(w0), K = 1, 2. (Indices K, L, M, ... run from 1 to 2; repeated indices K, L, M, ... are to be (5)

K(x1, x2) {xv

xK

summed from 1 to 2.) Let us introduce the function y3(w) by x3(w) - f(x1(w), x2(w)),

Y3(w)

(6)

w E Sr(wo).

Then we have the boundary condition "X (w) E S, w e I" transformed into Y3(w) = 0

(7)

for any w e 1,(wo),

and (5) can be written as

xK = - K(xl, x2)y.

(8)

on I4(wo)

for K = 1, 2.

Moreover, from (6) and AX = 0, we derive the equation 4Y3 = -fKL(x1, x2)DaxKDaxL

in Sr(wo).

Thus we have the two boundary value problems (*)

Ay' = -fKL(xl, x2)DaxKDaxL

in Sr(wo), y3 = 0

on I,.(wo)

with fKL := fxxxL, and (**)

AAK = 0

in S,(wo),

xK = -SK(x1, x2)Yv

on I,.(wo), K = 1, 2.

Now we are going to bootstrap our regularity information by jumping back and

forth from (*) to (**), assisted by the relation (6). To this end, we note that fECmorCm'S;fK, yKeC"-1 or C'" " ;fKLECm-2 or Cm-2,P ifSEC'"orC"`,9, b respectively.

We begin with the information X E C1(S,(wo), 183) assuming that S E C2. Then we infer from (*) that AY3 E L.(Sr(wo)),

y3 = 0 on I4(w0)

whence y3 E Cl,a(S(wo)) for any a c- (0, 1) and p e (0, r). In the following, we shall

always rename a number p with 0 < p < r in r; thus we actually obtain a sequence of decreasing numbers r. Now we can infer from (8) that xK E Co,a(1,(w0)), and it follows from (**) that xK E CI"'(S,(wo)), K = 1, 2. By virtue of (6), we have (9)

x3 = Y3 + J (xl, x2)

whence X E C1,°`(S,(w0), 183) for any a c- (0, 1).

Suppose now that S E C2,a holds for some l4 E (0, 1). Then we infer from (*)

that dy3 E Co,'(S,(wo)),

y3 = 0 on I4(wo),

7.8 Higher Regularity in Case of Support Surfaces with Empty Boundaries

105

whence Now it follows from (**) that xK E xK E C2,%,(wo)), K = 1, 2. Then we obtain from (9) that X E C2'1(S,(wo), p3). whence y3 e

Next we assume S E C3, whence 4y3 E C"(S,(wo)), and (*) yields y3 E CZ'a(S,(wo)) for all a e (0, 1). Now (**) implies xK E C1'a(1,(wo)), and therefore K X E C1'°(S,(wo)) for any a E (0, 1) whence X E C2'Q(S,(wo)), taking (9) into account. In this way we can proceed to prove the proposition.

Recall that any stationary point X of Dirichlet's integral in '(T, S) is a stationary minimal surface in ''(F, S), provided that X is of class C1(B u 1, R') (cf. Section 5.4, Theorem 1). Hence from Proposition 1 we obtain the following result, by taking also Theorem 4 of Section 7.7 into account : m >: 3, Theorem 1. Let S be an admissible support surface of class C' or Q E (0, 1). Then any stationary point of Dirichlet's integral in '(r, S) is of class Cm-',': (B u 1, R3) for any a e (0, 1) or of class u 1, l3) respectively.

Remark 1. The result of Theorem 4 in Section 7.7 is a by-product of the general discussion of that section, the main goal of which was to deal with surfaces S having a nonempty boundary. If 8S is void, we can use a different method that avoids both the derivation of L2-estimates and the use of the LP theory. This approach is more in the spirit of Section 7.3 and uses results which are closely related to those of Sections 7.1 and 7.2. To this end we choose cartesian coordinates x = (xt, x2, x3) in the neighbourhood of 0 = X(wo) E S in such a way that S is given by a nonparametric representation t(x1, x2) = (x1, x2, f(x1, x2))

Moreover, we introduce the distance function d(x) := dist(x, S)

and the foot a(x) of the perpendicular line from x onto S which has the direction n(x), I n(x) I = 1. Then, for all x in a sufficiently small neighbourhood of the origin 0, we have the representation x = a(x) + d(x)n(x).

(10)

If x e S, then clearly x = a(x) = t(x1, x2). Note that a(x), d(x), n(x) are of class Cni1 if S E C', i.e., their degree of differentiability will in general drop by one. Let now X be the stationary point that we want to consider, and let wo E 1, 0 < r - 2 thus

obtaining X E CS-2.2(B u I, R3). Since one has to assume S e CS+1 to keep this

method going, we essentially lose 2 derivatives passing from S to X. These derivatives can only be regained by potential-theoretic methods such as used in the beginning of this section. For details, we refer to Hildebrandt [3]. Analogously to Theorem 1, we obtain Theorem 1'. Let S be an admissible support surface of class C°` or C"fl, m >_ 3, P E (0, 1), and let B be the unit disk. Assume also that X : B -- 683 is a minimal surface of class C'(B u y, 683) which maps some open subarc y of aB into S, and which intersects S orthogonally along the trace curve X : y -> 683. Then X is of class Cm-','(B u y, 683) for any a e (0, 1), or of class C',O(B u y, 683) respectively.

Now we come to the second main result of this section. Theorem 2. Let S be a real analytic support surface. Then any stationary point of Dirichlet's integral in '(f, S) is real analytic in B u I and can be extended across I as a minimal surface.

Note that in Theorem 2 the parameter domain B is the semidisk {Im w > 0, w I < 1 } and I is the boundary interval {Im w = 0, 1 w I < 11.

7.8 Higher Regularity in Case of Support Surfaces with Empty Boundaries

107

Analogously we have

Theorem 2'. Let S be a real analytic support surface in R3, and let B be the unit disk. Assume also that X is a minimal s u r f a c e o f class C1(B u y, l 3) for some open

subarc y of 8B which is mapped by X into S, and suppose that X intersects S orthogonally along the trace curve X : y -- P. Then X is real analytic in B u y and can be extended across y as a minimal surface.

Since both results are proved in the same way, it is sufficient to give the

Proof of Theorem 2. By Proposition 1 we already know that X is of class C'O(B u 1, R3). Let X*(w) be the adjoint minimal surface to X(w) in B, and let f(w) := X (W) + iX*(w) = (f 1(w), f 2(w), f 3(w))

be the holomorphic curve in C3 with X = Re f and X* = Im f, satisfying < f '(w), f '(w)> = 0

on B.

We have to show that, for any uo e 1, there is some 6 > 0 such that f(w) can be extended across I6(uo) = 1 n B6(uo) as a holomorphic mapping from B5(uo) into C3. Without loss of generality we can assume that uo = 0. Set B6 := B6(0), 16 = 16(0) and Sa := B n B6. We can also achieve that f(0) = 0 holds true. Moreover, by a suitable choice of cartesian coordinates in l3, we can accomplish that S in a suitable neighbourhood °l1 of 0 is described by S n O& = {x = (x1, x2, x3) : x3 = ifr(xl, x2), Ixl I, x21 < R}

for some R > 0, where

0(0, 0) = 0,

/' i(0, 0) = 0,

l' 'x2(0, 0) = 0.

Then there is some So > 0 such that

Ix'(u)I < R,

for all u with IuJ < So.

Ix2(u)I < R

The vector fields TK(x) defined by T1 :_ (1, 0, 0x01

T2 := (0, 1, 1&x2)

are tangent to S. Moreover X, is orthogonal to X,,, and X. is tangent to S along I. As X,, is orthogonal to S along 1, we have 0

on 1bo

for K = 1, 2,

r

116

7. The Boundary Regularity of Minimal Surfaces

holds true, and we can apply Lemma 2 to

or

= y,' respectively,

obtaining

<Mr2 J

IV

2dudv.

rx.

Thus the function C(r), defined by (6), will satisfy

fi(r) < M f V2y1I2dudv for all r e (0, p), T. and we infer from formula (7) of Lemma 3 that

l

IV2YI2dudv}

IV2YI2dudv < c{rl+a + J

J

(

s.(w0)

sxr(wo) s.(wo)

)))

holds true for some S e (0, 1) and for all r c- (0, p). Adding the term cJ

IVYI2dudv

r S (wo)

to both sides of the last inequality, and dividing the result by 1 + c, it follows that

IV2yl2dudv + r"

IV2YI2dudv < BJ ( 52.(wo)

fs,(W())

holds true for some 6 E (0, 1) and for all r c- (0, p), where 0._

c

I +c that is, 0 < 0 < 1. Hence, by Lemma 6 of Section 7.6, we infer the existence of positive numbers k and a < 1 such that V2YI2dudv < kr2a for 0 < r < p,

(16) S,.(wo)

whence by IV2Xi2 < c{IV2YI2 + IQYI4}

and (10) we obtain IV2XI2dudv < k*r2a for 0 < r < p

and some constant k* depending on p but not on r. By virtue of Morrey's "Dirichlet growth theorem" we infer that X E C1'"(Zd, f183), for any d n (0, 1). F-I

7.10 Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points

117

7.10 Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences We have seen that a minimal surface X: B --> IE$3 can be extended analytically and as a minimal surface across those parts of 3B which are mapped by X into an analytic arc or which correspond to a free trace on an analytic support surface. At a branch point of such a part of 8B, the minimal surface X, therefore, possesses an asymptotic expansion as described in Section 3.2. In this section we want to derive an analogous expansion of X at boundary branch points, assuming merely that T or S are of some appropriate class C. Our main tool will be a technique developed by Hartman and Wintner that is described in chapter 8 in some detail. Presently we shall only sketch how the Hartman-Wintner technique can be used to obtain the desired expansions at boundary branch points.

Since in the preceding sections we have discussed stationary points of Dirichlet's integral in W(J, S), that is, stationary minimal surfaces with a partially free boundary on I, we shall begin by considering such a minimal surface X. Thus we can assume that we have the same situation as in Section 7.6:

S is assumed to be an admissible support surface of class C3; wo E I, xo := X(wo); {GII, g} is an admissible boundary coordinate system centered at xo, h = g-1 Y = (yl yz y3) := g(X), Y(wo) = 0; p > 0 is chosen in such a way that I Y(w) I < R for all w E S2p(wo); in addition, {JIi, g} is chosen in such a way that (32) of Section 7.7 holds true. We have yV2 = 0

and

y3 = 0 on I2p(wo)

and

dyt

+ jk(Y)DDY'D.y = 0 in B.

Moreover, on account of ggk(Y)YwYw = 0

in B,

it follows that (1)

IVy'12 < c{Ioy2I2 + IVy3j2}

and (2)

Idy2I + Idyll < c{IVy2j2 + IVy3I2}

holds in S2p(wo) for some constant c > 0.

In Section 7.7 we have also proved that y2 and y3 are both of class C"a(S2r(wo)) and of class Hp(S2r(wo)) for any a e (0, 1), p E (1, oo), and r e (0, p). Then the mapping Z(w) = (z'(w), z2(w), z3(w)) defined by z'(w) := 0 and by

if Im w > 0,

z2(w) := Y2(w),

z3(w) := y3(w)

z2(w) := Y2(W),

z3(w) :_ -y3() if IM w < 0,

118

7. The Boundary Regularity of Minimal Surfaces

f)83) and of class C2 in w = u + iv e BP(wo), w = u - iv, is of class BB(wo) - 1p(wo). Furthermore, for some constant c > 0, we have in BB(wo) - I (wo).

IZwwl < cIZWI

(3)

Let Q be an arbitrary subdomain of B,(wo) for some r e (0, p) which has a piecewise smooth boundary 8Q, and let 0 = (q 1, Cpl, (p3) be an arbitrary function

of class C'(Q, C3). Then, by an integration by parts, we obtain that 2i J an

(4)

dw = J n ( + ) d2w

where dw = du + i dv, d2 w = du A. Combining (3) and (4), we arrive at the inequality J

(5)

an

wo.

This implies that the surface normal N(w), given by

tends to a limit vector No as w -+ wo :

lim N(w) = No = Ia n fil_l(a n fi).

(9)

w-WO

Consequently, the Gauss map N(w) of a stationary minimal surface X(w) is well defined on all of B u I as a continuous mapping into S2. Therefore the surface X (w) has a well defined tangent plane at every boundary branch point on I, and thus at every point wo e B u I. Consider now the trace curve X: I -* 183 of the minimal surface X on the supporting surface S. We infer from (8) that

asw-wo,we1 and, writing w = u, wo = uo for w, wo e I, we obtain for the unit tangent vector t(u) := I

the expansion (10)

t(u) =

a (u - uo) " lallu-uol"+o(1)

asu

Therefore the nonoriented tangent moves continuously through any boundary branch point uo e 1. The oriented tangent t(u) is continuous if the order v of uo is even, but, for branch points of odd order, the direction of t(u) jumps by 180 degrees when u passes through uo. Finally, by choosing a suitable cartesian coordinate system in R', we obtain the expansion (11)

x(w) + iy(w) = (xo + iyo) + a(w - wo)"+i + o(Iw - woI"+i) Z(w) = zo + o(Iw - woI"+i)

as w --+ wo, where X(wo) = (xo, yo, zo) and a > 0; see Section 3.2, (6).

The same reasoning can be used for the investigation of X at a boundary branch point wo e int C. We obtain again an expansion of the kind (7) with some v >- 1 and some A E C3, A 0, = 0. As X : C -+ F is a monotonic mapping, the tangent vector t((p) := I Xw(e`4')I -1X,,(e`q )

7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces

121

of this mapping has to be continuous, and we infer from (7) that v is even, provided that F is of class C2.

The same result can be proved for minimal surfaces X E t(T) which solve Plateau's problem for a closed Jordan curve r' of class C2; cf. chapter 4 for the definition of ce (I'). Thus we obtain

Theorem 2. Let T be a closed Jordan curve of class C2 in R3, and suppose that X E''(T) is a minimal surface spanned in F. Then every boundary branch point wa E 8B is of even order v = 2p, p > 1, and we have the asymptotic expansion

Xw(w) = A(w - wo)2F + o(Iw - wol2p)

as w - wo,

where AnC3,A00,and=0. For details we refer to Chapter 8.

7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces In Section 1.4 we derived the Gauss-Bonnet formula (1)

x

r

for regular surfaces X e C2(6, R3) defined on a simply connected domain 0 c C which map 3Q onto a Jordan curve F. The result as well as the proof given in Section 1.4 remain correct if X does not map 00 bijectively onto a Jordan curve in R3 provided that we replace formula (1) by

I KdA+

(2)

Jx

ds=27r fox

or, precisely speaking, by (3)

A

as

f an

ic9IdXI =2n.

Now we shall drop the assumption of regularity and, instead, admit finitely

many branch points in the interior and on the boundary of the parameter domain 0. To make our assumptions precise, we introduce the class 99P(6) of pseudoregular surfaces X : Q --1, IR as follows:

A surface X is said to be of class YA(S) if it satisfies the conditions (i) X e C2(Q, R3) and (4)

IX41=1Xvi,

<XL, X.> = o.

7. The Boundary Regularity of Minimal Surfaces

122

(ii) There is a continuous function .*(w) on 0 such that

AX = 2 fX A X,.

(5)

(iii) There is a finite set E° of points in 0 such that XW(w) 0 0 for all

w e 0 - E°. For any point w° e E° there is an integer v > 1 and a vector A e C3 satisfying A 0 and = 0 such that A(w - w°)° + o(Iw - w°!")

(6)

as w-+W°.

We call 1° the singular set of X e 9_q(6). Remark 1. The set 0° := {w e Q : Xx,(w) 0} of regular points of X in Q is open e C°(0°) and, by Section 2.6, equations (4) yield the existence of a function such that (5) holds true on 0°. Moreover, the function .ye is the mean curvature of X I n.. Thus condition (ii) is a consequence of (i) if we assume that .*'(w) can be extended from 0° to 6 as a continuous function. This extension is possible if, for some reason, we know that X is a solution of

AX = 2H(X)X, A X

(7)

in 0, where H e C°(R3) If, on the other hand, X C C2 (0, R3) is a solution of (4) and (7) for some H E C' (tR3 ), it is sometimes possible to extend X to a function of class C2( 6, U83) .

For instance, the extendability can follow from suitable boundary conditions (e.g. from Plateau-type conditions or from free boundary conditions) as we have seen in the previous sections. Finally if X(w) is a nonconstant surface such that (4) and (5) hold for some

.f e

0 < a < 1, then the set of branch points of X defined by 1° := {w e Q : X,,(w) = 0} is finite (and possibly empty), and, for any w° e 1°, the mapping X has an asymptotic expansion (6) as described in (iii). For minimal surfaces we have stated this result in Section 7.10. The general theory will be developed in Chapter 8 using Hartman-Wintner's technique. Now we can formulate the Gauss-Bonnet theorem for pseudoregular surfaces; we shall immediately state it for multiply connected domains. Theorem 1. Let Q be an m -fold connected domain in C bounded by m closed regular curves yl, ..., ym of class C°°, and let X: Q --> l3 be a pseudoregular surface with the area element dA = I X. A Xt, I du dv, the singular set E°, the Gauss curvature K

in Q - E°, and the geodesic curvature K9 of X I an-z0. Suppose also that the total curvature integral f n IKI dA exists as a Cauchy principal value. Then we obtain the generalized Gauss-Bonnet formula

KdA = 2n(2 - m) + 27c E v(w) + it E v(w) - I

(8)

SQ

W ET,

Wea"

x9I dX I

Jon

where o' := 1° n Q is the set of interior branch points, a" := E° n 00 the set of boundary branch points, and v the order of a branch point w e E.

7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces

123

For the proof of (8) we shall employ the reasoning of Section 1.4. To carry out these arguments in our present context, we need two auxiliary results.

Lemma 1. Let a > 0, 1 = (0, a], and be f a function of class Ct(1) such that If(r) < m holds for all r c- I and some constant m >_ 0. Then there is a sequence of numbers rk e 1 satisfying r, -+ 0 and rk f'(rk) --> 0 as k --* CO.

Proof. Otherwise we could find two numbers c > 0 and e e (0, a] such that

rlf'(r)l >-c forallre(0,e]. Then we would either have

f'(r) >- c/r for all r e (0, e]

(i)

or

f'(r) < -c/r

(ii)

for all r e (0, s].

In case (i) we obtain rE

c

log- = c r

Jr I

dr r

E


= 0

(see Section 7.3), we infer from

hold

for

solutions

7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces

K< H2 < h2,

127

h:= sup H(X ) WED

that $ n 1 K I dA and f r x9 ds are finite, and that we have formula (19) as well as the estimate

2-m+

(21)

V(W): 4, p e (0, 1), bounds at least one minimal surface X

of class Cm'"(B, 683).

As a given boundary Tmay be spanned by many (and, possibly, by infinitely many) minimal surfaces, this regularity result by Hildebrandt [1] is considerably weaker than Theorem 1 of Section 7.3 whose global version can be formulated as follows: Let X e C°(B, 683) be a minimal surface, i.e.,

dX = 0,

IXXI2

= IX.I2, <Xv, X,,) = 0 in B,

which is bounded by some Jordan curve F of class Cm' with m > 1 and u e (0, 1). Then X is of class 683). Assuming that m >- 4, this result was first proved by Hildebrandt [1] in 1966. Some of the essential ideas of that paper are described in Step 1 of Section 7.7. Briefly thereafter, Heinz and Tomi [1] succeeded in establishing the result under the hypothesis m 3, and both Nitsche [16] and Kinderlehrer [1] provided the final result for m 1. Warschawski [6] verified that X has Dini-continuous first derivatives on B, if the first derivatives of F with respect to arc length are Dini continuous; cf also Lesley [1].

These results on the boundary behaviour of minimal surfaces hold for surfaces in 68", n > 2, and not only for n = 3; the proof requires no changes. For

130

7. The Boundary Regularity of Minimal Surfaces

n = 2 these results include classical theorems on the boundary behaviour of conformal mappings due to Painleve, Lichtenstein, Kellogg [2], and Warschawski [1-4]. (Concerning the older literature, we refer to Lichtenstein's article [1] in the Enzyklopadie der Mathematischen Wissenschaften; the most complete results can be found in the papers by Warschawski.) As Nitsche has described his technique to prove boundary regularity in great detail in his monograph [28], Section 2.1, in particular pp. 283-284 and 303-312

we refer the reader to this source or to the original papers by Nitsche and Kinderlehrer quoted before. Instead we presented a method by E. Heinz [15] which needs the slightly stronger hypothesis m > 2. By this method, Heinz could also treat H-surfaces, and Heinz-Hildebrandt [1] were able to handle minimal surfaces in Riemannian manifolds; cf. Section 7.3. The basic tools of Heinz's approach are the a priori estimates for vector-valued solutions X of differential inequalities 14X1 < aIVXI2

which we have derived and collected in Section 7.2. They follow from classical results of potential theory which we have briefly but (more or less) completely proved in Section 7.1. The results of Section 7.2 and, in part, of Section 7.1 are taken from Heinz [2, 5], and [15]. Closely related to this method is the approach of Heinz-Tomi [1] and the very useful regularity theorem of Tomi [1]. The first regularity theorem for surfaces of constant mean curvature was proved by Hildebrandt [4]; an essential improvement is due to Heinz [10]. The method of Heinz [15], described in the proof of Theorem 2 in Section 7.3, can be viewed as the optimal method. A very sharp result was obtained by Jager [3]. 5. The possibility to obtain asymptotic expansions of minimal surfaces and, more generally, of H-surfaces by means of the Hartman-Wintner technique was first realized by Heinz (oral communication). A first application appeared in the paper by Heinz and Tomi [1].

6. In Theorem 2' of Section 7.8 we proved that any minimal surface X, meeting a real-analytic support surface S perpendicularly, can be extended analytically across S. The proof basically follows ideas from H. Lewy's paper [4], published in 1951. There it was proved that any minimizing solution X of a free boundary problem can be continued analytically across the free boundary if S is assumed to be a compact real-analytic support surface. In fact, Lewy first had to

cut off a set of "hairs" from the minimizer by composing it with a suitable parameter transformation before he could apply his extension technique. (Later on it was proved by Jager [1] that the removal of these hairs is not needed since they do not exist.) 7. Combining Lewy's theorem with new a priori estimates, Hildebrandt [2]

proved that the Dirichlet integral possesses at least one minimizer in (J', S) which is smooth up to its free boundary provided that S is smooth and satisfies a suitable condition at infinity which enables one to prove that solutions do not

7.12 Scholia

131

escape to infinity. (A very clean condition guaranteeing this property was later formulated by Hildebrandt-Nitsche [4].) 8. The first stationary regularity theorem for minimal surfaces with a merely smooth but not analytic support surface S was proved by Jager [1]. He proved for instance that any minimizer X of Dirichlet's integral in IK(T, S) is of class C'-"(B u I, R3),1 being the free boundary of X, provided that S E and m >_ 3, µ e (0, 1). Part of Jager's method we have described or at least sketched in Section 7.8. We have not presented his main contribution, the proof of X E C°(B u 1, R'), which requires S to be of class C2. Instead, in Section 7.5, we have described a method to prove continuity of minimizers up to the free boundary that needs only a chord-arc condition for S. Because of the Courant-Cheung example, this result is the best possible one. The approach of Section 7.5 follows more or less the discussion in Hildebrandt [9]. The sufficiency of the chord-arc condition for proving continuity up to the free boundary was almost simultaneously discovered by Nitsche [22] and Goldhorn-Hildebrandt [1]. Later on, Nitsche [30] showed that Jager's regularity theorem remains valid if we relax the assumption m >_ 3 to m >_ 2 Moreover, if we assume S c- C2, then every minimizer in '(1, S) is of class C' (B u 1, R3) for all a e (0, 1). 9. The regularity of stationary surfaces in "1(1', S) up to their free boundaries

was - almost simultaneously - proved by Gruter-Hildebrandt-Nitsche [1] and by Dziuk [3]. Both papers are based on the fundamental thesis of Gruter [1] (see also [2]) where interior regularity of weak H-surfaces is proved. The basic idea of Griiter's paper consists in deriving monotonicity theorems similar to those introduced by DeGiorgi and Almgren in geometric measure theory. We have presented the method used in Gruter-Hildebrandt-Nitsche [1]; it has the advantage to be applicable to support surfaces with nonvoid boundary S. Moreover we do not have to assume that

lim dist(X(w), S) = 0 for any wo E I w-.wo

as in Dziuk [5]-[7]. On the other hand, Dziuk's method is somewhat simpler than the other one since it reduces the boundary question to an interior regularity

problem by applying Jager's reflection method. This interior problem can be dealt with by means of the methods introduced in Grater's thesis. 10. The results in Section 7.7 concerning the C', "'-regularity of stationary minimal surfaces with a support surface S having a nonempty boundary 8S are taken from Hildebrandt-Nitsche [1] and [2]. 11. The proof of Proposition I in Section 7.8 is more or less that of Jager [1], pp. 812-814. 12. The alternative method to attain the result of Step 2 in Section 7.7, given

in Section 7.9, was worked out by Ye [1], [4]. Ye's method is a quantitative version of the L2-estimates of Step 2 in Section 7.7 which is based on an idea due

to Kinderlehrer [6]. 13. Open questions: (i) The regularity results for stationary minimal surfaces X with a free boundary are not yet in their final form. In particular one should

132

7. The Boundary Regularity of Minimal Surfaces

prove that X is of class C"' up to the free boundary if S e C*'", and that X e Co"' for some a e (0, 1) if S satisfies a chord-arc condition (this is only known for and if minimizers of the Dirichlet integral). Here we say that S E C " if S E S satisfies a uniformity condition (B) at infinity (see Section 7.6). Dziuk [7] and Jost [8] proved that X is of class C1-,' up to the free boundary, 0 < is < 1, if S is of class C2 and satisfies a suitable uniformity condition. (ii) It would be desirable to derive a priori estimates for stationary minimal surfaces, in particular for those of higher topological type. As in general there are no estimates depending only on the geometric data of the boundary configuration (cf. the examples in Section 7.6), one could try to derive estimates depending also on certain important data of the surfaces X in consideration such as the area (= Dirichlet integral) or the length of the free trace.

Such estimates could be useful for approximation theorems, for results involving the deformation of the boundary configuration, for building a Morse theory, and for deriving index theorems. Note, however, that a priori estimates depending only on boundary data can be derived in certain favourable geometric situations, for instance if the support surface is only mildly curved. Results of this kind were found by Ye [2]. Let us quote a typical result: Suppose that S is an orientable and admissible support surface of class C3'", a e (0, 1), and let no be a constant unit vector and a be a positive number, such that the surface normal n(p) of S satisfies

>_a for allpES.

(2)

Then the length 1(1) of the free trace . ' of a stationary point X of Dirichlet's integral in '(1, S) without branch points on the free boundary I is estimated by the

length of r via the formula 1(1) < 1(f)/a,

(3)

and the isoperimetric inequality yields the upper bound

D(X) < 41(1 + a 2)l2(r)

(4)

for the Dirichlet integral of X.

Let us sketch the proof of (3), which is nothing but a simple variant of the reasoning used in Section 6.4. By means of Green's formula we obtain (5)

0=f a B

dudv = - ft X,,

+ fc

with w = u + iv = re`°, where B stands for the usual semidisk. Because of (2)

7.12 Scholia

133

and of

X = IX,In(X) on I (where we possibly have to replace n by - n),

IX,I=1X,1 onC=BB-1, on 1,

IX,,I= IX,I

we then obtain rI

ov1(E)=6J 1XuIdu=J QlXjdu I

( I X'vK 0, n e t\1, with X

D as follows:

I e,, < 1/4, and then we define sets A,,, B,,, C,,',

n=1

A,, :=5(x,Y, z): z=O,IxI 0 and >y < 0 on 852. Consider the convex set of comparison functions K,, := {v e H2 '(Q): : v in Q, v = 0 on 8Q} and a solution u e K,, of the variational problem D(u) := 2 J

I Vu(x)l2 dx -> min

in K, .

n

One readily verifies that a solution u e K,, satisfies the variational inequality (9)

ID

DjuD,(v - u)dx >_ 0 for all v n K,,.

Lewy and Stampacchia [1] used the method of penalization together with suitable a priori estimates to show that u is of class C1.", a < 1 (at least if t' is smooth and strictly concave). It is in fact true that u is of class H.', (Q); cf. Frehse [1], Gerhardt [1], and Brezis-Kinderlehrer [1]. The set 92 may now be divided into two subsets, "the coincidence set"

.5 = JI(u) = {x E 0: u(x) = fi(x)} and its complement

Q-5={xeQ:U(x)>c(x)}. Of particular importance is a careful analysis of the boundary 8_0 of the set of coincidence J. Such investigations were initiated by H. Lewy and G. Stampacchia [1] and continued by Kinderlehrer [7] and Caffarelli-Riviere [1]. It was proved that the free boundary 85 is: (i) An analytic Jordan curve if i/i is strictly concave and analytic; (ii) a C1 -Jordan curve, 0 < Q < a, if, is strictly concave and of class Cl,'; (iii) a Cm'1."-Jordan curve if t// is strictly concave and of class

Cm with m >2 and 0 < a < 1. The investigation of 8.0' is more difficult if we span a nonparametric surface as graph of a function u over some obstacle graph i// such that it minimizes area. In other words, if 0 is a strictly convex domain in fR2 with a smooth boundary, if c/i is given as above and K,, is the convex set of functions v e satisfying v t', we consider solutions of the variational problem

+jVuj2dx-min

in K,,,.

The existence of a solution u e K,, was proved by Lewy-Stampacchia [2] and by Giaquinta-Pepe [1]. Moreover, these authors showed that the solution u is of class HQ n C1'"(S2) for every q e [1, oo) and any a c- (0, 1). Thus the set of coincidence J = {X E 0: U(x) = t/i(x)}

140

7. The Boundary Regularity of Minimal Surfaces

is closed, and we have

ou

div

71

-0 inQ-J

as well as (10)

(l+IVu12)-I"2 0

and for all nonnegative µ e Z. We are thus in a position to repeat the argument given in the proof of Lemma 4. Inequality (21) with wo = 0 now reads as (27)

27rJ2(0) < 11(0) + 12(0) + 13(0),

where

11(0)=JJ S0, forall0 Inserting the relations IG(w)I = 0(1) and

IG(w)I = 0(Iw1µ-1),

µ 0}, i.e., XE

C°,a(B R3) n C2(B l3)

for some a > 0. Assuming the usual three-point condition, the points (1, 0) and (-1, 0) are mapped onto the corner points P1, P2 E T n S respectively. Hence our interest is concentrated on the behaviour of VX(w) when w --r + I respectively. We first mention a (local) result concerning the Plateau problem. Theorem 1. Let T+, F- c 683 be pieces of regular Jordan arcs of class meet at a point P e 683 forming a positive angle /3 < it. Suppose that

which

X E C°'a(Ba , R3) n CZ(Ba - {0}, J 3),

where Ba := {w = (u, v): Iwi < 6, v > 0} is a minimal surface which satisfies the boundary conditions X : la -+ Tt with 1,, := {(u, 0): 0 < ±u < S} and X (O) P. Then we obtain the asymptotic relation I VX(w) I = O(I w

la-1)

as w -+ 0.

For the free boundary problem we shall prove Theorem 2. Let T be a regular Jordan curve of class C2," which has only its two endpoints P1i P2 in common with a regular closed surface S of class C3. Suppose that X e W(T, S) solves the partially free minimum problem P(I', S) and that F, S satisfy some chord-arc condition. Then X(u, v) is of class C°'a(B, 683) n

{1, -11) for some a > 0 where B= ((u, v) : u2 + v2 < 1, v > 0), and /there holds the expansion (1)

IVX(w)I = o(Iw + l la-1) as w-i ± 1. We shall only prove Theorem 2 since the proof of the first theorem is similar.

Note that we only have to show the asymptotic relation (1) since the asserted regularity properties of X were already proved in Chapter 7. Also, it will be convenient to replace the semi-disk B by the upper halfplane

164

8. Singular Boundary Points of Minimal Surfaces

H={(u,v)EIEB2:v>0}. We may further assume that the point (u, v) = (0, 0) is mapped into the corner point Pl e Fn S. Observe that this simplification is without loss of generality since the conformal map

w=w(z)=

-rl - zlz LL

I + z

J

maps the semi-disk B = {(u, v) : u2 + v2 < 1, v > 0} conformally onto H, and the point (1, 0) into (0, 0). (Note that w(z) is not conformal at the boundary point z = 1.) Furthermore, if X is of class n C2(B), then Y(w) := X(z(w)) is and if Y satisfies an asymptotic relation of the type of class IVY(w)I =

0(Iwla12-1)

as w

0,

then also VX(z)I

0(IVY(w)I

I

dz

)= 0(11

-zIa-2.I 1 -zl)

= 0(11 -z"I°-) asz -+ 1,zeB. Since we only deal with local properties of X we may throughout this section require the following Assumption (A) to be satisfied by the minimal surface X.

Assumption A. Let S > 0 be some positive number and put

Ba :={w=(u,v)el}82: IwI 0}, 161- :={w=(u,0): 0 0 such that the estimate IVX(wo)I < e ' -const sup 19(w) - X(wo)I

(7)

B.(wo)

holds true for all woe D n Ba(0) and for all e > 0 with BB(wo) c D n B,(0). Proof.. We put Y(w)

1

= 2a

[9(W) - X (0)], w E D, and choose b > 0 so small that

sup I Y(w) I < 1. Then Y satisfies (3) on D n Ba(0) with a = 2. Applying Prop-

D nB,(o)

osition A to the function y C C2(BE(wo)) and to M = 1, a = Z we get the estimate IVY(wo)I B (wo)

sup IX(w) - X(wo)I B,(wo)

as required.

In order to state our results in a convenient way, we make the following Assumption B. For some fixed angle it >- y > 0 we denote by Dp the domain

DP:={w=re'9:0_ 0}

by

x(u, v) = Re(w°+2k),

w=U+iv6B

y(U, v) = Im(wa+2k)

z(u, v) - 0.

i Y

r-

r+

X

Fig. 1

Then X(u, v) is a minimal surface (i.e. AX = 0, <X, X,,> = 0) which maps the intervals I+ = (0, 1), 1- = (- 1, 0) onto the straight arcs

T+={(x,y,z)eR3: z=0,arg(x+iy)=0,0<x2+y2 0, and satisfies the boundary conditions X : 18 -+ Ft li), and X (O) = 0. Moreover, we assume that there exist functions h, h2 e e > 0, such that

r+

_ {(t, h (t), hz (t): t c- i,_'}

and

I'- = {(t, hi (t), hz (t): t e

and that furthermore

hf(0) = 0, hold true.

j = 1, 2, and hi'(0) _ ±cot

an 2

h? '(0) = 0

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

175

We note that Assumption A is quite natural and not restrictive since, by performing suitable translations and rotations, we can achieve that any pair of piecewise smooth boundary curves T+, F- will satisfy this assumption. Also, by the results of Chapter 7 any minimal surface bounded by T+, !-'- has the desired regularity properties. The main result of this section will be Theorem 1. Suppose that Assumption A holds. Then there exist Holder continuous complex valued functions 01 and -P2 defined on the closure of some semidisk Bs +, 6 > 0, such that the following assertions hold true: (1)

(2)

01(0)001

02(0) s 0,

012(0)

Y,(w) = W'02(w),

xw(w) = wlPl(w),

+' (0) = 0,

and

Izw(w)I = O(Iwlz),

where y = a - 1 + 2k or y = I - a + 2k for some k e C\l u {0} and A > y. Furthermore there exists some c e C - {0} such that wa+2k[c (3)

x(w) + iy(w) =

+ o(l)l

or

as w --* 0,

2-a+2k[c + 0(1)]

and

z(w)= O(Iwlx+') asw-*0. Finally, the unit normal . 4,(w) =

(X" A X,,) (W)

I(X A

lim 'V(w) =

(4)

W-0

tends to a limit as w-* 0:

0 +1

Remark. Theorem 1 extends without essential changes to conformal solutions X(w) of the system AX = f(X, OX), where the right-hand-side grows quadratically in I VX I. Also two-dimensional surfaces in R', N >- 3, can be treated. In the case of polygonal boundaries we can say more:

Theorem 2. Suppose that Assumption A holds where T+, T- are straight lines. Then there exist holomorphic functions H; and HH, j = 1, 2, 3, which are defined on a disk Bs for some 6 > 0, such that the following holds true: (5)

wHl (w) + 4H2(w)H3(w) = 0, 1

(6)

Xw(w) = wa-1H2(w)

-i 0

1

+ w-2H3(w)

i 0

176

8. Singular Boundary Points of Minimal Surfaces

x(w) + iy(w) = w°H2(w) + wt H3(w), z(w) = Re(wH1(w))

(7)

where w e Ba - {0}. Furthermore, (3) holds true as well.

The idea of the proof of Theorems 1 and 2 is to eventually apply Theorem 3 of Section 8.1 to a certain set of functions involving the gradient X. Here it is necessary and convenient to use first a reflection procedure followed by a smoothing argument. The new function of interest is then defined on a neighbourhood Ba - {0} of zero, and it turns out that Theorem 3 of Section 8.1 can be employed. A further essential ingredient is Theorem I of Section 8.2 which provides the "starting regularity" and thus makes our argument work. Proof of Theorem 1. Because of the continuity of X it is possible to select S > 0 so small that x(Ba) c [-a, a]; this will be assumed henceforth. By Assumption A we have on Ia the equality X(u, 0) = [x(u, 0), hi (x(u, 0)), h2 (x(u, 0))], whence

X (u, 0) = (1, hi'(x(u, 0)), h2'(x(u,

0).

The conformality conditions (which also hold on 1b) imply that 0), (1, hi'(x(u, 0)), hi'(x(u, 0)))> = 0

(8)

on 1, 1.

Now we put a±(t)

[1 + h;'(t)Z +

h2'(t)2]-tn(1, hi

'(t), hz '(t)),

t e [-e, a],

and consider the linear mappings S±(t)y

2a±(t)

-y

which are defined forte [-e, e] and y e R3. Then, using (8), we infer S±(x(u, 0))X (u, 0) = X (u, 0), S±(x(u,

0)

0) a I4±.

This may be rewritten as (9)

S±(x(w))XW(w) = Xw(w)

for all w e la .

Since S±(t), t e [-a, e], is a family of reflections, there exist orthogonal matrices Ot(t) such that

S±(t) = Ot(t) Diag[-1, -1, 1]Ot(t)` where we have used the notation

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

a0 Diag [a, /3, y] =

177

0

0

f

0

0

0

y

and A` denotes the transpose of the matrix A. Furthermore we define

T-(t) := 0f(0)Ot(t)` with

O+(0) = lim 0+(t)

0-(0) = lim 0-(t).

and

r-0-

1-01

Now put T(t) :_

JT+(t) if 0 < t < e

T-(t) if-e _

0 we easily conclude (14) taking the orthogonality of the matrices Tt into account. Relation (15) follows from the estimate G(w) I < c6 I VX I and from Theo-

rem I of Section 8.2. Finally, to prove (16), we calculate by means of (11) that lim G(u, v) = g(u, 0) = S-(0)4(u, 0) = S-(0)S+(0)S+(0)4F(u, 0) = S-(0)S+(0) lim G(u, v),

where we have used that S+(0)S+(0) = Id.

The function G(w) itself is not yet accessible to the methods which were developed at the end of Section 8.1, because of the "jump relation" (16). To overcome this difficulty, we have to smooth the function G which will be carried out in what follows. Recall that the jump of G at Ia is given by

cos 2na S-(0)S+(0) = I sin 2na

l I

0

- sin 2na

0

cos 2na

0

0

1

We can diagonalize S-(0)S+(0) using the unitary matrix

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

U=

0

1

1

0

-i

i

0

0

1

-,G

f2

179

and obtain S (0)S+(0) = U Diag[l, e2ni(a-1) e-2nia]U* where U* = U`. We define a new function F(w), w e Ba - i6-', by (17)

F(w) := Diag[1, w1-", w"] U*G(w)

or, more explicity,

F1(w)

F(w) =

F2(w)

F3(w) 2

We claim that F is continuous on the punctured disk Bb(0) - {0}, In fact, we infer from (16) the relation

lim F(u, v) = Diag[1, u1-"eu"ein"] U* lim G(u, v) = Diag[l, u1-"ein(1-") Vein"] U*S-(0)S+(0) lim G(u, v) = Diag[1

u1-aei(1-a) uaeina]U*U

Diag[1, e2ici("-1), a-2ac"] U* lim G(u, v))

= Diag[1, u1-ae-in(1-a) u'e ""]U* lim G(u, v) w°= lim F(u, v). v-o-

- Ia , C3 ), and by Assumption (A), it follows that F is even Lipschitz continuous on the punctured disk BB - {O}. Since G e CO, 1(B6

Lemma 2. The function F(w) = (F, (w), F2(w), F3 (w)) defined by (17) belongs to the class {0}, C3) and satisfies (18)

F?(w)w+2F2(w)F3(w)=0 for weBa-{0}, IF1(w)I = O(IwI"-1) as w

0,

and (19)

IF2(w)I = O(IwI"-")

as w -* 0,

IF3(w)I = O(Iw!"-e)

as w - 0,

180

8. Singular Boundary Points of Minimal Surfaces

where v denotes the Holder exponent of X and /3 = 1 - a. Furthermore, the following differential inequalities hold true: c{IwI-21IF2(w)12

IF1W(w)I 0 if a - 2/3 = -1.

Now relation (201) yields FiW(w)I = O(Iwl-2"I

as w -+0,

which is the desired assertion. (/3) F2(0) 0 0, F3(0) = 0: Here we obtain (i) of Section 8. 1, Theorem 3 with m = 0. Thus we can proceed as in case (a).

(y) F2(0) = 0, F3(0) 0 0: In this case we obtain (ii) of Theorem 3 in Section 8.1 with m = 0. In particular,

1F3(w)l=0(1) as w-+0, IF2».(w)I =

O(Iw11-2a)

as w

0.

But 0 < a < ? < /3 < 1 and /3 - 2a = I - 3a > - z; therefore we conclude from C) for all p < rnin(1, I + /3 - 2a). Since F2(0) = 0 we have that JF2(w)I = O(Iwl°), w --+0, u < min(1, I + /3 - 2a), and relation (201) implies Lemma 6 of Section 8.1 that F2 e

182

8. Singular Boundary Points of Minimal Surfaces 0(IwI-26+2" + IwI-2a)

IF,,jw)I =

0(IWI-2a)

=

asw-+0,

if we choose p in such a way that 2p - 2/3 >- 0. (b) F2(0) = F3(0) = 0: In this case we find

asw- 0,

F2(w) = awm + o(Iwlm) (22)

F3(w) -bwm + o(lwlm) asw-+ 0,

with a, b e C not both equal to zero and m >_ 1. A direct consequence of (20) is the following system : f1w(w)I 0, we have

ifm > 1,

O(Iwl-112)

I.f1 (w)I =

ifm > 1,

I.f3 n(W)1=0(1)

IF1(w)I = 0(Iwl°-1)

IF3(w)I = 0(1w1°

B)

Thus we may apply Lemma 6 of Section 8.1 which yields that the functions wl-"fl (w) and w1-2af3 (w) are of class C°1µ(B5, L) for all 1C < min(1, m + a). Let

t(w) = (1(w), lP2(w), 4/3(w)) := w-mU

t' W 1-2a] F(w)

Diag[w1-a

(25) 1

//

1

725

J2 (W)

-l

0

1

+ 'W1-2.f'-(W)

`

0

i

-}

0

W1-a{m(w) J

0

L

1

then i F(w) e C°'µ(B5, C3) for all y < min(l, m + a). We claim that 01(0) 0 0, we infer from Lemma '2(0) # 0 but 1/13(0) = 0. In fact, since I f 6 of Section 8.1 that 0(1wlm-21),

if m - 2/3 < -1

0(Iwlm+1-2P)

lfi (w)l =

O(IwJ-2)

foralle>0 ifm=0,/3=21

ifm-2$> -1.

0(1) as w

0. Hence

ifm - 2f < -1,

O(Iwlm+a)

1k3(W)1 = lw1-" f1 (w)I =

0(Iwl' -") for all c > 0 ifm = 0, fl = Z,

ifm - 2f > -1,

O(Iwl'-") and in all cases we obtain for some e > 0 that I13(w)I = 0(Iwle)

asw-+0.

In particular we have 1/13(0) = 0, and from (24) we deduce

184

8. Singular Boundary Points of Minimal Surfaces

w1-2afm(W) _

[W1-afm(w)]2

2f2 (w)

whence also [WI-22f3,](0) = 0 and 01(0) 0, t/i2(0) o 0. Because of T(0) = Id, also the function O(w) := T(x(w))*>G(w) satisfies (P1(0) 0 0, 02(0) 54 0 and c3(0) = 0. Since T is Lipschitz continuous and t(i and X are Holder continuous, also 0 is of class Co.v,(BB , C3) where v1 := min(v, µ). Because of (25) we have X,(w) = Wa-l+mO(w) = Wa-1+mT(x(w))*O(w),

that is, xw(w) = wa-1+mc1(w), Yw(w) =

(26)

Wa-1+m02(w),

IZw(w)I =

01(0)00, 'P2(0) #

0,

IWa-1+1I03(W)I = O(IwI'),

> a - 1 + m. This proves relation (2).

as w -). 0, with

On the other hand, let us assume that (ii) of Lemma 3 occurs; then we argue with the function 0

1

=f3(w)

i

+

W2a-1f2m(W)

0

t

+ waft (w)

0

0

instead of 1', and similar arguments show that (w) is Holder continuous on Ba and that q, (0) 0 0, 2(0) 0, > 3(0) = 0. In this case we have y = m - a because of wm-a,i,(w)

= U Diag[1,

wa-1, w-a]F(w)

and

Xw(w) = wm-aT(x(w))*,(w);

thus (2) holds with y = m - a. From the conformality condition

xw(w)+y ,(w)+z ,(w)=0,

WE1 Ula ,

we infer, using (26), that

0 = w2y[0i(w) + 0i(w)] + O(IwI2z) as w -> 0, where y = a - 1 + m or m - a. Letting w - 0, we obtain

0 = bi(0) + s(0) which proves (1).

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

185

Relation (3) follows by integrating formula (26), using the fact that r

X(w) = 2 Re 0

Thus o

(27)

x(w) + iY(w) = 10"', [c +(o(1)] as w -, 0

in the two cases respectively. Relation (27) and the boundary conditions imply that m = 2k in the first and m = 2k + 1 in the second case. Finally we have to consider the normal X

AX

J X. A X,,

Since IX,, A X0I = 21X.12 = 21w12y[Ip1(w)12 + 102(w)I2] + o(iwl2z),

).> y,

and

X. A X,, = 2(Im(y zw), -Im(x.zw), lm(xwy )) we find by means of (26) that I wlY+7

lim .Nl (w) = lim const w-0

w-+0

= 0 since A > y,

2

IwI Y

and

lim .N2(w) = 0. w-O

Finally lim .N'3(w) -= 2 lim W-0

W-0

Im(xw(w)Yw(w)) I Xw(w)12

_ = 2 Im(010)(P2(0)) -+_1 101(0)12 + 1(P2(0)12

since 01(0) =±'02(0), and Theorem 1 is proved.

Proof of Theorem 2. If F and F are straight lines, then the matrix T is the identity y Id

1

R3,

whence g(w) = Xw(w) and

dG (w) = Xw(w) = 0 almost everydw

where in Ba - {0}. According to Theorem 1.15 in Vekua [1] (or Satz 1.17 in Vekua [2]) we see that G and hence F are holomorphic on Ba(0). By the definition

of F we obtain Xw(w) = G(w) = U Diag[1, wac-1, w-°`]F(w)

186

8. Singular Boundary Points of Minimal Surfaces

r1

wa-1F2(w)

w-aF3(w)

1

i

0

Putting H1 := F1, H2 := (,,/2)-'F2 and H3 := (..,/2)-'F3 we obtain representation (6). Finally (7) follows by integration, and (5) is a consequence of (18). Thus Theorem 2 is proved.

8.4 An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem The aim of this section is to prove an analogue of Theorem 1 in Section 8.3 for minimal surfaces with partially free boundaries. Here the point of interest is the intersection point of the boundary arc T with the supporting surface S. Let us again start with an instructive example: Let S be the coordinate plane {x3 = 0} and

T={(x,y,z): z=xtan(an),y=0,0<x -1, and P(w) = (it(w), (P2(w), 03(w)) denotes some Holder continuous complex valued function with 01 (0) 0 0, 03(0) 0 0, and 02(0) = 0 if a # 2. From the representation (1) we deduce that the surface normal tends to a limiting

position as w --> 0. If in particular a 0 2, then the tangent space of X at the corner P E Fn S is spanned by the normal to S at P and the tangent to F at P. Thus the solution surface X must meet the point P at one of the angles at, (2 - a)n, (1 - a)it and (a + 1)n depending on whether X behaves like fi, f2, f3, or f4i respectively. In each of these cases X may penetrate S and can wrap P k-times.

Let us recall some notation. We define the sets 1 , Ib and Ba as in Sections 8.2 and 8.3, and we formulate Assumption A similar as in Section 8.2:

Assumption A. Let S be a regular surface of class C3, and F be a regular arc of class C2," which meets S in a common point P at an angle an with 0 < a < 2. We assume that P is the origin 0, that the x,y-plane is tangent to S at 0, and that the tangent vector to Tat O lies in the x,z-plane. Moreover, let X (u, v) be a minimal surface of class CO v(BB , l83) n C2(B8 - {0}), 8 > 0, which satisfies the boundary conditions

X: 1; --T,

(2)

X: 1,,+ -> S,

X(0) = P.

We also suppose that X intersects S orthogonally along its free trace X 1,+.

The main result of this section is Theorem 1. Suppose that Assumption A holds. Then there exists an R > 0 and a Holder continuous function O(w) = (0t (w), 02(w),'i3 (w)) defined on BR such that (3)

Xw(W) = W70(W)

holds true on BR - {0} with either y = a - 1 + m or y = - a + m for some integer m -> 0. Moreover, we have Ot(0), 02(0), i03(0) E V8 and (4)

0t (0) _ ±i03(0)

0,

02(0) = 0

fa36 2,

that is, (5)

0t (0) + 02 (0) + 03 (0) = 0

and at least two cj(0) 0 0 if a = 2. The unit normal vector

188

8. Singular Boundary Points of Minimal Surfaces

V(w) = (-41,(w), K2(w), X3(w)) =

Xu A

X'

I Xu A Xj

(w)

satisfies 01

1-

lim A'(w) = I± 1 W-0

if a

1

2

0

(6) Cl

liraX(w)=

c2

,

ifa=i

where c1,

0

For the trace X (u, 0), u E IR , we find

X(u, 0) = uY+'ii(u)

(7)

with some Holder continuous function i such that 1i (0) = (01(0), 02(o), 0). Further-

more, the oriented tangent vector t(u) =

(8)

Xa(u, 0) 0)I

+

, u c IR , satisfies

lira t(u) _ U-01

and

rd11 (9)

li m t(u) =

if a = 2 , where di + d2 = 1.

d2

[oJ If, in addition, S is a plane and if T is a straight line segment, then there exist functions Hl, H2, H3, holomorphic on BR(O), such that 0 (10)

Xw(w) = w'-'H, (w)

0

+ w-'H3(w)

0

+ w- 1/2 H2(W)

1

0

1

holds true on BA - {0} and (11)

HZ(w) + 4H1(w)H3(w) = 0 on BR(0).

Corollary 1. If a 94 2, then there exist some c c- C - {0} and some integer k > 0 such that one of the following four expansions holds true:

w - 0, (12)

(x + iz)(w) =

w -- 0,

w-+0,

w-*0.

8.4 An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem

189

Moreover I y (w) j = 0 (I w j 2 +1)

as w -+ 0,

for some A. > y

where y is the exponent in the expansion (x + iz)(w) = w1[c + o(1)] stated in (12).

The proof of Theorem 1 consists in an adaptation of the method which was developed in Section 8.3 for the proof of the corresponding result, see Theorem 1 in Section 8.3. So from time to time our presentation will be sketchy and leave the details to the reader as an instructive exercise. We begin the proof of Theorem 1 with a description of a reflection and a smoothing procedure. To this end let us henceforth assume that S is locally described by

z=f(x,y),(x,y)eBB(0)_{(x,y)eX82: x2+y2<e}, where f C C3(Be(0), R), and f(0, 0) = 0, Of(0, 0) = 0. Also, T may locally be described by two functions h,(t) and h2(t) of class C2'µ([0, a], ER) such that (h, (t), h2(t), t) e F for t e [0, e], and h, (0) = h2(0) = h2(0) = 0 while h'(0) = cot WE.

The unit tangent vector of Tat zero is then given by (cos an, 0, sin air). Because of the continuity of X we can select a number R > 0 such that X(BR) C

(0) _ {(x, y, z) E ER3: x2 + y2 + Z2 < e}.

We define the unit vector a(t), t e [0, a], by a(t) := [hi(t)2 + h2 (t)2 +

and the reflection across T by RrAt)Q := 2a(t) - Q

for Q e E3, t e [0, a]. Similarly, we define reflections across S by

Rs(x, y)Q = Q - 2N(x, y), for all Q e R3 and (x, y) e BE(0) c E2, where

N(x, y) = [I + f (x, y) + f2(x, y)]-1/2

-fx(x, y) -fv(x, y) 1

is the unit normal of S at the point (x, y, f(x, y)). Identifying the reflections Rr and Rs with their respective matrices Rr(t) and Rs(x, y), we may construct orthogonal matrices O, (t) and Os(x, y) with the properties'

R1.(t) = Or-(t) Diag[-1, -1, 1]0.(t), R5(x, y) = Os(x, y) Diag[l, 1, - I]Os (x, y). 1 As the symbols t and Tare used otherwise, we presently denote the transpose of a matrix A by A*.

190

8. Singular Boundary Points of Minimal Surfaces

We put T,4t) := Or(0)O,*(t) and T5(x, Y) := Os(0, 0)Os**(x, A.

Thus we have obtained matrices R. and Ts which are of class C2(Be(0), R9), BE(0) c R2, while Rr and Tr are of class C''" ([0, e], O9). If we extend a(t), t e [0, e] by a(t) = a(- t) for t e [ - e, 0] and call the extended functions again e]). Now let K, denote the cone a, Rr and Tr, then also a, Rr, Tr e with vertex 0 and opening angle r whose axis is given by x = z cot an, z > 0, y = 0. We assume that z is so small that the vertex 0 is the only point of K2, n S in the ball Y ,(O). Next we choose a real valued differentiable function defined on the punctured ball (0) - {0} = {0 < x2 + y2 + z2 < e} which satisfies ?I (X, y, z) = =

i

onK,n[.f(0)-{0}],

0

on.*^,(0) - {0} - K2,,

and

IV (x, y, z) I < const[x2 + y2 + z2]-'12

on Y, - {0}.

We extend g (noncontinuously) by defining n(0, 0, 0) = 0, and denote by T = T(x, y, z), (x, y, z) E A;(0), the matrix-valued function T (x, y, z):= ?1(x, y, z) [Tr(z) - Ts(x, Y)] + Ts(x, y).

Then T is continuous at zero because

lim

T(x, y, z) exists and in equal to

(x, Y, z) - o

Id ss3. In fact, T is even Lipschitz continuous on Y ,(O) c R3 because

ITr(z) - T5(x, y) I < const[x2 + y2 + z2]1/2 and hence IVT(x, y, z)I stays bounded as (x, y, z) -+ 0. Defining

g(w) := T(X

for w e RR'- - {01

we then obtain Lemma 1. The function g(w) is of class properties: (13)

{0}, C3) and has the following

Rr(0)g(w) = g(w) for allwEIR

,

Rs(0)g(w) = g(w) for allwEIR

,

and (14)

where Rs(0) := Rs(0, 0).

Proof. The Lipschitz continuity of g(w) is an immediate consequence of the Lipschitz continuity of T and of the regularity properties of X. Relation (13)

8.4 An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem

191

follows similarly as equation (11) in Section 8.3 using the fact that T(X(w)) _ Tr(z(w)) if w c- IR. To prove (14), we let w e IR ; then fy(x, Y)Y.,(w))

(x (w), and

<X.(w), N(x(w), y(w))> = 0.

From the transversality condition we infer that <Xjw), N(x(w), y(w))>N(x(w), y(w)),

for all w E IR whence X.(w),

Rs(x(w), and

Rs(x(w), Y(w))Xjw) = -X,,(w)

or equivalently (15)

Rs(x(w), y(w))Xw(w) = XW(w),

W E IR.

Now, using (15) and the definition of T, we obtain g(w) = T(X (w))X-w(w) = Ts(x(w), Y(w))X-w(w)

= Ts(x, Y)Rs(x, Y)Xw

= Os(0)OS(x, y)Os(x, y) Diag[1, 1, -1]OS(x, y)Xw = Os(O) Diag[1, 1, -1] Os(O)Os(O)OS(-x, y)X. = Rs(0)Ts(x,Y)Xw = Rs(0)T(X(w))Xw(w) = Rs(0)g(w),

where the argument of X, x, y is always w c- IR. We now reflect g(w) so as to obtain a function G(w), (16)

G(w):=

g(w)

if w E BR - {0},

Rs(0)g(w)

if w E BR ;

then G E CO, 1(BR(0) - IR, C3) and lim G(w) = Rr(0)Rs(0) lim G(w) for all w = U-0-

V-01

(u, v) with u c- IR. Furthermore, G satisfies (17)

IGw(w)I 0, the asymptotic

I Xu(U, 0)1

behaviour

lim t(u) = [(Re 0,(0))2 + (Re

02(0))2]-1/2

Re 01(0) Re 02(0)

u-+o'

0

which proves relations (8) and (9). If S is a plane and T is a straight line, then T = Id R3 and g = X.. Hence G is holomorphic on BR - {0} and F is holomorphic on BR. Finally (10) and (11) follow from (20) if we take H1

1

F1,

H2 := F2,

1

H3

F3 1

and Theorem 1 is proved.

8.5 Scholia The basic idea of this chapter, the Hartman-Wintner method, was described and developed in the paper [1] of Hartman and Wintner in 1953. Its relevance for

8.5 Scholia

197

the theory of nonlinear elliptic systems with two independent variables was emphasized by E. Heinz. In particular, he discovered the use of this method for obtaining asymptotic expansions of minimal surfaces at boundary branch points, and of H-surfaces at branch points in the interior and at the boundary.

The results of Sections 8.2-8.4 concerning minimal surfaces with nonsmooth boundaries are due to Dziuk (cf. his papers [1-4]). His work is based on methods by Vekua [1, 2], Heinz [5], and Jager [1, 3]. Earlier results on the behaviour of minimal surfaces at a corner were derived by H.A. Schwarz [3] and Beeson [1]. The boundary behaviour of conformal mappings at corners was first treated by Lichtenstein, and then by Warschawski [4]. The continuity of minimal surfaces in Riemannian manifolds at piecewise smooth boundaries was investigated by Jost [12].

The proofs in the paper [1] of Marx based on joint work of Marx and Shiffman concerning minimal surfaces with polygonal boundaries are somewhat sketchy and contain several large gaps. Heinz [ 19-24] was able to fill these gaps

and to develop an interesting theory of quasi-minimal surfaces bounded by polygons, thereby generalizing classical work of Fuchs and Schlesinger on linear differential equations in complex domains that have singularities (see Schlesinger [1]). In this context we also mention the work of Sauvigny [1, 2, 3, 6]. The papers of Gamier are also essentially concerned with minimal surfaces having polygonal boundaries, but apparently these results were rarely studied in detail and did not have much influence on the further progress. This might be both unjustified and unfortunate.

Chapter 9. Minimal Surfaces with Supporting Half-Planes

In Chapter 7 we investigated the regularity of stationary minimal surfaces in the class '(r, S). Such stationary surfaces had been introduced in Section 5.4. We have shown that, for a uniformly smooth surface S with a smooth boundary aS, the stationary surfaces X belong to the class C"' (B v 1, ER3). 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 7, the non-oriented tangent of the free trace E = {X(w): w E I} of a stationary minimal surface X in 49(r, 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 __ 0, y = 0} of the x, z-plane and that as coincides with the z-axis. Then we may assume that the projection of F 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 I'is monotonically increasing. In this case, the free trace of a soap film spanned in (I', S> is depicted in Fig. 2. Let us now define the arc Tin such a way that its endpoints on S are kept fixed and the projection of r onto the plane is only slightly altered, whereas the z-component of the representation of I' 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

200

9. Minimal Surfaces with Supporting Half-Planes

(a)

(b)

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

seen by looking at the free trace in various stages of the bending procedure; cf. Fig. 3. Let us deform the arc I' 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 2.

It is interesting to contrast the situation depicted in Fig. 4 with another, related experiment where T 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 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 T stated above are essential for the following mathematical discussion, but they are by no means essential for the

9.1 An Experiment

201

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

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

experiment. The supporting surface S can be an arbitrary smooth surface, planar

or not, and T 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

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.

9. Minimal Surfaces with Supporting Half-Planes

202

9.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface In the sequel, B will not denote the unit disk { wI < 1} but the semidisk

B:= {weC:Iwl0}, and I denotes the interval

I:= {ueO:Jul B keeping the three points 1, -1, i fixed.

As in (9.1), we consider the half-plane

S= {(x,y,z)El 3:x>-0,y=0} as supporting surface.

In Sections 3.4 and 3.5 we have seen how Schwarz's formula solving

Bjorling's problem can be used to construct stationary surfaces X : B -+ l3 which intersect S perpendicularly in a given curve £ 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:

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

9.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface

Fig. 2. Another view of Henneberg's surface in a configuration .

(b)

(a)

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

x = cosh(22u) cos(2).v) - 1

(1)

sinh(3),u) sin(31v)

y=

sin(Av) -

z=

cos(Av) + 3 sinh(3)u) cos(3),v).

3

It follows from X(u, 0) = (cosh(22u) - 1, 0, -sinh(Lu) + 3 sinh(3Au)) _ (2 sinh2(Au), 0,

sinh3(Au)) 3

that (1) intersects the plane y = 0 in Neil's parabola

(2)

2x3 = 9z2,

y = 0.

203

204

9. Minimal Surfaces with Supporting Half-Planes

For small values of w we have the expansion x(w) = Re{222w2 + ... },

y(w) = Re{2i w2 +

... ; ,

z(w)=Re{3A3w3+-..i Let us denote by # the portion of (1) which corresponds to the closed semidisk B = {w: I wi _< 1, v >_ 0} in the parameter plane. The surface .if is bounded by a configuration 2,

since Vx(u') = 0 and x (u) = 0 for u, < u < u2. From the fact that u = u' is a local maximum of x(u) on I we conclude that K < 0 and v = 2n, n >- 1. A neighbourhood of w = u' in B is divided into 2n + 1 - at least three - open sectors v,, a2, 62,r+1 such that x(w) < m in al, 63, ... , Q2n+1, and that x(w) > m in a2, a4, ..., Now consider two points w1 and wen+1 in of and Q2n+1 respectively. As we know from Lemma 2, the set B-(m) is connected and contains the points w1 and

w2.+1 Thus we can connect w1 and w2,+1 by a path y contained in B-(m). Connecting w1 and wen+1 with u' in a, and a2n+1 respectively we obtain a closed curve which separates the component 02 of B+(m) containing the sector a2 from the components Q1 and Q2 that were introduced in the proof of the preceding lemma. In other words, the case (y) would imply that B+(m) has at least three components, which is not true. Having ruled out case (y), we shall now prove that (/3) cannot hold unless xo = 0. In fact, the inequality xo > 0 would imply 0 on I, and then the unique continuation principle would yield x(w) _- xo in B if (/3) were true. This is again not possible.

9.5 Nonparametric Representation, Uniqueness, and Symmetry of Solutions

213

Therefore the relation xo > 0 implies that we are in case (a), and the proof is completed.

9.5 Nonparametric Representation, Uniqueness, and Symmetry of Solutions The following representation theorem which will be proved in Section 9.9 is the

key to all the other results of this chapter. It states that all stationary minimal surfaces X in '(I', S) are graphs (cf. also Section 4.9). Theorem I (Representation theorem). Let X be a stationary minimal surface in cq(r, S), and let xa be the lowest x-level of the free trace of X, that is, xo :_ min{x(u): u a I}. Moreover, denote by D = D(xo) the two-dimensional domain in the x, y-plane which is obtained from the interior of the orthogonal projection y of r by slicing this interior along the x-axis from x = xa to x = a. In defining the boundary aD of the slit domain D, both borders of the slit xo < x < a will appear, with opposite orientation. Then the functions x(w), y(w) provide a C'-mapping of a onto D u aD which is topological, except in case II, where the interval of coincidence

12= {ual:x(u)=0} corresponds wholly to the point (0, 0) on D. Moreover, the minimal surface -0 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 xo < x < a, and (1)

lim a Z(x, y) = lim Y-

aay Z(x, y) = 0

for 0 < x < a.

Z(x, y) is continuous on D u aD in cases I and III. In case II, Z(x, y) is continuous on D u aD - {(O, 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(r, S) which are normed in the same way, say, X1, X2 aW*(F, 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 W*(I', 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

WV, S). Then also

214

9. Minimal Surfaces with Supporting Half-Planes

X(u + iv) := (x(-u + iv), -y(-u + iv), -z(-u + iv)) is a stationary minimal surface in *(r, S), and the surfaces X and same lowest x-levels. Then the uniqueness theorem implies that X(w) B, and we obtain

have the X(w) on

Theorem 3 (Symmetry theorem). Every stationary minimal surface X e c'*(r, s) is symmetric with respect to the x-axis. More precisely, we have (2)

x(u + iv) = x(-u + iv)

(3)

y(u + iv) = -y(-u + iv)

(4)

z(u + iv) = -z(-u + iv).

In cases I or III we have

x0=x(O) and xo < x(u) W-3(t) C(P2 - Pt)' + (q2 - qt)']. Since (W2(t))" > 0, we obtain

f(t) > (max{W, '1'2})-3L(P2 - pt)' + (q2 - q1)2]

(8)

For 8 > 0 and a > 0 we now introduce the set Da,E consisting of all points in D the distance of which from (0, 0) and (a, 0) exceeds s, and whose distance from OD is greater than S. Let Q be an arbitrary compact subset of Db,e, and set m(Q) := max{ W, (x, y), W2(x, A: (x, y) E Q}, and

[P2 - P)2 + (q2 - qt )2] dx dy.

1(Q) := 1 J

Q

Invoking (7) and (8), we arrive at

1(0:5:

m3(Q)

J

f(1)dxdy < m(Q)

JDo,,

Inserting [q2 f-2 - P1] + (q2 - ql) [ W2 _ qt f()1 _ (P2 - PI) [W2 Wt W

and applying a partial integration, we obtain that 1((?) 1.

0 appearing in the general expansion formula

X(w)=Xo+Re{Bw"+...} satisfies = 0. Therefore we obtain, in conjunction with the formulas (2), (3) and the relations 0, y(u) = 0 on I, the following local expansion for Y(x')

y(w) = Re { ± iKw2 +

Here the plus sign must be chosen because

... } . 0 for 0 < u < 1. This follows

from E. Hopf's lemma if one notes that y(w) < 0 on the boundary of the set Q = {w: I w < 1, u > 0, v > 0}, so that by virtue of the maximum principle y(w) < O for w e Q.

Proof of Theorem 1. Because of Section 9.5, (5), y(iv) vanishes for all v e [0, 1].

Since x(0) = xo > 0, and x(i) = -b < 0, there exists a smallest number v1 in [0, 1) such that x(iv1) = 0. Suppose now that X (w) is a solution of the minimum problem in *(r, 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 = ivl. Furthermore, let w = T(C) be the conformal mapping from B onto B',

218

9. Minimal Surfaces with Supporting Half-Planes

leaving the three points w = + 1, -1, i fixed. Then, Y(C) = X

is again of class 16*(1', S) since y(iv) = 0 for all v e [0, 1]. From the invariance of the Dirichlet integral with respect to conformal mappings we conclude that is also a solution of the minimum problem in '*(F, S), but of type I, by virtue of the Theorem in Section 9.4. By (1), Y(C) = (y' (Z'),

sion near

y3

possesses an expan-

= 0 of the form y' (C) = Re {rcl'2 +

... }

Re{iic 2 + ...I

(5)

y3(() = Re{µC2+t + ... }

where K > 0, u 0 0 and n > 1. Let ( = a + i(3. We infer from (5) that the images of suitable segments (- E, 0) and (0, e), E > 0, on I under the mapping Y(1;) are

different, that is, y3(-a) 0 y3(a') if 0 < a, a' < E. On the other hand relation (5) in Section 9.5, z(iv) = 0 for 0 < v < 1, implies that y3(a) = 0 for 0 < I a I < e',

a e 1, if s' is a sufficiently small positive number. Such a discrepancy is not possible, and X(w) cannot be of type III. Finally we shall give another proof of Theorem I without using the expansion formula. The symmetry theorem of the previous section shows that the minimum X in W* (T, S) maps the interval {w = iv:0 < v < 1} onto the x-axis. If X is of type III, that is, if xo > 0, then also the value

vl := inf{v >- 0: x(iv) < 0}

is positive. Now let z be the conformal mapping from B onto the slit semidisk B - {iv: 0 < v < vt } mapping each of the points i, 1, -1 onto itself. Since the Dirichlet integral is conformally invariant, we conclude that

Xoz=:Y=(Yt,Y2,Y3) is another minimum for the Dirichlet integral in '*(r, S), but Y is of type I. Because of formula (5) in Section 9.5, the third component z(w) of the minimum X vanishes for w = iv, 0 < v < 1. Therefore the third component y3(w) of Y(w) satisfies

y3(u, 0) = 0

and y'(u, 0) = 0

on certain intervals (- S, 0) and (0, S), b > 0, which are mapped by z onto the slit {iv: 0 < v < vl }. The reflection principle implies that Y(w) _- 0 on B, which is impossible.

9.7 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 le*(J', S)

which is of type II, and let [ut, u2] be its set of coincidence 12, -1 < ul < u2 < 1.

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

219

(It follows from formula (2) of Section 9.5 that u2 = -u1 > 0.) Then there are positive numbers K and p, and a real number zl 0 0, such that x(w) = Re{iK(W - U1)312 + (1)

... }

y(w) = Re{ - iy(w - u 1 ) + ... }

near w = u l ,

z(w) = Re{zl - (sign zl)µ(w - ul) + ... } and

x(w) = Re{K(w - u2)312 + ... (2)

}

y(w) = Re{iic(w - u2) + ... }

near w = u2,

z(w) = Re{-z1 - (sign z,)p(w - u2) + ...}. Moreover, no point on I is a branch point of X(w).

Proof. Let h(w) be the holomorphic function in a neighborhood of w = u, in B satisfying h(u,) = 0 such that x(w) = Re h(w), and g(w) = h'(w) = x(w)

If u e I is close to u,, we have Re g(u) = 0 for u > u,, and Im g(u) = 0 for u < ul . Consider the transformation w = ul + y2, and set g(ul + t'2). The func-

tion f(C) is holomorphic near

= 0 in {t': Re C > 0, Im C > 0}, and Re f(C)

vanishes on the positive real -axis, while Im f(1;) is zero on the positive imaginary axis. The C'-character of x(w) in B allows us to extend f (C) by a twofold reflection analytically to a holomorphic function in a full neighbourhood of the point = 0, with an expansion f(C) = ao + a,C + a2 y2 +

The relations the expansion

...

near C = 0.

x,(ul) = 0 imply that ao = f (O) = 0. For v > 0 we then get

g(w) = al(w - U1)112 + a2(W - u1) + a3(W - U1)312 + ... .

(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)312 + b2(w - u1)2 + b3(w - U1)5t2 + ...}

with complex coefficients b; = p; + iqq. From the relation x(u) = 0 for u > ul it follows that po = pt = p2 = ... = 0; we may also assume that qo = 0. The condition 0 for u < u, allows us to conclude that q2 = q4 = ... = 0. Denoting the first non-vanishing coefficient of the remaining ones by iK, we arrive at x(w) = Re{iK(w - U1)"+1/2 +...} where (-1)"ic < 0, and n >- 1. By virtue of formula (3) in Section 9.5 we also have

220

9. Minimal Surfaces with Supporting Half-Planes

the expansion x(w) = Re{K(w - u2)"+t/2 + ... }

for w e R near the value u2. Arguments similar to those employed in the proofs in Section 9.4 show that we have n = 1 in the above expansions. Thus we obtain

nearw=u1

x(w)=Re{iK(w-u1)312+...}

(3)

x(w) = Re{K(W - u2)312 + ...}

near w = u2 .

The harmonic function y(w) vanishes on I as well as for w = iv, 0< v < 1, while y(e'B) < 0 for 0 < 0 < 2 and y(e I') > 0 for

71

< 0 < 7r. Consider the two sets

2

Q-={w:IwI0,v>0} and

Q+= {w:lwj - 0 for w e OQ+ and y(w) < 0 for w e 8Q-, the maximum principle implies that y(w) > 0 for w e Q+ and that y(w) < 0 for w E Q. It then follows from E. Hopf's lemma that 0 for 0 < u < 1, 0 for -1 < u < 0 and and hence 0, y,,(u2) < 0. Because y(u) = 0 for all u e 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{-iu(w-u1)+...} with a constant u > 0. The conformality relation I X.1' = I 2 yields zu (u1) = y' (u l) so that z (u1) _ ±µ,while z (u1) = 0. We set z1 = z(u1)andz2 = z(u2).Sinceu1 = u2, formula (4) of Section 9.5 implies that z1 = -z2. Hence,

z(w)=z1 ±Iz(w-u1)+... near w=ul z(w) = z2 ± jt(w - u2) + ... near w = u2 . The conformality relation I X. I2 = I

also implies that

zu(u) = xV(u) + yV(u)

for u E I21 because

0 for u e I2 and

0 for u e I. Assume that

0 for some u' c- (u1, u2). Since x(w) can be extended as a harmonic function across 12, we would then obtain an expansion of the form

x(w) = Re {a(w-u')"+...},

n>-2

valid in a full neighbourhood of the point w = u2. Arguments similar to those employed earlier in conjunction with the properties of the expansions (3) show that this is impossible. Thus, 0 0 for u1 < u < u2; in fact, we see from (3) that 0 for u1 < u < u2. It now follows that the derivative cannot

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

221

vanish in the interval of contact, so that z1 0 0. Since z2 = -z 1, we have z (u) > 0 for u e '2 if z1 0 (loop).

From the expansion formulas Section 9.6, (1), and Section 9.7, (1) and (2), it is apparent that the three generic forms of the trace X (u), u e I, for a solution X of the minimum problem in 16*(F, S) look as depicted in Fig. 1. In conclusion, let us describe a situation in which the trace curve X (u), u e I, is free of cusps. Theorem 2. Suppose that the open subarc of the arc 1' with the end points P1 and P3 lies in the half-space {z < 0}, and that the open subarc of T between P3 and P2 is contained in the half-space {z > 0}. Then there exists exactly one stationary

222

9. Minimal Surfaces with Supporting Half-Planes

minimal surface X in W*(I, S). This surface is of type II, and its trace X (u), u E I, on the half-plane S is a regular curve of class Ct, t"2 and has the form of a tongue.

Remark 1. The expansions (1) and (2) of Section 9.7 show that the regularity class of a stationary surface of type II is exactly C1,112(B v 1, ER3) and no better on I, and Theorem 2 guarantees that there are surfaces of type II. Thus the principal regularity theorem from Chapter 7 cannot be improved. Remark 2. The assumptions of Theorem 2 are satisfied if the z-component p3(s) of the representation P(s) of I' changes monotonously from z = - c to z = c as s moves from 0 to L; cf. Fig. 2. The situation is altered if T 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. Y

Fig. 2

Fig. 3

Proof of Theorem 2. We introduce the two arcs

C+:=1w=e`o:0 0.

Clearly, a whole interval on I around u' is also contained in OR n I. Then, by E. Hopf's lemma, z (u') > 0, in contradiction to the relation z (u) = 0, which is valid for all u E I. Since Q+ and Q- are the only components of the set Q, we conclude from Section 9.5, (5) that

Q+={w:Iwl0,v>0} and Q- = {w: Iw(< I, u < O, 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 < u < 1. In cases I or III the expansion (1) of Section 9.6 shows that a neighbourhood of u = 0 in B is divided into 2n + 2

(and at least four) open sectors 61 62, ... , a2n+2 such that z(w) > 0 in al , a3, ... , cr2n+,, and z(w) < 0 in 62, 64, , 072n+2 It can be demonstrated as before that this is impossible. Thus it follows from the above that the solution X(w) must be of type II. Hence, by the uniqueness theorem of Section 9.5, 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.

9.9 Proof of the Representation Theorem Now we want to supply the proof of the representation theorem, stated in Section 9.5, 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 e V (F, S). For this purpose it is useful to recall the results of Sections 9.3 and 9.4 as well as the definitions of the subsets B(µ), B+(µ), B-(µ) of B and of the arcs C; (p.), C2 (µ),

C-(µ) given in Section 9.4. (i) We shall first pursue the discussion of case I assuming that 12 =

{uo}.

By Lemma 2 of Section 9.3, 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(uo) = 0 and x(u) > 0 for u uo, the function x(w) must have an

expansion x(w) = Re{x(w - uo)2rt + .. . }

near w = uo where x > 0, n >_ 1. A neighbourhood of w = uo in B is divided into 2n + 1(and at least three) open sectors al, a2, ..., c2,+1 such that x(w) > 0 in a,, 073, , c2n+1, and that x(w) < 0 in a2, a4, ..., a2ri. Denote by Q1, Q2, . .., Q2e+1

the components of the set B(0) which contain the sectors a,, a2, ..., c2,+1, respectively. These components are mutually disjoint for topological reasons

9. Minimal Surfaces with Supporting Half-Planes

224

and because of the maximum principle. Then, by virtue of Lemma 2 of Section 9.4, it follows that n = 1 and that B(0) consists of three different components Q 1, Q2, Q3. Clearly, Q2 = B-(0). According to the remark following the same lemma we may assume that Ci (0) c 8Q1, C2'(0) c 3Q3. Since x(u) > 0 for u e I, u 0 uo,

and since x(1) = x(- 1) = a > 0, the interval (uo, 1) belongs to 8Q, while the interval (- 1, uo) is part of 8Q3. Then, by our standard reasoning, the gradient of x(w) cannot vanish on 1 except for u = uo. On the other hand, x,(u) = 0 on 1, so that x (u) 0 for u 0 uo. Therefore, the function x(u) increases strictly from the value 0 to the value a as u increases from uo to 1, or decreases from uo to 0 for uo < u < 1. We -1. Furthermore, x (u) < 0 for -1 < u < uo, and expansion of x(w) near the point observe finally that xn(uo) = 0, and that the w = uo must have the form

x(w)=Re{K(w-u0)2+...},

(1)

K>0.

We assert that I Vx(w) I > 0 for all w e B. Otherwise we would have Vx(wo) = 0

for some wo e B. Then, according to Radb's reasoning (cf. Lemma 2 of Section 4.9), the set B(µ) consists of at least four different components. This contradicts Lemma 2 in Section 9.4.

Next we consider the harmonic function y(w). We have y(e1e) < 0 for

0< 0< 2 and y(eie) > 0 for

it, as well as y(u)=O for -1 < u 0

225

for 0 < 0 < rr.

The are rmeets the halfplane Sat right angles; therefore X (e`B) = (0, y,(e`B), 0) for 0 = 0 and 0 = It. Consequently, we have y,(± 1) 54 0; more precisely, y,(1) < 0

and y,(-1)>0,since y(e`B)0 and x(w")=0, y(w") > 0. Condition (C2). We have

G c U

s, ««2

2'(s),

and secondly:

6n 2'(s) n.(s') is empty for all s, s' E [s,, s2],

s

s'.

Fig. 5. The condition (C2).

Now we consider the

Problem --v(F, S): Find a mapping X: B-* R' of class H2' n C2(B, R3) which satisfies in B the equation AX = 0 and the conformality conditions

<X,,, X> = 0, which maps C continuously and one-to-one onto r such that X(- 1, 0) = Pl , X (l, 0) = P2, and which is stationary with respect to the configuration (F', S>.

By the results of chapters 6 and 7, any solution X of 9(T, S) is continuous on 9 and of class C2 '(B v I, R3) for all a e (0, 1) if S is of class C. The assump-

9. Minimal Surfaces with Supporting Half-Planes

238

tion "S e C,3k" means that S E C3 and that S satisfies a suitable uniformity condi-

tion at infinity (as formulated in Section 7.6; see Definitions 1 and 2) if it is noncompact. This will be assumed once and for all in the sequel. Moreover we shall suppose that F is piecewise of class C" (by the regularity result of Kinderlehrer/Nitsche, it would even suffice that T is piecewise of class C1"3). The free boundary condition along I can then be formulated as X(1) C S and

X(u, 0)

(6)

for all u E I

where )(u) is a continuous real function on I. If we know in addition that X(w) lies on the same side of S as T, it follows that ,(u) < 0. Suppose that X(w) = (x(w), y(w), z(w)) is a solution of S). Then f(w) _ (x(w), y(w)) = it o X (w) defines a harmonic mapping f : B -+ i82 with the following properties : _

feC°(B,R2)nC'(BvI,R2); (ii) f(I)=E,f(C)=F,f(-1,0)=Pt,f(1,0)=P2; (i)

(iii) We have (6')

f,(u, 0) = A(u)v(f(u, 0))

for all u e I.

Finally we denote the Jacobian of f by

Jf = det(f,f) =

a(x, Y)

a(u v)

We begin our discussion with an inclusion result for the projection (n o X)(I) of the free trace X (I) of a solution X of the problem e(T, S) onto the plane {z = 0} which is identified with Flt. Proposition 1. Let f := it o X be a harmonic map B -> F2 obtained from a solution X of 9(F, S), and suppose that the boundary configuration satisfies condition (Cl). Then we obtain that f(I) = E.

Proof. Set £ := f(I). Because of E c E, there are two numbers s' and s" such that s' < s1 < s2 < s" and f = {Q(s): s' < s < s"}. Then one of the two relations of (C 1), (iii) has to be true, say,

{U(T): S' < T < s} c int

-(s").

In conjunction with (Cl), (ii) it follows that {Q(s): s'< s < s"} c int.*' (s"), and

(C 1), (i) implies that t c int.-(s").

_ Let us now introduce the function cp(w), w e B, by (P(w) := s2. Then we easily see that cp attains its maximum M at some point u* e I such that f(u*, 0) = v(s"). Because of lu(u*, 0) = 2(u*)v(f(U*, 0)) = /t(U*)IZ(s")

we obtain QPv(u*, 0) _ 0. On the other

0), A(so)> = ,t(uo) 0, 0, or else the opposite set of fying either 0, inequalities.

Let W be the closed connected component of 2 n G such that po e W. Then the inclusion f (BO) c . n G implies that f (Z) c 4' whence f(2) c W. Since the boundary map f IaB: 8B - 8G is (weakly) monotonic, we obtain by the definition q; := f(w,), 1 < j < 4, four different consecutive points q1, q2, q3, q4 on 8G. By construction, these points lie in le, and therefore the connected component W of 2 n G contains at least four points of 8G, the points q1, q2, q3, q4 We can assume

that the straight segment [q1, q4] := {p: p = (1 - t)ql + tq4, 0 < t < 1} lies in ', and that q2, q3 e [q1, q4]. Then we infer that the subarc r* of 8G containing q1, q2, q3, q4 must either be a subset of the halfspace Yf + :_ {p e R2 : 1(p) > 0} or of the halfspace .- := {p e R': 1(p) < 0}. However, this contradicts the fact that each of the open halfspaces int 0' and int .*E°- contains at least one of the points f(Cj), j = 1, 2, 3, which lie on r* between q1 and q4. Thus we infer that Jf(w) 0 0 for all w e B.

Hence the mapping f 1B is open and locally one-to-one. In conjunction with f(B) c 6 we infer that f(B) c G. Moreover, the set

9.10 Scholia

243

G, :_ { p c- G : There is some w c- B such that f (w) = p}

is open, and an elementary reasoning yields that also G2 := {p E G : There is now c- B such that f(w) = p} is an open set. Since G = G, u G2 and G, n G2 = 0, it follows that G2 is empty.

Hence we have G = G1, and therefore f(B) = G. Invoking the monodromy principle we now infer that f IB maps B bijectively onto G, and therefore f IB is a real analytic diffeomorphism of B onto G. If f IOB is one-to-one, then f clearly yields a homeomorphism from ,6 onto G. As an immediate consequence of the previous result we obtain Proposition 7. Let.' be a minimal surface in l3 represented in the form A' = X(B) where X(w) = (x(w), y(w), x(w)) is a harmonic mapping of class C°(B, 683) n C2(B, 683) satisfying the conformality relations. Assume also that f = it o X maps B into G and aB monotonically onto aG. Then the function Z(x, y), (x, y) e G, defined by Z := x o fG' is a real analytic solution of the minimal surface equation (1 + ZY2)ZXX - 2ZXZYZXY + (1 + Z. ,2)Z,, = 0

in G,

and the mapping (x, y) --* (x, y, Z(x, y)) yields an equivalent representation of X in the form.' = graph Z. If f dB provides a topological mapping of 8B onto 5G, then Z can be extended continuously onto G such that .' = graph Z. Moreover, if X is a solution of 1(f', S) and if the Jacobian if satisfies

Jf>0 on I, then Z is of class C' (G u ±) and satisfies the boundary condition (8)

a Z = 0 on!.

(Remark. Since we use x and y both as dependent and as independent variables, we denote the components of X(w) by Script symbols.)

Proof. The first part of the assertion follows directly from Proposition 6. If in addition Jf > 0 holds on 1, then f maps I topologically onto I = f(1), and f turns out to be of class C' (G u I, 682) whence Z = z of -'is of class C' (G u I). Since X satisfies the free boundary condition (8')

X (u, 0) = )(u)(v(f(u, 0)), 0)

for u e 1,

the surface normal N:= IX,, A X,I-'X A X,, is well defined and continuous on B u I and perpendicular to (v of, 0) along I. On the other hand, we have

Nof-' = ±{l +ZZ +Z;,}-'12(ZX,ZY, -1}, and thus it follows that = 0 on I. This completes the proof of Proposition 7.

244

9. Minimal Surfaces with Supporting Half-Planes

The preceding result demonstrates that it may be useful to establish the relation if > 0 on the free boundary I. The next Proposition will show that we can verify this relation for any solution X of the free boundary value problem 1(T, S). Proposition 8. Let X be a surface of class C3(B, 683), f = t o X, and suppose that f(1) c E, X E Cz(B u 1, 683), Jf > 0 on B, and W = I X A X, I > 0 on I. Moreover, we assume that the boundary condition (8') holds and that

dN+NIVNIZ=0 inB.

(9)

Then we obtain that if > 0 on I.

Proof. Note that N = (N1, N2, N3) = W-'X. A X, is of class C2(B, 683) n C1(B L) I, 683),

and that

WN3 = if = det(f , f).

(10)

The assumption if > 0 on B implies that

N3 > 0 on B u I,

(11)

and by virtue of (9) we infer that

AN3 < 0 in B.

(12)

Suppose now that Jf(uo, 0) = 0 for some uo e I. Then N3(uo, 0) = 0, and, by means of E. Hopf's lemma, it follows from (11) and (12) that N3(uo, 0) > 0.

(13)

Differentiating both sides of (8) with respect to u and noting that X,(u, 0) (f (u, 0), 0), we see that

on1

(14)

where v' denotes the tangential derivative of v(p) along E. On the other hand, we obtain from (10) that

onBuI.

(15)

We infer from (14) and (15) that

NEW=

on I.

Since Nv (uo, 0) W(uo, 0) > 0 and N3(uo, 0) = 0, it follows that f (uo, 0) :0.

Moreover, we obtain from X. A X 0 0 on I and X = (f 0) that &10 0 0. Finally, the boundary conditions f(1) c E and (8) imply that = 0

on I

9.10 Scholia

245

whence

J I = IJ.I - If,,l

on

I.

Consequently we obtain Jf(uo, 0) 54 0, contrary to our assumption Jf(uo, 0) = 0. Hence it follows that

Jf(u,0)>0 on I, as we have assumed that Jf > 0 on B. Now we obtain the following result from Propositions 6-8. Theorem 1. Let X be a solution of Y(T, S), and suppose that f = it o X satisfies f(B) c G, and that f 11 maps I monotonically onto X. Then we obtain f(B) = G and Jf > 0 on B u 1. Moreover, f IB maps B diffeomorphically onto G, and therefore X18: B --> R' defines an embedded minimal surface := X (B). By means of the function Z = z of -1: G -4 R, the minimal surface can be written as 4' = graph Z. The function Z is a solution of the minimal surface equation which is of

class C'(G u E) and satisfies the boundary condition a Z = 0 on E'. Remark. We recall that Proposition 3 furnishes conditions which guarantee the basic condition f (B) c G that has played an essential role in the preceding results. It should be interesting to find other conditions ensuring the property f (B) c G. Finally we turn to the discussion of uniqueness results. Theorem 2. Let G be convex, S e C,3k, and suppose that satisfies (Cl) and (C2). Then, up to conformal equivalence, there is exactly one solution of the problem c(T, S). This solution can be represented as a graph over the domain G. Proof. The existence of a solution follows from Section 4.6. To prove uniqueness,

we assume that X1 and X2 are two solutions of 9(T, S); then we have X1, X2 E C2 (B u I, lv). Let f1 := 7c o X1,f2 := 7c o X2,Z1 := xl of1 `,Z,:=x, o f2 1. By virtue of Proposition 3 we obtain f(B) c G, j = 1, 2, and Proposition 4 yields that both f1 11 and f2I r map 1 monotonically onto .E. Hence we can apply Theorem 1, thus obtaining that Z1 and Z2 are real analytic solutions of the minimal surface equation in G which are of class C1(G u £) and satisfy

a Z;=0 on Z, j= 1,2. Now, suppose in addition that T is a graph above r. Then we realize that both Z1 and Z2 are continuous on G, and that

Z1(p)=Z2(p) for allpeT.

246

9. Minimal Surfaces with Supporting Half-Planes

Consider the function Z := Z1 - Z2. We have

Z=0 on F, az 8v

0

on E,

and Z e C°(G) n C1(G u t). Moreover, Z(x, y) satisfies an elliptic equation a(x, y)Zxx + 2b(x, y)Z., + c(x, y)Z,,, + d(x, y)Zx + e(x, y)Z3, = 0

in G

which is uniformly elliptic in every compact subdomain of G U E. 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 its maximum M and its minimum m on the free boundary E. By means of E. Hopf's lemma we would then arrive

at a contradiction to the boundary condition aZ = 0 on If T is merely a generalized graph, the function Z might be discontinuous at points p of F corresponding to vertical pieces of r, and VZ could possibly be unbounded. Therefore, the preceding reasoning cannot be carried out as it stands; instead we shall apply an argument already used in Section 9.5 which is more flexible as it requires less boundary regularity and allows at(ax least a finite number of

exceptional points on G. Set W :=

1 + ivZ1i2, vZ; =

zi1I, j = 1, 2.

Z;, y

J

Since Z1 and Z2 satisfy the minimal surface equation, we have

div[W,-'VZ1 - WZ'VZ2] = 0 in G. Let G' cc G be a subdomain of G with a smooth boundary 7G'. Then we obtain (p div[ Wi ' VZ1 - WZ 'VZ2] dxdy = 0 SG

for any cp e C1(G), and a partial integration yields