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
analogous regularity results for minimal surfaces with free boundaries on a support surface S. If the boundary as of S is empty, the reasoning is considerably simpler than for as 0 0; in fact this second case has to be viewed as a Signorini problem (or else, as a thin obstacle problem). For a survey of the results on the boundary behaviour of minimal surfaces with free boundaries we refer the reader to Section 7.4. Finally, in Section 7.10, we shall derive an asymptotic expansion for any minimal surface at a boundary branch point which is analogous to the expansion at an interior branch point that was obtained in Section 3.2. The results of Section 7.10 are based on the discussion in Chapter 8. Regrettably, Chapter 7 has become somewhat lengthy as we have tried to give a self-contained presentation of most of the material. The p.d.e.-expert will be able to skim through various parts of our discussion. To obtain a comprehen-

7.1 Potential-Theoretic Preparations

7

sive view of the results, the reader should start by looking at Sections 7.3, 7.4 and 7.8 before studying the details.

7.1 Potential-Theoretic Preparations In this section we want to supply some results from potential theory which will be needed in Section 7.3 for investigating the boundary behaviour of minimal

surfaces which are bounded by smooth Jordan arcs. The reader who is well acquainted with Schauder estimates may skip this part at a first reading. Although a large part of the material can be found in the treatise of GilbargTrudinger [1], a brief presentation may be welcome because it will enable the reader to study the essential results of Section 7.2 on solutions of differential inequalities without consulting additional sources. In what follows we shall use the following notation: We write w = u + iv,

= + iri, d = d + idq, and d2 = dgdrl denotes the two-dimensional area element. Moreover, we set

l

a

a

a

a

a2

+

a2

a

aw-2 au+`av

aw=2(Ou-`av) d=

a

1

= 4 a- -a

,

BR=BR(O)={wEC:IwI
2GR(w, ) =

(1)

21r

log'

R(C - ) = Re"' e aBR, is defined'

and the Poisson kernel 9R(w, (p) = 9R (w, C), w = re`°, by

R2 -r2

I

gR(w' (P)

1

- 27r R2 - 2rR cos(O - (p) + r2 =

(2)

=

I 27r

Red+w

=

C-w

I

_w

I

27r

I2--R 8

w 12

C

27r

Re

-

av;

R+

R-

re`te-40

re`la-w)

GR(w,

where vv denotes the exterior normal to aBR at . One computes that (3)

aw

GR(w, ) =

47r

G1

w

R2

WC

1 Note that often the expression R YR(w, (p) _ -- GR(w, 1') is called Poisson kernel; cf. for instance T11;

Gilbarg-Trudinger [1], formula (2.29).

8

7. The Boundary Regularity of Minimal Surfaces

whence it follows that as a

(4)

aw'

I

1

GR(w, ) =

-

I(C - w)s

4rc

s

(R' -

-

A straight-forward estimation shows that

RIB-wI
for all

s-1' 2n IC _ wIs

for all

which implies as (5)

aws

I

GR(w, C)

w BR with w

The following result is a direct consequence of Green's formula and can be found in any textbook on partial differential equations.'

Proposition 1. Any function x e C°(BR) n C'(BR) with q := Ax e L. (BR) and x((p) := x(Re`') can be written in the form (6)

x(w) = h(w) - J

GR(w, C)q(C)d2C BR

where (7)

h(w) :=

Zz I0

9R(w, (p)x((p)dc

denotes the harmonic function in BR which is continuous on BR and satisfies h = x on aBR.

Proposition 2. Suppose that x((p) is of class C'(i8) and periodic with the period 21t, and let q(w) be of class LA(B). Assume also that

sup I q l< a,

sup WI
B

R

holds for some numbers a, P. Then the function x(w), w e B, defined by (6) and (7)

for R = 1 can be extended to B as a function which is of class C1' (B) for any

R E (0, 1) and satisfies x(e"P) = x(q ). For suitable numbers c1(a, f) and c2(a, fl, p) depending only on the indicated parameters and not on q and x, we have (8)

IVxIo,e <-

c, (a, A

CVx]a,s <- c2(a, A, 9)

-

If q e C°'°(B) holds for some a e (0, 1), then we have x E and the equation Ax = q is satisfied on B. Moreover, for any R, R' with 0 < R' < R < 1 the function y(w) defined by (9)

y(w):=

GR(w, JBR

2Cf. for instance Gilbarg-Trudinger [1], p. 18; John [1], p. 96.

7.1

Potential-Theoretic Preparations

9

is of class C2,Q(BR) and satisfies IYI2+o,BR- < c(R, R',

(10)

o)lglo+7,BR.

Here and in the following we use the notation I

IXIO,B=SBPIXI,

IXI.s,B =

S

x(w

-

)I:

[x]",B=sup { Iw - w( I

w,w'eB,w

w'

,

Ixls+",B = IXIs,B + [Vsx]",B.

IDkxlo,B,

k=0

Moreover, we shall use the notation V. =

0) in order to distinguish the

a

Gu 8v

of a function flu, v) from its complex derivative real gradient Owf = (f Jw = 2(Ju - 'Jv) The reader will find more complete results on Schauder estimates in Gilbarg{{

Trudinger [1], chapters 2-4 and 6; Morrey [8], chapters 2 and 6; Stein [1]; Agmon-Douglis-Nirenberg [1], [2]. We shall use some of these refined results later on. For the present the reader might welcome to see how one can obtain Schauder estimates in the simple situation at hand. Proposition 2 and related results will be proved by a sequence of auxiliary results. Lemma 1. Let H(w, t') be a C2-kernel on the set {w, c e BR : w b

1

(11)

C} such that

IH(w, 0)1 < b log -

IowH(w, C)I <

r

b

IV2H(w, C) I <

,

holds for r = I w - t; I and some constant b > 0. In addition we assume that q e L,,,(BR). Then the function y(w) defined by Y(w) =

(12)

J BR

H(w, 0)q(0)d21;,

w e BR,

can be extended to a function of class C1'"(BR) satisfying IYII,BR < cl(b, R)IRIO,B,,,

(13) (14)

[VY],z,BR

5 c2(b, R, lu)Ig10,BR,

with constants c1, c2 depending on the parameters b, R and b, R, It, respectively. Moreover, we have Vwy(w) =

(15)

f V H(w, ()q(C)d2t' for w e BR . BR

Proof. As 1 e L1(BR) and q e L. (BR), the integrals (12) and (15) are well defined.

r

We choose a cut-off function qh e C`°(08) with 0 < qh < 1, rl,,(r) = 0 for r < h, ?1,,(r) = 1 for r >_ 2h, and n '(r) < h. Then we set

7. The Boundary Regularity of Minimal Surfaces

10

Hh(w,0:=fh(r)H(w,4) withr=1w-CI Then we have IHhl < IHI and H = Hh for r >- 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
IHId2 BRrmB2h(w)

< const h , 0 as h -. 0. Thus we infer that y e C°(BR). Now we define w c- BR.

VwH(w,

z(w) := J BR

We want to show that y E C1(BR) and Vwy = z. In fact, we have VYh(w) = I (w) + 12, (W)

with

I'(w) := J

flh(r)VWH(w, C)q(()d2C,

Vwnh(r)H(w,

Iz(w) J BR

BR

As before we show

ash-+0, and a straight-forward estimate yields IIZ(w)I < const h-'h 2

for any a > 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


- au(w2)I =

I

a (w,, 0 - 8 ,

(w2,

01

C

7.1 Potential-Theoretic Preparations

d2C+al


.) BR'

+a

a (wl, C) IBR-B2p(w)

-

au

2p(WI)

au

(W2, C)

11

d 2 C

(w2, C) d2C.

Note that
-CI-1,

8u


(w2, C)

and the mean value theorem implies 2bp

au (w1, C) - a (w2, C)

< Iw* - CI2

for some w* _ (1 - t)wt + tw2, 0 < t < 1. If I C - wt j > 2p, we infer that

IC - w*I -IC - wll - jwl - w*I> IC - wll, and therefore OH au

(wi, Y)

8H

- at!

(w2, C)

Thus we arrive at ay

8u(wl) -

-U


jw1 -Cj-1d2C B2p(`01)

+ 5B3p(W2)

Iw2 - C1-1 d2C + 8p

< ab 47cp + 6irp + 161rp log

R

P]

J BR-B2p(wl)

Iwl

- CI-2 d 2C ]

< ac(b, R, µ)p"

for any µ e (0, 1) and p = Iw1 - w2j, and (14) is proved. Estimates (13) and (14)

imply that y can be extended to BR as a function of class C""(BR) for any 0 µe(0, 1). Lemma 2. Let H(w, C) be a kernel of the form

H(w,()=K(w-C)=K(u-C,v-rl) for some function K(C) which is of class C2 on {C

0}, and suppose that H(w, C) satisfies the growth condition (11). Furthermore we assume that q(w) is of class C°''`(BR), 0 <,u < 1. Then the function y(w) defined by (12) is of class C2(BR), and we have (17)

Io2ylo,BR, <_ c1IgIµ,BR,

YI2.BR, < X21 gIk,BR

for 0 < R' < R. Here cl and c2 denote constants depending solely on b, µ, R and R'.

7. The Boundary Regularity of Minimal Surfaces

12

Moreover, if

is of class C3 fort * 0 and if also IVx3,H(w, C)l -< biw - 1-3,

(11*)

then y(w) is of class C2 "(BR) and satisfies IYl2+",BR. < c4lgI",BR

IV2YI".BR, < C3IgI".BR,

(18)

for 0 < R' < R. Here the numbers c3 and c4 only depend on b, p, R', and R.

Proof. We set again Hh = rjhH where nh is chosen as in the proof of Lemma yh_2 for some constant y > 0. Then but in addition we arrange that I nZ(r)I < zh(w):= JOR

is of class C'(B R2), and for D = Dzh(w) =

a

au

or

a

we can write

av

DVWHh(w, JBR

=f

DVWHh(w, ()[q(O) - q(w)] d2C + q(w) J

DVWHh(w, BR

BR

By partial integration, we obtain

f

D,VWHh(w, C)d2C

C)d2b =

BR

-fDR

V wHH(w, ) cos ads(C), aBR

If 2h < 1w - (I we obtain

where ds is the line element on aBR and cos a

Dzh(w) = 0h(w) - q(w) faBjt V.H(w, C) cos ads(C)

with cosa=llor = and A(w) := J

DVWHh(w, C) [q(C)

- q(w)]

BR

Similarly we set (19)

DVWH(w, ()[q(()

O(w) BR

Forr=lw-i;landa:=lql",BRwehave

-

7.1 Potential-Theoretic Preparations br_2

IDV.H(w, )I S

and

13

lq(C) - q(w)I < ark,

whence 1 O(w)1, Ioh(w)I <- 2nabp 1Rµ.

(20)

By a similar reasoning we obtain (V = Vw and 0 < It << 1): A(w) - O(w)I <-

f

Inh(w) - 11 IDVH(w, C)I Iq(C) - q(w)I d2 +f"

1

+logh

I-0 ash-0.

Thus Dz,,(w) tends uniformly to

VwH(w, ) cos

O(w) - q(w) J aBR

as h --* 0, for w e BR, and 0 < R' < R. On the other hand, if yh is defined by (16), we know that Dzh = DV yh

zh = VYh,

,

zh e CI

,

and, as shown in the proof of Lemma 1, we also have

limly-yhlt,BR,=O for0
Consequently we have y e C2(BR) and (21)

VwH(w, ) cos «(t")ds(C) for Iwl < R.

DVy(w) = ¢(w) - q(w) J aBR

Now inequalities (17) follow from (20) and (21). Finally, taking assumption (11*) into account, we derive from the representation formulas (19) and (21) that y is of class C2. (BR) and in conjunction with (17) that the estimates (18) are satisfied. Since we may proceed in the same way as in the last part of the proof of Lemma 1, we shall skip this part of the proof. F-I

Proposition 3. Suppose that x e C°(BR) n C2(BR) and that Vx e L2(BR) and Ax e Then we obtain the following representation formulas which are satisfied

forweBR: (22)

(23)

x(w)

a xx(w)

3w

f.. 1 Re' + w)x(C) - I GR(w, C)dx(C)d2 \ f xg) dC (' = - J BR 3w a GR(w, C)dx(C)d2( W)2 2nt J

1

27t

1

aBR ( -

14

7. The Boundary Regularity of Minimal Surfaces

(24)

x (0) = R2

('(25)

x (u, v)dudv - Zn j

u[- R2]dx(uv)dudv

x(u, v)dudv

v

J

Proof. Formula (22) is merely a reformulation of (6) and (7). Differentiating (22), it follows in conjunction with Lemma 2 (in particular, with (15)) that (23) holds if we take

0 C+w

2C

-w)2

+w

0

0 R2-Iwl2

Re w t aw OwC-w2 Ow-

IC-w12

for 4 e OBR into account.

By applying (23) to w = 0 and noting that 8w GR(0, ) =

42c

G RZ)

we infer that xw(O)

21Ci

faBR SS)ds

JBR

nII2 -

)4xg)d2.

Because of t'-2 d( = -R2 d, it follows that 1

r

_

1

x(C)dC

27ciR 2 J 'Bit 1

x(C)(d - idyl)

2niR2

it

1

(-ix4 -

z BR

= _R2 J x;d2C. BR

Replacing t' by w, we arrive at (24) and (25) by separating the real and imaginary parts. Actually we first prove (24) and (25) for BR-, R' < R, instead for BR, and then we let R'-+ R.

Now we prove Schwarz's result concerning the boundary continuity of Poisson's integral. Lemma 3. Let x((p) be a continuous, 2n-periodic function on R, and let h(w) := f o"gR(w, (p)x((p)dgp be the corresponding Poisson integral, which is a harmonic

7.1 Potential-Theoretic Preparations

15

function of w e BR. Then we obtain h(w) -+ x(cp) as w -+ Re"'. Thus h(w) can be extended to a continuous function on BR such that h(Re`') = x(cp) for all (p e R.

Proof. It suffices to treat the case R = 1. Then we have to prove lim h(re`a) _ r-1-0

x(O) for any 0 e R. We can write _ 2

rz

h(re'a) = 27r

f-

el' -rr12 x(0 + (p)d(p.

l"

rz

Because of the identity

f'_

1

I 27r

1 - r2 rzIe`w-rI2dcp= 1

it follows that

h(re`B) - x(0) =

1

2n

_ r2

rz

Iel - r12

_rz

[x(0 + (p) - x(0)] dcp

whence

s

h( re`B) - x(0)I

LIe

_rr121 x(0 +

-a 1

< 2n

)

- x(0)I dcp

a 1

+

2n f-6

-rz

rz

I1+12+13

1

+

27c

b

for any S e (0, 2) Fix some s > 0 and choose 6 > 0 so small that .

x((p) - x(0)I < e for all (p with I (p - 01 < 6. Then we obtain

12<e-2

a

-a

Moreover, by setting M := max Ix I we obtain a

I1' 13 <

< sin2,(1 - r)

_(n - 8)(1 - r)(1 + r)sin2

since Ie`' - r12 >- sin2 S for b < IcpI < x. Thus we arrive at I h(re`B) - x(0)I <_ e +

sin 8(e) (1 -

and therefore

lim h(re`B) = x(0). r-1-0 As a by-product of this proof we have found :

r)

for r e (0, 1)

16

7. The Boundary Regularity of Minimal Surfaces

Lemma 4.

Let h e C°(B) n C'(B) be harmonic in B and suppose that

I h(e") - h(eie)I < E holds for all co with 19 - 61 < 6, b E (O, 2) Then it follows .

that Ih(re`B) - h(eie)I < E +

(26)

41hlo,ae(1

- r)

sin'6

holds for all r e (0, 1).

Lemma 5. Let h e C°(BR) n C'(BR) be harmonic in BR, and suppose that the boundary values x(cp) of h defined by x((p) := h(Reie) are of class C'(l) and satisfy I x"((q)I <_ k for all 9 e R. Then we obtain

loh(w)I < cR-1k for all w e BR

(27)

where c is an absolute constant independent of h and R.

Proof. By virtue of an obvious scaling argument we can restrict our attention to the case R = 1. Then we have to prove I Vh(w) I< const k

for w e B.

Let h* be the conjugate harmonic function to h. Then f(w) := h(w) + ih*(w) is a holomorphic function of w = u + iv, and we have the convergent power series expansion CO

f(w)=Ec,w' forawl<1. !=o

Set co = I(ao - ibo), c, = a, - ib, if l >- 1, a,, b, a R. Then we have for w = re's that h(w)

2+

r'(a,cosl(p +b,sinIrp)

whence a

x((p) = 2 + 1

a, =

(a, cos lcp + b, sin l(p) ,

fO2,,

fO'

1

x(rp) cos lrpd(p = -BIZ

1f n1 2a

n

x"((p) cos lkpdcp

O2

b, =

x(g) sin lcpdcp = - n1 2

f

o

Because of f' = h + ih,* = h - ih we infer that 00

loh(w)I = If'(w)l =

l=1

l(a, -

ib,)wl-1

x"(q) sin lcpdrp.

Potential-Theoretic Preparations

7.1

II

<_tCO

"&p)2dcp]tz

ai +b?< f OD

l

17

z}12[fO2 ))

o

< k iflwl<1, taking Schwarz's inequality into consideration as well as Parseval's relation for the Fourier series expansion of x". The next result is known in the literature as Theorem of Korn and Privalov.

Lemma 6. Let f(w) = h(w) + ih*(w) be holomorphic in BR, h = Re f, h* = Im f, and suppose that h* e C°(BR) n C°-"(8BR) holds for some y e (0, 1). Then f is of class C°'"(BR) and we have (28)

[f]",aR
Proof. We can assume that R = 1 applying a scaling argument. Set H Then we have (29)

[h*]", as

I h*(eto) - h*(e`()l < Hl e`a - e"°I"

for all 9, rp e R. Fix some cp e [0, 2n) and consider the function

,fr(w) = Re(1 - we-`')" which can be viewed as a univalent harmonic function of w e B. Introducing the angle a between the rays {te'' : t > 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"°) <

cos--

An cos 2

holds for all w E B. Applying the maximum principle, we obtain that (30) holds for all w E B and in particular for all w e Bi_r(rej') if 0 < r < 1. Hence we infer that (31)

(1 - r)" for 1w - re`Pl < 1 - r

Ih*(w) - h*(e`')I < cos

is satisfied.

Now we set w°:= re`s, 0 < r < 1, and ho := h*(wo). Applying Gauss's mean value theorem to the harmonic function h, and to the ball BB(wo) with some radius p e (0, 1 - r), we obtain

18

7. The Boundary Regularity of Minimal Surfaces

npnp B,(wo) (h*-h)dzw,

hu(wo)=12 JBP(wO) h(w)d'w= 1f and a partial integration yields

u-uo(h*-ho*)ds.

1 nP2

h.(wo)=

L,(wo)

P

Analogously, hV(wo) =

v - vo(h*

1 2

P

- h*)ds.

P

J aB,(wo)

Thus we obtain IVh*(wo)I < 2p-' Ih* - ho lo, aB,(wo)

(32)

If we let p --* 1 - r and combine the resulting inequality with (31), it follows that 4H

IVh*(wo)I <

(1

n

- r)

1+'

.

cos 2

Since we can choose rp arbitrarily, we obtain that

If'(w)l <` c(y)H(1 -

(33)

Iwl)-1+µ

for all w e B,

if we set c(µ) := 41 cos 2 I-1. Then it follows from (33) for 0 <- r < 1 that

I f'(w)I < c(p)H(r -

(33')

Iwl)-1+µ

for all w e Br .

For any r and r' with 0 < r < r' < 1 we now conclude ('r era

f'(w)dw < c(p)H

If(r'ew) - f(re`o)I = J

('r

(r' -

re1B

whence (34)

I f(r'e'e) - f(reie)I < c(u)AC'H(r' - r)" for 0:!9 r < r' < 1.

We infer that lim f(re`B) exists for any 0 e R. Setting (q') := lim fire") we

"-0

r-+1-0

extend f(w) from B to B by defining f(e4) :_ (cp). We now want to show that f e C°'"(B). In fact, setting c*(,u) := µ-' c(µ) we obtain from (34) that (34')

I f(r'e`B) - f(re`B)I < c*(p)H(r' - r)"

for 0 < r < r'< 1, 0 e ff8,

and in particular

I (B) - f(reie)I < c*(p.)H(1 - r)" for 0 < r < 1 Then it follows for 01 < 02 that

and 0 e R.

7.1 Potential-Theoretic Preparations

l

(0l) - (02)I -< <-

19

(0r) - f(re`o')I + I x(02) - f(re`B')I

I

I x(01) - f(re'B')I + 1 (02) - f(Ye'BZ)I + I f(re`B') - f(re`B2)I o2

< 2c*(µ)H(1 - r)" + J

1 f'(re'o)I rdO. e,

Moreover, we derive from (33) that 02

fo

f'(re'B)IrdO < Hc(µ)r(1 - r)-1+"(02 - 01)

Suppose that 0 < 02 - 01 < 1, and choose r = 1 - (02 - Ol ). Then it follows that {2c*(u) + c(µ)}H102 - 011"

if 101 - 021 < 1. Renaming 8c*(µ) + 4c(µ) by c(µ), we arrive at S(02)I < c(µ)HI01 - 021" for all 01, 02 E R.

(35)

Applying the maximum principle to the modulus of the holomorphic mapping f (e'aw) - f (w), w E B, we see that

max If(e'aw) - f(w)I < max If(e`aw) - f(w)I we,B

weB

holds for all a c- R, and in view of (35) we obtain

I f(e'aw) - f(w)1 < c(µ)HI al"

for w E B and ac-R.

This estimate is equivalent to (35')

If(re`o2) -f(re`o')I < c(u)H101 - 021"

for 0 < r < 1, 01, 02 E R.

Combining the estimates (34') and (35') we arrive at

If(r1e"') -f(r2e`o2)I < If(r1e`o')

-f(r2eiet)I

+

If(r2eie,) -f(r2e`o2)I

< c*(u)HI r1 - r21" + c(µ)H101 - 021"

for arbitrary rl, r2 E [0, 1] and 01, 02 E R.

If wl = rle'o', w2 = r2e'B2, 2 < rl, r2 < 1, lee - 01I < 1r, then there is a constant K such that rl - r21" + 101 - 02 I" < K I w1 - w2 I" .

Consequently we have

If(wr) - f(w2)I <- c(y)HI w1 - w2

for all w1, w2 e B - B112 and for some constant c(µ), and because of (33) the same estimate holds for any wl, w2 E B112. Then we easily infer that

If(wl) - f(w2)I -< c(p)HI wl - w2 I" holds true.

for all wl, w2 E B

20

7. The Boundary Regularity of Minimal Surfaces

Lemma 7. Suppose that h e C°(BR) n C2(BR) is harmonic in BR and that its boundary values x((p) := h(Re`9) satisfy x e C2(R) and I x"(cp)I < k for all cp e R. Then we obtain h e C',"(BR) for every y e (0, 1) and [Vh]µ,a,e <_

(36)

c(u)R-1-"k

where the number c(µ) only depends on ,u.

Proof. It is sufficient to prove the result for R = 1. Let us introduce the tangential difference quotient (Teh)(re`4') :=

[h(re`(w+e)) - h(re` °)]

and note that (Teh) (w) is a harmonic function of w e B which is continuous on B and has the boundary values (TBx)(4) := 8 [x(cp + 0) - -((p)7

By assumption the boundary values (Tex)(4?) tend uniformly to x'((p) as 0 - O_ Then, on account of Harnack's first convergence theorem, we easily infer that the functions (Teh)(w) tend uniformly on B to the harmonic function hq,(w) with the boundary values x (cp) = a -h(e`') on 8B which, by assumption, are Holder continuous for any exponent y < 1, and (37)

IX'],, R <- 2nk.

Consider a holomorphic function f(w) on B with f = h + ih*, that is, h = Re f,

h* = Im f. Then g(w) := iwf'(w) = of (w), w = re`*, is another holomorphic function on B with

Re g and x'((p)

(e`4)), x e C°'"(ll) for any u E (0, 1

Hence we can apply Lemma 6 to the holomorphic function ig(w) _ w e B, and we obtain that ig(w) is of class C',"(B). This implies f' E C°. "(B - B112),

and inequalities (28) and (37) yield (38)

[f']",B-B,12 < const - k.

Moreover, (27) implies If'Io,B <- const k,

and Cauchy's estimate for holomorphic functions then gives

If"Io,BZ <- constk,

-wf'(w),

7.2 Solutions of Differential Inequalities

21

whence

[ f']µ B, < const k. Combining this estimate with (38), we arrive at the desired inequality

[ f']µ B < const k. If we now recall that f' = h. - ih, we find that the lemma is proved.

11

Proof of Proposition 2. We now see that Proposition 2 is a direct consequence of Lemmata 1-7 in conjunction with Proposition 1 and with formulas (1)-(5).

Remark. We have formulated the estimates and the regularity results of Proposition 1 in a global way. Analogous local results can be derived by similar methods, but certain changes will be necessary to obtain local estimates at the boundary. A very simple approach to local C1'"-estimates is based on a "reflection method"; it will be described in the next section.

7.2 Solutions of Differential Inequalities In this section we want to derive a priori estimates for solutions X(u, v) = X(w) _ (x' (w), x2(w), ..., x"(w)) of differential inequalities

I4XI < alVXI2,

(1)

which can equivalently be written as (1')

IXwwI 'G

a denotes a fixed nonnegative constant.

Lemma 1. Let X E C2(0, Di') be a solution of (1) in the open set 52 of satisfies IXlo.,2 < M. Then we obtain (2)

AIX12 > 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

<

IVXI dude

14)

max

I X (w) - X (w°) I

R 1 - aM we mt(wo)

log -

Bp(w0)

P and

4rtMz

1

IVXlzdudv 5

(5)

R1

am

B,(wo)

log P

Section 7.1 to the Proof. Choose some p e (0, R) and apply Proposition I of domain BR(wo) instead of BR = BR(0), assumfunction x(w) := IX(w)I2 and to the C2 on BR(wo). Then formula (6) of Section 7.1 ing in addition that X is of class yields I

Jo"' [x(w+ Re") - x(wo)] d=

R IBR(WO )

log Iw - woI

Axdz w.

Because of

Ix(w) - x(wo)I < 2MIX(w) - X(wo)I we infer that log

I

27t

Axdzw <-2M max IX(w) - X(wo)I

R

we3BR(wo)

IW --w 0 l

BR(wo)

On the other hand, Lemma 1 gives

2(1 - aM)IVXI2 < Ax, whence

(I - aM)n-' J

log BR(WO)

IVXIZdzw R 1w - wol

< 2M max IX(w) - X(wo)I. we BR(WO)

Moreover,

logR f P

B,(WO)

IVXI2d2w <

log BR(wo)

R IVXlzdzw 1w - wol

if 0 < p < R, and (4) is proved. The additional hypothesis can be removed if we first apply the reasoning to p and R' with 0 < p < R' < R, and then let R'- R - 0. Inequality (5) is a direct consequence of (4). Proposition 1. There is a continuous function K(t), 0 < t < 1, with the following property: For any solution X e C2(BR(wo), R') of the differential inequality (1) in

7.2 Solutions of Differential Inequalities

23

BR(wo) satisfying

X(w)I < M for w e BR(wo)

(6)

and for some constant M with aM < 1, the estimates IVX(wo)I < K(aM)

(7)

R

and

IVX(wo)I S

(8)

K(Ram)

sup

I X(w) - X(wo)I

w e BR(wo)

hold true.

Proof. Fix any R' e (0, R), and consider the nonnegative function

f(w):=(R'-Iw-wol)IVX(w)I on BR.(wo) which vanishes on 8BR,(wo). Then there is some point w1 e BR,(wo) where f (w) assumes its maximum K, i.e.,

f(w,) = K := max { f(w) : w e BR,(wo)}. Set r = I w - w1 I for w = BR.(wo) and p := R' - I w, - wo I. Clearly, we have

0
By formulas (24) and (25) of Section 7.1, we obtain for any 0 e (0, 1) that 2o2

Xdudv

J

2r

Bpe(w)

and, an analogous formula holds for we infer that

By means of Schwarz's inequality t/z

IVX(w1)I -<

1

p9 Bvq(wl)

p9

+

IVX12dudv

i

2it B,9(w)

IdXldudv. r

Applying Lemma 2, (5) to pB and p instead of p and R, we also obtain IoXlZdudv <

J

c(a, M)

Bve(w ,)

log

1

9

Taking 14X1 < a I VX I2 into account, we arrive at

(+

IVX(wi)I <

/pO log e)1/Z

af 2i

IVXIZdudv

r

24

7. The Boundary Regularity of Minimal Surfaces

and

a

lIVXlzdudv
27z

IVXI2.

Be(wi)

BO(w1) r

On account of

K =f(wt) = pIVX(wt)I we obtain

IVX(wi)I = K/p Moreover, if r = 1 w - wt I < p6, it follows that

R'-lw-wol>R'-Iwo-wll-lw-wtl=p-r>(1-6)p. Thus we infer from

IVX(w)l(R' - lw - wol) < K for all w e BR.(wo) that IVX(w)I < (1 K6)P

for all w e Bpe(wt)

holds true. Thus we conclude that a

1

27r

r

B

IVX12dudv

f

whence

K

<

a6K z

0 - 6)zP aOK 2

1/2

+(1-6)zp,

"p6(109 e ) and finally

c/

K<

a6Kz

tiz + (1 - 6)z log l

6/

0C Set aO6)z

a(6)

,

(1

/c17

fl(6):=

t/z

6 l log B l

Then we have

aKz-K+fi>-0,

7.2 Solutions of Differential Inequalities

25

or equivalently 1

1 - 4al

2

CK - 2a

>

4x2

.

Note that

a IC/"

0400 0) =

c = c(a, M).

1)1/2

(1 - 0)2 Clog

6

Hence there exists a number 00(a, M) e (0, 1) such that

4x(0)/3(0)
i

4x(0)/3(0)

if 0 < 0 <- 00

.

Set _

(0)._

m

1-

1 - 4x(0)/3(0)

+

m+(0)

1 - 4x(0)/3(0)

2x(0)

2x(0)

Then we infer for any 0 e (0, 00] that either

(i) K < m-(0),

or

(ii) K >- m+(0)

holds true.

Moreover, the functions m-(0) and m+(0) are continuous on (0, 00] and satisfy

m-(0) < m+(0)

for 0 < 0 < 00

and

lim m+(0) = oo .

0-+0

The last relation yields that case (ii) cannot occur for 0 close to zero; hence we have K < m-(0) for 0 near zero, and a continuity argument then implies

K<m-(0) forall0a(0,00], in particular, K < m-(00). Finally, for 0 e (0, 00], we also obtain that

m-(0) =

1-

1 - 4x(0)/3(0)

_

2/3(0) 1 +

1 - 4a(0)p(0)

1 - (1 -

1

2x(0)

2x(0) 1 +

-

2/3(0) 1 +1/2

whence

m (00) <

33(00)

1 - 4a(0)fl(0)

- 4 Q(0) 3

26

7. The Boundary Regularity of Minimal Surfaces

Consequently, K < 3 fl(O0) =

4

c/1 li2

:= c*(a> M).

300 I loge o

Because of

R'IVX(wo)I =f(wo)
IVX(w0)I < c*(a, M)/R'

for any R' E (0, R)

whence

IVX(wo)I <- c*(a, M)/R.

(7*)

This estimate is close to (7). We now introduce the function K(t) := c*(t, 1). A close inspection of the previous computations shows that K(t) can assumed to be an increasing and continuous function on the interval [0, 1). In order to prove (7) we assume that M > 0 because this inequality trivially M_1 holds true if M = 0. Then Z(w) := X(w) satisfies both IZ(w)I < 1 and 1421 <

aMIVZ12.

Applying the estimate (7*) to Z, we arrive at IVZ(wo)1 < K(aM)/R.

Multiplying this inequality by M, we obtain (7). Estimate (8) is now an easy consequence of (7). To see this we introduce the quantity m := sup {IX(w) - X(w0)I: w e BR(wo)}.

If m = 0 or m = oo, the estimate (8) is true for trivial reasons. If M < m < oo, (8) follows directly from (7). If 0 < m < M, we introduce Z := m-1 [X - X (w0)] and obtain as before IVZ(wo)I <_ K(am)/R < ic(aM)/R,

and this implies (8).

El

Corollary 1. Suppose that X e C2(BR(w0), R''') is a solution of (1) in BR(wo) satisfying

IX(w)I < M for w e BR(w0) and for some constant M with aM < 1. Then we have (9)

I VX(w)I < K(aM)

M P

for all w e BR_p(wo),

0 < p < R.

7.2 Solutions of Differential Inequalities

27

Proposition 2. Let X E C°(BR(wo), RN) n C2(BR(wo), 1N) be a solution of the differential inequality (1) in BR(wo), and suppose that IX(w)I < M holds for all w E BR(wo) and for some number M with 2aM < 1. Moreover, set x(w) := IX(w)12, and let H E C°(BR(wo), R') n C2(BR(wo), RN) and h c- C°(BR(wo)) n C2(BR(wo)) be the solutions of the boundary value problems (10)

dH = 0

inBR(wo),

H = X on 8BR(wo),

Ah = 0 inBR(wo),

(11)

h = x on 8BR(wo).

Then for any w e BR(wo) and w* E 8BR(wo) we have the inequality (12)

X(w) - X(w*)

2(1

a2aM) h(w) - h(w*)l + 11- 2aMI H(w) - H(w*)I

Proof. Inequality (2) implies IVX12 <2(1

1aM)dx,

which in conjunction with 14X1
J.XI <

a

2(1 - aM)

Ax.

Pick some constant vector E E 6FN with I E I = 1 and consider the auxiliary function z E C°(BR(wo)) n C2(BR(wo)) which is defined by z(w) .

2(1

a aM) [x(w) - h(w)] +

and vanishes on 3BR(wo). Because of

dz = >

a

2(1 a

2(1-aM)

aM)dx -

dx-JdXI>-0,

we see that z is subharmonic on BR(wo). Then the maximum principle yields

<2(1 aaM)[h-x] onBR(wo) for any unit vector E of 1V.", and we conclude that (13)

IH-XI<2(1 aaM)[h-x]

28

7. The Boundary Regularity of Minimal Surfaces

holds on BR(wo). Moreover, the inequality IX(w)I < M in conjunction with the maximum principle gives

IH(w)I < M for all w e BR(wo). Then we obtain

h-x=(IHI2 -IXI2)+(h-IHI2) <2M(IHI - IXI) + (h - HI2), whence

IH-XI <1 aamIH-XI+2(1 aam)(h-HIZ). Since

am -1-2aM<1, 0<1- 1-aM 1-aM it follows that IH - X I

(14)

-<

2(1

a2aM)(h - IHI2)

on BR(wo).

For w* e aBR(wo) we have

IX(w*)12 = x(w*) = IH(w*)I2 = h(w*),

and therefore

IX(w) - X(w*)I <- IX(w) - H(w)I + IH(w) - H(w*)I

<

a

IH(w) 2(1 - 2aM)(h(w) - IH(w)I2) +

- H(w*)I.

Because of

h(w) - IH(w)I2 = h(w) - h(w*) + IH(w*) 12 - IH(w)I2 Ih(w) - h(w*)I + 2MIH(w) - H(w*)I, we may now conclude that (12) holds for any w E BR(wo) and for any w* E aBR(wo).

Remark 1. Note that the differential inequality (1) remains invariant with respect to conformal transformations of the parameter domain. Thus we can carry over

Proposition 2 from BR(wo) to any bounded domain 0 in C which is of the conformal type of the disk and has a closed Jordan curve as its boundary.

-

Proposition 3. For any a >- 0, R > 0, M >- 0, and k> 0 with 2aM < 1, there is a number c = c(a, R, M, k) >- 0 having the following property:

7.2 Solutions of Differential Inequalities

29

Let X e C°(BR(wo), RN) n C2(BR(wo), RN) be a solution of (1) in BR(wo) satis-

fying IX(w)I < M for all w e BR(wo). Suppose also that the boundary values X(9):= X(wo + Re`') are of class C2(IR) and satisfy I.I"(qi)I < k for all cp E R. Then we have

IVX(w)I < c(a, R, M, k) for all w c-BR(wo).

(15)

Proof. It suffices to treat the case wo = 0 and R = 1, that is, we consider the parameter domain B = B, (0). Let w = reie, 0 < r < 1, be an arbitrary point of B. By formula (8) of Proposition 1 we have 1VX(w)I <

(16)

c(a,

M) sup {IX(w') - X(w)l: w' c B,_,(w)}.

Moreover, for w, We B and w* e 8B it follows from Proposition 2 that IX(w) - X(w')I <- I X(w) - X(w*)I + IX(w') - X(w*)I 2(1

(17)

a2aM) {Ih(w) - h(w*)I + Ih(w') - h(w*)I}

+ 1 -am [I

1 - 2aM

H(w) - H(w*)I + I H(w') - H(w*)I ]

holds true where H and h are harmonic in B and have the boundary values X and x := IXI2 respectively on B. By Lemma 5 of Section 7.1 we obtain that IVH(w)I < ck

for all w e B

whence

I H(w,) - H(w2)1 < ckIw, - w21

for all w,, w2 E B.

Therefore we have

IH(w) - H(w*)I + IH(w') - H(w*)I < 3ck(1 - r)

(18)

for w = rete, w* = e`9, w' e B,-r(w).

Furthermore, the boundary values >)(cp):= I-'(9)I2 of x(e`') satisfy rl" _ 21-T'12 + 2<.

',

'">, hence

In"I <21T' 12 +21<,T,. '"> <21X'12 + 2Mk. Let E e R' be a constant unit vector. Then we have 2n

<E, t'((q)>dr0 = 0.

J0

Consequently, there is some cpo E [0, 21r] such that <E, X'((po)> = 0

30

7. The Boundary Regularity of Minimal Surfaces

and therefore <E,.T'(ro)> = J

<E,9",((p)>d(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 <- 8n2k2 + 2Mk. Then we infer from Lemma 5 of Section 7.1 that

IVh(w)I
Ih(wl) - h(w2)I < c*(1 + k2)Iwl - w2I

for all w1, w2 e B

and consequently

Ih(w) - h(w*)I + Ih(w') - h(w*)I < 3c*(1 + k2)(1 - r)

(19)

for w = re`°, w* = e°, w' e B, -,(w). Combining (17), (18), and (19), we arrive at

IX(w)-X(w')I
IVX(w)I < c(a, M, k) for all w E B, taking (16) into account.

Theorem 1. Suppose that the assumptions of Proposition 3 are satisfied. Then X is of class C''(BR(w°), 18N) for all p e (0, 1) and we have [VX],.EA(wo> <- c(a, R, M, k, p).

(20)

Proof. This result is an immediate consequence of Proposition 2 of Section 7.1 in conjunction with Proposition 3 that we have just proved.

Now we come to the proof of the most important result with regard to the next section. Theorem 2. For w° e 8B, we introduce the set So(w°) := B n Bp(w°). Assume that, for some p e (0, 1), X E C°(SS(w°), RN) n C2(Sp(w°), RN) is a solution of the dif-

7.2 Solutions of Differential Inequalities

31

ferential inequality (1) in SP(wo) that vanishes on aSP(wo) n B. Then we obtain X C- C1""(SP,(wo), LN) for every y e (0, 1) and every p' e (0, p).

Proof. It suffices to show that for any w* = eie e aSP(wo) n aB there is a S > 0 such that X e C1,1(S6(w*), 68 N), where Sa(w*) denotes the circular two-gon B n Ba(w*). We may also assume that a > 0. Thus, having fixed an arbitrary w* = e`° E aB n aSP(wo), we first choose an e > 0 such that S3e(w*) e Sp(wo) holds and that sup { IX(w)I : W E S3e(W*)} < 4 . a

Then the mapping

Z(w) := 4aX,

w e S3e(w*),

satisfies the inequalities

IZI < 1

and

IAZI <-

41VZI2

in S3e(w*).

We now consider the functions H(w) and h(w) which are harmonic in S3e(W*) and which have the boundary values X and IXI2 respectively on aS3,. As h and H vanish on the circular arc C := aB n aS3e(w*)

we can extend h and H to harmonic functions in B3e(w*) by reflection at C, applying Schwarz's reflection principle. Hence there is a number c(e) such that IVH(w)I + Vh(w)I < c(e)

for all w e B2E(w*)

whence (21)

IH(w1) - H(w2)l + Ih(w1) - h(W2)1 <- c(E)Iwl - W21 for all w1, w2 C B2e(w*).

Fix some w = re`' e SS(w* ). Then we have I w I = r > 1 - E and, for I w - W1 < 1 - r, we have 1 w' - w* I < I w' - w l + 1w - w* I < E + E = 2s, and consequently B1-,(w) C S2e(W*) C S3e(W*) C Sp(WO)

By Proposition 2 and the subsequent Remark 1 we obtain for Q := S3e(w*),

We 0and e`0eaBnS3e(w*) that IZ(w')l = IZ(w') - Z(e`4)I < Ih(w') - h(e")I + H(w') -

iI In connection with (21) we infer for any w' a BI _,(w) that

IZ(w')I <'-ac(a)I w' - e"PI < ?c(e)(1 - r) < 4c(e)(l - r).

32

7. The Boundary Regularity of Minimal Surfaces

In other words, we have sup {IZ(w')I : w' c- B, _jw)

r)

4c(s) (I

for any w e SE(w*) with wl = r. Moreover, we infer from Proposition 1, (8) that

IVZ(w)I <

21

(1/4) sup {IZ(w')I : w' e B,(w)}

for any w e SE(w*) with wl = r. This implies

IVZ(w)l < c*(e)

for all w e SE(w*).

Since X = 4a Z, we conclude that I dX IO,s,(W*) _< const,

and

I VXIO.S,(w*)

< const.

Now we choose a cut-off function rl e 0(682) with q(w) = 1 for w e Ba(w*), S := 2, and with q(w) = 0 for I w - w* I >_ 4r. Then the mapping Y:= qX on SE(w*) satisfies

dY=s14X

+dr1 X

and therefore (22)

Id Y(w)I < const for all w e SE(w*)

(23)

Y(w) = 0

(24)

Y(w)=O for all w e SE(w*) with

on 8SE(w*)

< Iw - w*I < s.

4s Consider a conformal mapping r of the unit disk B onto the two-gon SE(w*).

We can extend r to a homeomorphism of B onto SE(w*), and it can be assumed that C = ± 1 are mapped onto the two vertices of the two-gon. By the reflection principle the mapping is holomorphic on ,6 - { -1, 1 }. Then it follows from (22) and (24) that the mapping Y*(C) := Y(r(C)) is of class C°(B, RN) n C2(B, RN) and satisfies

const

for all e B.

Moreover, we infer from (23) that

Y*(C)=0 for all CeBB. Thus we can apply Proposition 2 of Section 7.1 and obtain that Y* e C',"(B, RN) for any It e (0, 1). It follows that Y e C',1'(SE(w*), 18N), and therefore X E C',µ(Sa(w*), 681) as Y(w) = X(w) for all w e SB(w*), 6 = 2.

7.3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

33

7.3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs In this section we want to investigate the boundary behaviour of minimal surfaces at smooth Jordan arcs. The results can be applied to minimal surfaces

bounded by one or several Jordan curves, to solutions of the partially free boundary problem, and to solutions of the thread problem (see Chapters 6 and 8-10). As all results are of local nature, it suffices to formulate them on simply connected boundary domains, say, for minimal surfaces X : B --> 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 <- s

for ItI < to

and

(6)

Ix3(w)i < to

for w e SR(1).

Consider the auxiliary function Y(w) = (y'(w), y2(w)) which is defined by yk(W):= xk(w) _ gk(x3(w)),

(7)

w e SR(1)

k = 1, 2,

where X(w) = (x'(w), x2(w), x3(w)). Clearly, we have Y E C°(SR(1), 482) n C2(SR(1), 482) and Y(w) = 0 for w e 8B n 8SR(1). Moreover, we infer from (1) and from the relations (8)

dyk = Axk

- 9k(x3)dx3 - gk(x3)IVX312,

k = 1, 2,

that

JAYI < aivXI2

(8*)

holds for some constant a > 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
1I <s.

Consider a conformal mapping T of the unit disk B onto the two-gon Se(1). We can extend r to a homeomorphism of B onto Se(1), and it can be assumed that C = ± 1 are mapped onto the two vertices of the two-gon. By the reflection

7.3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

37

principle the mapping is holomorphic on .6 - { -1, 1 }. Then it follows from (23) and (25) that the function y(t(C)), e B, is of class C'(B) n C2(B) and satisfies

Ay*I
E C "(OB);

here the radial derivative y* is the normal derivative of y* on B. According to Section 7.1, Proposition 2, the solution of the boundary value problem

dp=dy*

in B,

p=0 onBB

is of class C'""(B) for any u e (0, 1). Hence h := y* - p is of class C'(B) n C2(B), harmonic in B, and h, is of class C°,,' on 8B. Therefore the conjugate harmonic function h* with respect to h is of class C' (B) too, and the equation h, = h* on aB implies that h* la, is of class C. Applying the Korn-Privalov theorem (see

Section 7.1, Lemmata 6 and 7) we infer that h e C'""(B), and therefore also y* e C',"(B). Returning to y = y* o T-' it follows that y E C','U(Se(1)). Since y(w) = x'(w) holds true for w E 9a(1), S = 2, we finally arrive at x3 E C'-(S6(1)),

and therefore X E C''"(Sa(1), R3) for any p n (0, 1). This concludes the proof of the first part of the theorem. (ii) The initial step (i) is the crucial part of our investigation whereas the further proof is essentially potential-theoretic routine. However, as our estimates in Section 7.1 are not quite complete, we only want to indicate how one can proceed. The reader should use the Schauder estimates (described for example in Gilbarg-Trudinger [1]) to derive higher regularity by "bootstrapping". and that r E C2"" for some p e (0, 1). Thus let us assume that H E Then .mo(w) := H(X(w)) is of class C°-"(B v y) and, using the notation of (i), the functions g' (t) and g2(t), I tI < 2t°, are of class C2.". On account of (8) and (2) the mapping Y satisfies d Y = Q

in SR(1)

(26)

Y=O on 8B n 8SR(1) with Q e C°'"(SR(1), R2).

Then a potential-theoretic reasoning yields Y e C2."(SE(1), 682) for O < s < R. Now we use the equations (cf. (1) and (20))

Ax' = 2H(X)(x,',x2 - xIx2) in SR(1) (27)

x: = - k pk(x3)y, on 8B n SR(l) k=1

to prove by a potential-theoretic argument that x3 e C2."(Se(1)) for 0 < s < R.

We only have to note that p' (x3)y; + p2(x3)y; is of class C'-(y') for y' _

38

7. The Boundary Regularity of Minimal Surfaces

8B n 8SR(l) because of the result for Y that we obtained before. By virtue of (7) we then infer X e C2'"(SE(1), 183), and therefore X e C2."(B u y, 183). Remark 1. Similarly one proves X e C',"(B u y, 183) as claimed in Theorem 1 if F e C'"," and H E Cm-2."(183). The proof is carried out by a bootstrap reasoning, considering the boundary value problems alternatingly. Since a similar idea is developed in detail in the following sections, we want to omit the proof of higher boundary regularity of X except for proving analyticity in the case that 1' is a real analytic, regular arc. This will be done next for a minimal surface. We shall present H. Lewy's regularity theorem. In the following we shall suppose B to be the semidisk {w = u + iv : I w I < 1, v > 0}, and I will denote the straight segment {u a R: I u I < 1} on the boundary of B.

Theorem 3. Let X e C°(B u 1, 183) n C2(B, 183) be a minimal surface which maps

I into a real-analytic and regular Jordan arc F in 183. Then X can be extended analytically across I as a minimal surface. Proof. Let X*(w) be an adjoint minimal surface to X(w) = (xl(w), x2(w), x3(w)) which is assumed to satisfy AX = 0 and (2) in B, and let (28)

f(w) = X (w) + ix *(w) _ (f t (w), f 2(w), f 3(w))

be the holomorphic curve in C3 with X = Re f and X* = Im f satisfying

= 0.

(29)

By Theorem 2 we know already that X e C2(B v I, 183) holds true. We have to show that for any u° e I there is some S > 0 such that f(w) can be extended as a holomorphic curve from Sa(u°) := B n B6(u°) to Ba(u°). Without loss of generality we can assume that u° = 0. Set Ba := Ba(0) and Sa = S6(0) = B n B. As in the proof of Theorem 2 we can arrange for the following: x1(0) = x2(0) = x3(0) = 0, i.e., X(0) = 0.

For sufficiently small p > 0, we can represent F n X,(0) in the form (30)

x1 = g'(t), x2 = g2(t),

x'=t,

t e I2Ro

where I8 := {t a R: I tI < 61, and gl (t) and g2(t) are real analytic functions on IZRO, R° > 0. Hence, choosing R° sufficiently small, we can assume that and are holomorphic functions of C e B2Ro; hence

g(() :=

g2(C), (),

ICI < 2R°

is a holomorphic curve on B2Ro. In addition, we may (in accordance with X(0) = 0) assume that

g1(0)=g2(0)=91(0)=92(0)=0

7.3 The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs

39

and z

2

2

< 2 for ICI < 2Ro

d

dC

are satisfied, and that If 3(w)I < Ro holds for w E SR

where R is a sufficiently small positive number. Consider now the holomorphic function

F(w, ) :=

(31)




of (w, ) e SR x BR,,, and note that F is of class C1 on SR x BR,, (of course, differentiability in the second statement is real differentiability). We claim that the differential equation xu(u) = F(u, x3(u))

(32)

for u E IR

holds true. In fact, the boundary condition X(1) c F together with the above normalization implies that X(u) = g(x3(u)) for u e IR,

(33) hence

X.(u) = g'(x3(u))x' (u) for u c- IR.

(34)

0 or 0 for some u e I. yields Xu(u) = 0, and (2) gives X.* (u) = 0 and consequently f'(u) = 0; therefore we also have F(u, x3(u)) = 0, and (32) is trivially satisfied. Thus we may now assume that x, (u) 0 0. Because of (34) and (2), we obtain on IR: Therefore

x = = <Xu, f'>

=<X.,X.+iX.>=1Xu12-r<Xu,X0>-IXu12 = (xu)2

whence

x3(u)

x3

<9 (x3), 9'(x3)>

(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 <

on SR.

m3 E Ioxa12 a=1

that the mapping As in the proof of Theorem I we infer from (43) and (45) y(w) := (x' (w), x2(w), ... , xn-1(W)) is of class C°(SR) n C2(SR) and satisfies the relations

on SR,

I4Y1 < m4jVYI2 (46)

Y=0 ony'.

Now we proceed as in the proof of Theorem 2. In fact, from (46) we infer that is of class C°(Sj). for any s e (0, R), whence (45) implies that Ye Therefore we obtain X e C'(S,). Moreover, from (45) and (462) it follows that iwx," +

San

lwxN

=

9nn

2

x'. +

gan

xl +

x + San xW 9nn

9nn

g2

=

o (Wxw)(Wxw) g nn

>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
C

((X, Y, Z):

z=0,x0, -e-x
E±

{(x, y, Z):

x>O,y=±1,-5
G:= {(x, y, z):

x>0,-l
x>O,y= ±e-x} Q+:={(x,y, -3): x - O,y= ±1}. Y± := {(x, y, 0) :

Now we connect each point (x, ±e-x, 0) on yt by straight segments with the corresponding point (x, ± 1, -3) on /3+, thus obtaining two ruled surfaces F+. Let

S, := E+ u E_ u F+ u F_ u G and denote by S* the reflection of S, at the plane {x = 0}. Then we define

S:= S, us* and

T:={(x,y,z):x=0,z=0, -1
on B u T. Then the trace Y(I) is compact and has to pass G continuously as Y(I) c S. By a projection argument we infer that the area of the part of Y(B) below the plane {z = - 31 is greater than or equal to the area of B2 which is 4, and thus it is larger than the area of C which is 2. Thus each solution Y of 9(1', S)

must have a discontinuous trace Y11. In fact, Y11 cannot be contained in the subregion S. := A',(0) (0) n S for any R > 0. (Otherwise, by Theorem 2 of Section 7.5, we would obtain Y E Co 1L(B, R3) for some µ > 0). Hence it follows that the

trace Y11 is unbounded, and a projection argument shows that Y has to be a parametrization of C or of its reflection C* at {x = 0}. In other words, if there is a solution Y of 9(1', S), it will be given either by C or by C*. As the existence theory of Chapter 4 yields the existence of a solution of 9(1', S), we infer that C and C* are the two solutions of 9(1', S) and that there is no other solution of this minimum problem.

By reflecting S at the plane {z = 01 we can extend it to a Lipschitz continuous noncompact surface S without boundary. Furthermore, by rounding off the edges of S and of S, we can even construct examples of smooth supporting

7 4 The Boundary Behaviour of Minimal Surfaces at Their Free Boundary

45

surfaces, with or without boundary, having the desired property that there is no solution Y E C°(B u I, 1183) of the minimum problem o(T, S). Thus we have an example of a boundary configuration consisting of a smooth arc F and a smooth support surface S for which the minima of area in 1(F, S) have a discontinuous (and even unbounded) trace on S. However, the reader will note that the surface S in the Courant example does not satisfy a uniform extrinsic Lipschitz condition in QB3, i.e., the quotient of the distance of two points on S divided by their air distance is unbounded. We say that S does not satisfy a chord-arc condition (the precise definition of this condition will be given in Section 7.5).

Surprisingly, the chord-arc condition suffices to enforce that all minima of a free or partially free boundary problem have a continuous free trace. In fact, we shall prove :

(i) Suppose that X minimizes Dirichlet's integral in the class '(T', S) and that D(X) > 0. Assume also that the support surface S satisfies a chord-arc condition. Then X is of class C°'"(B u I, 1183) for some µ e (0, 1).

This result is the main statement of Theorem 1 in Section 7.5. The proof is based on an adaptation of Morrey's idea to compare any minimizer locally with a suitable harmonic mapping. To make this idea effective one constructs such a mapping by exploiting the chord-arc condition in order to set up its boundary values on S. Several variants of the assertion (i) are given in Theorems 2-4 of Section 7.5. In particular, Theorem 4 of Section 7.5 provides a regularity theorem analogous to (i) holding for minimizers of a completely free boundary problem. In Section 7.6 we shall prove regularity of stationary points of Dirichlet's integral at their free boundaries. At present it is not known whether the free trace £ of any such surface X is a continuous curve provided that the support surface S satisfies merely a chord-arc condition. However, assuming that S is of class C2 we obtain the desired result. More precisely, we have: (ii) Let S be an admissible support surface of class C2, and suppose that X is a stationary point of Dirichlet's integral in the class '(I', S). Then there is some a E (0, 1) such that X E C°'"(B u 1, 1183).

This result is the content of Theorem 2 in Section 7.6; a similar statement can be obtained for solutions of completely free boundary problems (cf. Section 7.6, Remark 2). The proof of (ii) is quite different from that of (i). Whereas in (i) we shall proceed by deriving a priori estimates for X, the approach in (ii) is indirect. Using the finiteness of the Dirichlet integral of X we shall first derive suitable monotonicity results for functionals that are closely related to Dirichlet's integral. Combining these results we shall infer that X has to be continuous on B U I if D(X) < oo. Once the boundary values X11 are shown to be continuous, we can apply

suitable techniques from the theory of nonlinear elliptic equations to obtain

46

7. The Boundary Regularity of Minimal Surfaces

X E C0'a(B u I, 683 ), a e (0, 1). For instance, Widman's hole-filling method (cf. Lemma 5 in Section 7.6) yields a direct way to this result; the details are carried out in the proof of Theorem 2 in Section 7.6.

Note that in all these cases the support surface S may have a nonempty boundary. If 8S is void, we can say much more on the free trace E = {X (w) : w e I}

of X on S. Roughly speaking I will turn out to be as good as the support surface S. We shall, in fact, obtain: (iii) Let S be an admissible support surface with OS = 0 which is of class Cm e, m >- 3, $ e (0, 1). Then any stationary point X of Dirichlet's integral in '(F, S) is of class Cm,ft(B u 1, E3). If S is real analytic, then X is real analytic on B U I and can be continued analytically across I. This result is formulated in Theorems 1 and 2 of Section 7.8. Starting from (ii),

we shall first verify that X is contained in C' (B u 1, R3 ). This can either be achieved by transforming the boundary problem for X locally into an interior regularity question for some weak solution Z of an elliptic system dZ = F(w)IVZIZ, which is derived by a reflection argument, and then applying Tomi's regularity theorem, or by playing the full regularity machinery for nonlinear elliptic boundary value problems. The first possibility is sketched in Remark 1 of Section 7.8, whereas the second approach is discussed in Section 7.7 in great detail and in a wider context (see in particular Theorem 4 of Section 7.7). Having proved that X is of class C1(B u I, 683), we use classical results from potential theory to derive X e u I, 683) by employing a suitable bootstrap argument. The reader can find this reasoning in the proof of Proposition 1 in Section 7.8.

In Theorem 2 of Section 7.8 we show that X can be continued analytically across its free boundary if the support surface S is real analytic. To this end, we set up a Volterra integral equation

Z(w) = J

F(w, Z(m))doo 0

which has exactly one solution Z in the space .sal(sa) of mappings Z: Ss -> C3 which are continuous on 96 and holomorphic in S,,:= {w: jwj < 6 < 1, Im w > 0}, and F is constructed in such a way that Z(u) = X (u)

for u E R with u j< 6

(assuming a suitable normalization of X).

Let X* be an adjoint surface of X and f = X + iX*. Then both f and g := f - Z are of class .sal(sa), and we have

Im Z = 0 and Re g = 0 on

la .

By Schwarz's reflection principle, we can continue both Z and g across la as

7.4 The Boundary Behaviour of Minimal Surfaces at Their Free Boundary

47

holomorphic functions, whence also f = g + Z and X = Re f are continued analytically across I. This approach to analyticity at the boundary, due to H. Lewy, is by far the easiest, but it cannot be carried over to H-surfaces or to minimal surfaces in a Riemannian manifold as it uses the holomorphic function f = X + iX*. This tool is, however, not available in those other cases. Here one can apply a general

regularity theorem due to Morrey [5] (cf. also Morrey-Nirenberg [1] and Morrey [8] ).

Let us now turn to the case when the support surface S has a nonempty boundary. Then we shall establish the following result (cf. Section 7.7, Theorem 1):

(iv) Let S be an "admissible" support surface of class C4 (by definition, this implies 8S a C4; cf. Section 7.6, Definitions 1 and 2). Moreover, let X be a stationary point of Dirichlet's integral in '(F, S). Then X is of class C1.1/2(B U I, 683). According to Remark 1 in Section 9.8, this is the best possible result which can, in general, be expected. This follows from the asymptotic expansions (1) and (2) in Section 7.10 around points u, and u2 on 1 where the free trace X 11 of X on S

lifts off the boundary 8S of the support surface S. We could interpret (iv) as a regularity result for a Signorini problem (or else for a thin obstacle problem). The proof of (iv) will be carried out in three steps. First, by applying Nirenberg's difference quotient technique, we shall derive L2-estimates for the second derivatives O2X up to the free boundary. For this purpose we need the Holder continuity of X on B u 1, established in (ii), as well as an important calculus inequality due to Morrey (cf. Section 7.7, Lemma 2) which implies that the Morrey seminorm is "reproducible". As a second step it will be shown that X is of class C' (B u I, 683 ). This follows

from LP estimates for solutions of the Poisson equation. In order to apply these estimates we introduce suitable local coordinates {qi, g} on S such that the boundary conditions for Y(w) = (y'(w), y2(w), y3(w)) = g(X(w)) become uncoupled. For y2 we derive a Neumann condition and for y3 a Dirichlet condition. Then we apply the L. -estimates to y2 and y3, thus obtaining Holder continuity of Dye and Vy3 up to the free boundary. Finally, the continuity of y1 up to the free boundary will be derived from the conformality relations.

Xe

As a third step we devise an iteration scheme, which allows us to attain C1.1i2(B u I, R3), by exploiting once again the conformality relations. The use of LP -estimates can be circumvented by a method which is developed

in Section 7.9. Here one derives directly that 02X satisfies a "Dirichlet growth

condition" (i.e., has a finite Morrey seminorm) up to the free boundary by alternatively applying one of two possible Poincare inequalities. This method is nothing but a skilful improvement of the estimates derived in step 1. Note that the regularity results (i)-(iv) are not directly meaningful for differential geometry, as the free boundary I may contain branch points. This can, at least partially, be remedied in the following way. First, by applying a technique due to Hartman and Wintner, we show that, for every branch point wo e 1, we have an asymptotic expansion

48

7. The Boundary Regularity of Minimal Surfaces

Xw(w)=A(w-wo)v+o(Iw-woIv) asw-wo with some ve JandAEC3,A 00,=0. This implies that there exists a limit tangent plane of X as w --+ wo with the normal No = lim N(w), where W- W()

A X,,),

N(w) = I and that the oriented tangent t(u) := IXu(u)I-tX.(u)

as u -w0 = u0 E 1

of the free trace X I, is either continuous or jumps by 180 degrees; the first case occurs if the order v of the branch point wo = uo E I is even, the second case, if v is odd.

Hence, if v is even, the representation X(s) of the trace f = X I, with respect to its arc length

s=

u

J uo

I Xu(u)I du

is of class C1, and therefore the trace I can be viewed as a regular Ct-curve in the neighbourhood of xo := X(uo). If v is odd, then 1 has a cusp at xo, and only the unoriented tangent is continuous at xo. We sketch the derivation of this result in Section 7.10; the details of the Hartman-Wintner technique are given in Chapter 8. Most of our results will be stated and proved merely for stationary points of Dirichlet's integral in W(F, S), that is, for solutions of a partially free boundary problem. Similar results hold mutatis mutandis for minimal surfaces with com-

pletely free boundaries, or for minimal surfaces of higher topological type spanned in a general boundary configuration
7.5 Holder Continuity for Minima Courant's examples indicate that one cannot expect a solution of a free or a semifree boundary problem to be regular at its free boundary, even if the support surface S is of class C. On the other hand, we shall see that a minimal surface is continuous up to its free boundary if it is minimizing and if S satisfies a kind of uniform (local) Lipschitz condition. Such a condition on S will be called a chord-arc condition.

7.5 Holder Continuity for Minima

49

Definition. A set S in f is said to fulfil a chord-arc condition with constants M and 6, M > 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: JwJ<1,v>0} the boundary of which consists of the circular are

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

I={uel:Iul<1} on the real axis. Consider a boundary configuration (F, S> 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:Jwl<1-d}={weB:dist(w,C)>d}, (0
Then we can prove: Theorem 1. Suppose that X e c'(r, S) minimizes the Dirichlet integral D(X) within the class W(I', S), and let e = e(F, S) := inf{D(Y) : Y e e(r, S)} be positive. Moreover, assume that S satisfies_a chord-arc condition with constants M and 6. Then, for any d e (0, 1), any wo a Zd, and for any r > 0, we have ('

IOXIZdudv < ( 2r

(1)

J s.(wo)

\d

Zµ

f IVXI2dudv Ja

where (2)

p := min {(1 + M2)-1, 62/(2e7r)}.

50

7. The Boundary Regularity of Minimal Surfaces

It follows that X is of class C°'"(Zd, R') and that (3)

[X]",Z,, < c(p)d-"

D(X) = c(p)d-"f (T, S)

holds true for some constant c(u) > 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) <_ (r/d)2µ-P(d, wo)

(6)

holds true for all r e (0, d]. To this end we fix wo E I with IwoI < 1 - d and set B,:= B,(wo), S,:= Sr(wo), and P(r) := O(r, wo). Introducing polar coordinates p, 9 around wo by w = wo + pe`B and writing somewhat sloppily

X(w)=X(wo+ pe`B)=X(p,0), we obtain (7)

O(r) =

{IX,(p, 0) 12 + p

r

fo

X9(p, 9)12}pdOdp.

fog

Fr om (4) we infer (8)

IXo12

= p-21XoI2,

<X,,, X8> = 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<7r-1S2

(12)

0

holds true. Then we infer from (13)

Q1(r) - Q2(r)l <

f

o

I Xe(r, 0)I dO -<

f." Xe(r, 0)I2d0} n

the inequality

l/z

JJ

I Qt(r) - Q2(r)I <- S.

Since S satisfies a chord-arc condition with constants M and 6, there exists a rectifiable arc

of length 1* = L(F*) on S which connects the points Q1(r) and Q2(r), and whose length L(F*) satisfies (14)

1* = L(I'*) < MI Q1(r) - Q2(r)I

We assume s to be chosen as parameter of the arc length on r*. Then it follows that Ic'(s)I = 1 a.e. on [0, 1*]. Introducing the reparametrization (0), 7r< 0< 27t, of r* which is defined by C(8) :=

(7t-1(B

- 7Z)1*),

we obtain I SB(0)I = const = l*/7t

a.e. on [rt, 21t]

and

I* -

ran I CeI de; n

therefore also

(15)

nranICe12d0=(j2nIC0Id6)2=I*2

From (13)-(15) we conclude that f.2%

(16)

10012 dB < M2 J

n o

I Xe(r, 0)12 d9.

7. The Boundary Regularity of Minimal Surfaces

52

Consider now the harmonic vector function values q(O) = H(wo + re") are defined by

H(w) in B,. whose boundary

0<8<_rt X(r 8) for 7E < 9 <- 21t {((e)

n(e)

n(17)

Because of (16), we have

IXa(r,B)IZd9.

Hn fo,

Ir1eI2d8<_(1+MZ)

o

Expanding H and in Fourier series we obtain (w)

cosnO + B,, sin n8)

(P_)(A

= 21 A o +

r

and

r1(B)= Ao+ 2

(An cosnO+Bnsin nO). n=1

From these expressions, we derive

IVHI2dudv =1t E n(IAn12 + n=1

Zn

I1igI2d6 = It

n2(1A,, 12 + IBn12), n=1

Jo

and therefore o2n

IVH12dudv <

(18)

Inel2dO.

J B, Relations (11), (17) and (18) imply that (19)

1VH1 2dudv

2(1 + M2)rW'(r).

Next we consider the mapping Y(w) on B u B, which is defined as Y(w):=

H(w) X(W)

for

w e Br

WEB - Br

Clearly Y is continuous and of class HZ on B u Br. Let T be the homeomorphism of .6 onto B u Br which maps B conformally onto B u Br, keeping the points 1,

- 1, i fixed. Then the mapping Z := Y o r is contained in W(r, S), and the minimum property of X( implies

r

f, B

IVX12dudv <

IVZI'dudv. J

7.5 Holder Continuity for Minima

53

On account of the conformal invariance of the Dirichlet integral we have IVXI2dudv < fQB, IVY I2dudv,

and the definition of Y now implies JVXI2dudv <

(20)

f IVHI2dudv.

fS,

B1

By virtue of (5), (19), and (20) we obtain the relation

0(r) < ?(1 + M2)r4P'(r)

(21)

for every r e (0, d) - V satisfying equation (12). On the other hand, if the equation X0(r, 0)12d8 > 7r-162 fo"

holds for some r e (0, d) - .N, then we trivially have (P(r) < 2D(X) = 2e < 2e7ra-2 fo

I X0(r, 0) 12 dO,

a nd the identity (11) yields

0(r) < 7rea-2rck'(r).

(22)

Defining the number y e (0, 1) as in (2), it follows that (23)

2yO(r) < rV(r) for all r in (0, d) - .N',

and an integration yields 0(r) < (r/d)2,,0(d) for all r E [0, d].

Thus we have established (6) for any d e (0, 1), w0 e I with I wo I < 1 - d, and r e [0, d]. Consider any w0 with I w0 1 < 1 - R and Im w0 >- R for some R e (0, 1). Then we have B,(w0) c B for any r e (0, R) and analogously to (18) we obtain zrz

IVXI2dudv <

o1X9(r, 0)I2dO

B.(wol

for almost all r e (0, R). By (5) and (11) we therefore infer 0(r,w0):!

rd0(r,w0)

for a.a. r e (0, R), and an integration yields (24)

d5 (r, w0) < (r/R)2(P(R, w0)

for all r e [0, R].

54

7. The Boundary Regularity of Minimal Surfaces

Finally we fix some d e (0, 1) and choose an arbitrary point wo e Zd = B r ; I wl < I - d}. Set uo = Re wo and vo = Im wo. We distinguish three cases : (i) vo >_ d/2.

Choosing R = d/2 we infer from (24) that

P(r, wo) <(2r/d)' fe I VXI2dudv for 0 < r < d/2

(25)

holds true.

(ii) 0
k(r, wo) < (P(2r, uo). Applying (6) we have also 0(2r, uo) < (2r/d)2µk(d, uo),

and therefore 0(r, wo) < (2r/d)2µ

(26)

I VX I2 du de.

fB

In p articular we have (27)

for any vo E [0, d/2].

(vo, wo) < (2vo/d)2SOX I2dudv lB

(iii) 0
Combining this inequality with (27) it follows that (28)

0(r, wo) <(r/vo)2(2vo/d)2µ

IVX12dudv fB

< (2r/d)2µ

IVXI2dudv. e

On account of (25), (26) and (28) inequality (1) holds true for any r e [0, d/2], and for r > d/2 estimate (1) is satisfied for trivial reasons. The bound (3) and X e C°µ(Z4, R3) now follow from Morrey's "Dirichlet growth theorem" (see Morrey [8], p. 79). M

Remark 1. Note that the assumptions of Theorem 1 do not require S to be a regular surface. In fact, S is allowed to degenerate to a rectifiable arc. Thus several variants of Theorem 1 can be proved. For instance we get:

7.5 Holder Continuity for Minima

55

Theorem 2. Suppose that S u T satisfies a chord-arc condition with constants M and 6, and let X n 1(T; S) be a minimizer of the Dirichlet integral in the class c'(T, S), that is, a solution of the minimum problem .l(T, S) considered in Section 4.6 which satisfies D(X) > 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 <- cL(F)

where the constant c only depends on M, S, and on the chosen three-point condition

of the family a(T).4

Similar results hold for solutions of minimum problems with a completely free boundary, i.e., for the minimizers of the Dirichlet integral within one of the classes W(a, S), '+(S), and W(H, S) introduced in Sections 5.1 and 5.2. As in Chapter 5 we now choose the parameter domain B as the unit disk in C,

B={we C:Iwl<1} and

C=aB={weC: 1wI=1}. `See Hildebrandt [3], pp. 55-59, for a sketch of the proof.

56

7. The Boundary Regularity of Minimal Surfaces

=WO

Fig. I

Moreover, we set Sr(wo) := B n B.(wo),

Cr(wo) := B n 3Br(wo)

Theorem 4. Let S be a closed, nonempty, proper subset of 683 satisfying a chord-arc condition with constants M, S. Moreover assume that for some µ > 0 the inclusion

S -- T. of S in Tµ induces a bijection of the corresponding homotopy classes: ft1(S) H it, (T,,).5 Finally, suppose that'' denotes one of the classes'W(a, S), le+(S), 1(H, S). Then for every minimizer X of the Dirichlet integral in the class W there is a constant c such that (31)

Jsr(WO) IVXI2d

udv
holds for any wo e B and any r > 0, where

v = (1 + M2)-1.

(32)

In particular, we have X e C°,°(B, 683) and (33)

lim dist(X(w), S) = 0 for all w° E 8B. w-wo

Sketch of the proof. Set So := 4nµ2; this constant is nothing but the number S which appears in Theorem 2 of Section 5.1. Then there is a number Ro E (0, 1) such that (34)

IVX12dudv <So

J Qo

holds true for the annular domain Slo For any point wo e B we define (35)

45(r) = P(r, wo)

{w e C : 1 - Ro < Iwl < 1}.

fs'(W")

I VX 12 du dv.

Introducing polar coordinates p, 0 around w° by w = w° + pe`a and writing s This is assumption (A) of Section 5.1.

7.5 Holder Continuity for Minima

57

X(w) = X(wo + pe`B) = X(p, 0), we obtain analogously to (9) that for f02(p)

(35')

O(r) = 2

p

-t I X9(p, 0)I2dOdp

0,(p)

holds for two angles 01, 02 with 0 < 02(p) - 01(p) < 21r. Consequently the absolutely continuous function O(r) satisfies e2(r)

I XB(r, 0)I2 d0 = 2 rW'(r)

(36) B, (r)

for all r e (0, oo) - A' where AV is a one-dimensional null set. Let wo e C and consider some positive number /3 which will be specified later. Moreover, let r e (0, Ro) - X. ez(r)

Case 1.

I XB(r, 0)12d0 >

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 <-

f

B

B

IXe(r,0)Id0<10'-0112 { f

e

e

whence

IP'-P1 <JB IX.(r,0)1d0
(38)

In particular, this estimate holds true for the two endpoints Q1(r) and Q2(r) of the arc X: Cr(wo) -4 Al which lie on S. Choosing /3 less than or equal to 6 (M and 6 being the constants of the chord-arc condition of S), we have IQ1(r)-Q2(r)1 <-b. Thus there is a rectifiable arc r* = {, (s) : 0 < s < 1* } of length 1* = L(I'*) which connects the points Q1(r) and Q2(r), and whose length satisfies

1* = L(r*) < MI Q1(r) - Q2(r)1-

(39)

Consider now the harmonic vector function H(w) in Br = Br(wo), the boundary values s7(0) = H(wo + re") of which are defined by (X (r, 0)

n(0) :=

f c(0)

for

01(r) < 0 < a2 (r) 0 e [0, 21rl - {61(r), 02(0)]

7. The Boundary Regularity of Minimal Surfaces

58

where t'(O) is a suitable reparametrization of F* proportional to the arc length. Then analogously to (19) we obtain IV HI2dudv

(40)

< (1 + M2)rW'(r).

Jflr

We now define the mapping Y(w) on B u B, by Y(w) := I X (w) for w

e

wE

Br- B

and set

Z:= Yoz where z is a homeomorphism of B onto B u B, which maps B conformally onto BV B,. Claim. The mapping Z is an admissible comparison surface, i.e., Z e 16, if we choose as

/3:= min{b, p, [(1 + M2)-tit60]1/2}.

(41)

Then analogously to (21) we arrive at

O(r) < ?(1 + M2

(42)

Combining the discussion of the cases 1 and 2 we infer from (37) and (40)

that (43)

O(r) < Zcr 45'(r) a.e. on (0, R°),

where (44)

c:= max{1 + M2, 2n/-2D(X)}.

Now we can proceed as in the proof of Theorem 1, and we obtain the Dirichlet growth condition (31) with v = 1/c. As this implies X e C°,"(B, R3), we can repeat the previous discussion in such a way that case 1 becomes void. For this purpose

we only have to choose R° > 0 so small that IQ1(r)-Q2(r)l
forre(0,R°)-.N'.

Then we obtain condition (31) with the desired exponent v = (1 + It remains to verify the claim. First of all we choose a radius p E (0, 1) so close to 1 that the closed curve Z: 8BP(0) -* l is completely contained in T,r,2 and represents the boundary class [ZIBB]. We shall show that this curve is homotopic in T. to some curve M2)-1.

X : BBP.(0) -+ 683 which represents the boundary class of X. Let Q1(r) = X(wl), Q2(r) = X(w2), w1, w2 e 3B. Since z is a conformal map-

ping of B onto B u B,(w°), the tangent of the curve C,(w°) := z-1(C,(wo)) tends to a limit as w tends along C,(w°) to one of the endpoints w1 and w2 of C,(w°), and

7.5 Holder Continuity for Minima

59

Fig. 2. This sketch illustrates the proof that the comparison surface used in the regularity proof is admissible.

this limit is different from the tangent of aB. This can either be seen by an explicit computation of r or from a general theorem of the theory of conformal mappings

(cf. Caratheodory [4], p. 91). Therefore the above number p can be selected in such a way that aB,(0) intersects Cr(wo) in exactly two points z3 and z4i and that the curve r(aBP(0)) is completely contained in QE := Br(wo) u B - B, _,(O) where e is chosen to satisfy 0 < e < r < Ro. Because of (34), it follows that 2Dne(X) < bo.

(45)

We can find a number p' E (1 - e, 1) such that the trace of the curve X : aBp,(0) - 183

is completely contained in T 12 and represents the boundary class [X aB] of X. We can also achieve that aBp.(0) - B,(wo) lies between aB and r(C*), where C* denotes that part of aBo(0) which is mapped by r into B - B,(wo). Set CZ := aBp(0) - C*. Moreover, note that the curve X: Cr(wo) --> R3 remains completely in TT,2 since its endpoints lie on S and its length is less than or equal to /1 (cf. (38)), and P:5 u on account of (41). Finally we infer from (36) and (40) that B2(r)

IVYI2dudv < (1 + M2) B, (Wo)

IXo(r, 0)12d0, B i (r)

and the right-hand side of this inequality is bounded from above by

(1 +

Mz)n-tflz

By virtue of (41) we arrive at (46)

f

B

r(xb)

I VYI2 dudv < So .

60

7. The Boundary Regularity of Minimal Surfaces

We infer from (34), (45) and (46) as well as from Theorem 2 of Section 5.1 that all curves to be considered in the following are contained in T,,2, and that we obtain the following homotopies (^ ). Here C; will denote the subarc of Cr(wo) which connects the intersection points w, and w4 of C,(wo) with -r(5Bp(0)): X I dBo,(0)

X I aBo.(0)-B,.(w0) X IC,.(w0)rBa,(0)

XIV(c;)' XIc,

= Y 0TIc,' YIc, = Y0TIC- -Y0TIC; =ZIOBo(0)-

This completes the proof of the claim and thus of the theorem.

11

Remark 2. An inspection of the proof of Theorem 4 shows that the constant c in (31) will depend on the number Ra which in turn depends on X. Hence (31) does not yield an a priori estimate of the Morrey seminorm or of the Holder seminorm of X. Yet such estimates for minimizers are correct, and we can derive them by invoking the free-trace estimates of Chapter 6.

7.6 Holder Continuity for Stationary Surfaces In the previous section we have proved that minimizers of the Dirichlet integral in various classes of admissible surfaces corresponding to free boundary prob-

lems are Holder continuous up to their free boundary. The proof has made essential use of the minimum property of the solution of the free boundary problem. In case of partially free problems we have even derived a priori esti-

mates for the Holder seminorm up to the free boundary. Now we want to establish Holder continuity of stationary minimal surfaces up to the free boundary. However, we shall have to use a completely different approach in this case as we are not able to derive a priori estimates for the Holder seminorm or even for the modulus of continuity. In fact, such estimates do not exist, as an inspection of the Schwarz examples discussed in Section 5.9 will show. Consider, for instance, the boundary configuration depicted in Fig. 1 which consists of a cylinder surface S and of a polygon F with its endpoints on S. For this particular configuration the corresponding semi-free boundary problem possesses infinitely many stationary solutions, all of which are simply connected parts of helicoids,

and it is fairly obvious that there is neither an upper bound for their areas (Dirichlet integrals), nor for the length of their free traces, nor for their moduli of continuity. For this reason we shall not approach the regularity problem by deriving estimates. Instead we want to use an indirect reasoning, first proving continuity up to the boundary by a contradiction argument. We shall constrain our atten-

7.6 Holder Continuity for Stationary Surfaces

61

Fig.1. The stationary solutions of the boundary value problem for the configuration cannot be estimated a priori.

tion to stationary surfaces in the class c'(I', S) defined for semi-free problems; see Section 4.6. Similar results can be obtained for stationary solutions of completely free problems without any essential alterations. We begin by defining Assumption (B) and the notion of admissible support surfaces.

Definition 1. An admissible support surface S of class C'", m > 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
for ally,bel3

(iv) There exists a number K > 0 such that

gi` (Y)

IC

is satisfied on 183 for i, k, I = 1, 2, 3. We call the pair {all, g} an admissible boundary coordinate system centered

at x0.

Let us recall the standard notation used for semifree problems and for the definition of '(I', S): The parameter domain B is the semidisc

B={w=u+iv: IwI< 1,v>0}, the boundary of which consists of the circular arc

C= {w=u+iv: IwI = 1,v>0} and of the segment

I={weR: I wI<1}. Moreover, we set

Zd={w=u+iv: IwI<1-d,v>0}, S.(wo) = B n Br(wo),

de(0,1),

I.(wo) = I n Br(wo),

Cr(wo) = B n aB.(wo)

Next we introduce some terminology with respect to a fixed admissible boundary coordinate system {all, g}. Given a minimal surface X : B -+ R3, we use the diffeomorphism g: If83 - R3 to define a new mapping Y e C3(B, R3) by (1)

Y(u, v) := g(X(u, v)),

whence also (1')

X(u, v) = h(Y(u, v)).

7.6 Holder Continuity for Stationary Surfaces

63

In other words, we have

Y=goX

and

X=hoY.

Quite often we use the following normalization: (2) (Let wo e I, and set xo := X (wo). Suppose that {all, g} is an admissible

boundary coordinate system for S centered at x0. Then Y(wo) = 0.

In case of this normalization, the following holds true :

(2)

Then we have I Y(w) I < R. Let 0 be a subset of B such that X (0) c If wo c- 1, d = 1 - I wo 1, r < d, X e C° (Sr(w), 11 3), and X: I,(wo) - S, then

we have y3(w) = 0 for w e 1r(wo). If 8S is nonempty and xo E 8S, then we have yt(w) > 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
al/Pcc0 for 0
°Iii ccQnK,(x*) for0
lim sup p-++0

i1

np

IVXI2dudv > 2. W,

This relation is, however, an immediate consequence of Proposition 2 in Section 3.2.

Lemma 2. For each X E C'(B, R'), for every wo e I, and for each p E (0, 1 - I wo 1),

there is a number r with p/2 < r < p such that )1/2

('

oscc.(wo)X 5 (7r/log 2) 112

I VX 12 du dv }

I

Sa(w0)

.

J

Proof. Let us introduce polar coordinates r, 0 about wo by setting w = wo + re`' and X(r, 0) = X(w). Then, for 0 < 01 <('02 < n, we obtain eZ

I X(r, 02) - X (r, 0,)1:!5 J

1 Xe(r, 0)I d0 < e,

where we have set p(r):= fz I X9(r, 0)12 d0. 0

If p/2 < r< p, it follows that

f

p

p/2

p(r)d-< r

IoXI2dudv. sr(wo)

np(r)

7.6 Holder Continuity for Stationary Surfaces

67

Consequently, there is a number r e [p12, p] such that 'o Jdt)p(r)

<J

t

V,2

IVXIZdudv sa(wo)

or

p(r) <

1

log 2

IVXIZdudv,

J s,(w0)

and the assertion is proved. Lemma 3. Let wo e I, r e (0, 1 - I wo I), and X E C'(B, R'). Assume also that there are positive numbers al and a2 such that osCCr(WO)X < at

and

inf IX(w) - X(w*)I < a2 .

sup

w* e 5,(w0) w e C,(w0)

Then we obtain oscs,(WO)X < 2a, + 2a2 .

Fig. 2. A domain used in Lemma 3.

Proof. Let w e S,(wo) and w', w" e C,(wo). Then we infer from

IX(w) - X(w')I < IX(w) -

X(w

)I +

IX(w")

- X(w')I

that IX(w) - X(w')I <

inf

W" E Cr(w0)

IX(w) - X(w")I + oscc,(W0)X.

Thus we have

IX(w)-X(w')I
68

7. The Boundary Regularity of Minimal Surfaces

This yields for arbitrary w1, w2 e Sr(w°) and w' e Cr(wo) the inequalities IX(w1) - X(w2)I

IX(wl) - X(w')l + IX(w2) - X(w')I< 2a1 + 2a2,

and the assertion is proved.

11

Lemma 4. Let X be a stationary point of Dirichlet's integral in the class c'(T, S). Suppose also that the support surface S is of class C2, and let R, K, m1, m2 be the constants appearing in Assumption (B) that is to be satisfied by S. Then, for R1 := R ,,/m-2 and for some number c > 0 depending only on R, K, m1, m2, we have: If for some r e (0, 1 - Iw0)) and for some number R2 e (0, R1) the inequality u2

IVXI2dudv

< R2/c

[JsWO)

holds true, then it follows that

IX(w) - X(w*)l < R2.

inf

sup

w* a Sr(wo) w e S,(wo)

Before we come to the proof of Lemma 4 which is the main result in step 1 of our discussion, let us turn to the Proof of Theorem 1. Since f B I VX 12 du dv < oc, we have

lim IVXI2dudv = 0 r+0 Js,(wo)

for every wo e I. Then Lemmata 2, 3, and 4 immediately imply that llm oscs'(wo)X = 0 r-+0

for w° e I. In conjunction with X a C°(B, R') we then infer that X is continuous

onBul. Proof of Lemma 4. Let w° e I and 0 < r < I - I wo 1. Then we have to prove the following statement: There is a number c = c(R, K, m1, m2) with the property that for any R2 with 0 < R2 < R1 and for any w* ESr(w°) with

inf

(16)

IX(w) - X(w*)l > R2

W e C,(wo)

the inequality 1/2

R2 < c

(17)

IVXI2dudvl S,.(wo)

holds true.

Thus let us consider some w* e Sr(w°), wo E 1, 0 < r < I 1 - w°1, and set

x* := X(w*),

6(x*) := dist(x*, S).

We shall distinguish between two cases, 6(x*) > 0 and S(x*) = 0.

7.6 Holder Continuity for Stationary Surfaces

69

Case (i): 5(x*) > 0.

Then we proceed as follows: Choose some function A e C'(R) with A' >- 0 and with A(t) = 0 for t < 0, and introduce the real valued function

A(p - IX - x*I)IVXI2dudv,

P(p):= 2

S (wo)

for 0 < p < min{a(x*), d2R2}, where R2 is some number with 0 < R2 < R1 R ,,/m2, and where we have set

d:= -

m2

0
Define a test function q(w) as (Z(p - IX(w) - x*I)[X(w) - x*] 10

n(w)

forw e S,(wo) for w e B - S,(wo)

We use rl to define a family {XE}ose«o of outer variations XE(w):= X(w) - sri(w).

On account of

IX(w)-x*1

- R2>p forweC,(wo)

we find that XE is of class HZ(B, R3). Furthermore, we obtain IX,.(w)I = IX(w)I

forweB-S,(wo).

Hence X and X, have the same boundary values on C. Moreover, for 271almost all w e I, we have X(w) e S and therefore IX(w) - x*1 >- 5(x*) whence p - IX(w) - x*I < 0. This implies >)(w) = 0 for 2t-a.a. W e I. Consequently we obtain X, e '(F, S) for 0 < E < so and for any so > 0. As ri e HZ n L. (B, R3 ), we conclude that X, is an admissible variation of type II in the sense of Definition 2, Section 5.4. By Section 5.4, (3) and (7), it follows that dudv < 0 J$(wo)

(in fact, even the equality sign holds true since we are allowed c e (-so, so)), and therefore

1S(wo)

IVXI2A(p - IX(w) - x*I)dude

<

2'(p - Ix - x*I)Ix - x*I-1 Sr(w0)

{<X., x - x*>2 + <X,,, X - x*>2} dudv.

to take

70

7. The Boundary Regularity of Minimal Surfaces

By virtue of the conformality relations IXL12 = IX"I2

<X., Xe,> = 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 <0. sr(WO)

Since

,'(P-IX -x*I)=0 ifIX-x*I?p it follows that

IX -x*I2'(p-IX -x*I)<-P).'(p-IX -x*I), and (18) yields (18')

2(P(P) - Pp'(p) < 0.

Thus, d dp

{P-,P(P)} > 0,

and it follows that (19)

P-2tp(P) S (P')-2QP(p')

for 0 < p < p' < R*

where we have set

R* := min{S(x*), d2R2}. Now we choose 2 in such a way that it also satisfies ,Z(t) = 1

for any t >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
p-2

2

Sr(x'o)nK,_.(xw)

where we have set

K,(x*) := {w e B: IX(w) - x*I < i}. Letting a --+ + 0 and then p' - R* - 0, we find that p-2 J

IVX12dudv <(R*)-2 J

IVX12dudv

Sr(wo)nKp(x')

taking al(t) < 1 and (19) into account. Now let p -* +0. Then it follows from Lemma 1 that (20)

2nR*2 <

IVXI2dudv. J S.(wo)nKx.(x')

In case that d2R2 < 8(x*), we have by definition of R* that R* = d2R2, and (20) implies (21)

R2 < 2 d4

)tie XI2dudv}

f

if d2R2 < 8(x*).

S,.(wo) I V

Now we treat the opposite case 5(x*) < d2R2 where we have R* = b(x*). Because of (20), we have already proved that (22)

21t 2(x*) < fSr(WO)nKM.-)(X*) IVXl2dudv.

( The

still missing case S(x*) = 0 is formally included and will be treated at the end of our discussion.) First we choose some point f e S which satisfies

If - x*I = dist(x*, S) = 8(x*) < d2R2 <-

aR2.

Then we choose an admissible boundary coordinate system {all, g} for S centered at xo := f as described in Definition 2, with the diffeomorphisms g and h = g-t. As before we define by ggk(y) the components of the fundamental tensor: ah'

9,k(Y)

oht

ayi (Y) ay,

(Y)

Let us introduce the transformed surface Y(w) by Y(w) := OX M) = (Yt(w), YZ(w), Y3(w)),

and set 11Y(W)II := {gJk(Y(w))Y'(w)Yk(w)}t12

.

72

7. The Boundary Regularity of Minimal Surfaces

For p with d-16(x*) < p < dR2, we define q(w)

A(p - II Y(w)II) Y(w) 0

for w e Sr(wo)

if weB-Sr(wo).

Firstly we prove that q E H2 '(B, R3 ). For this it suffices to show that 'i vanishes on Cr(wo). For this purpose, let w be an arbitrary point on Cr(wo). By assumption (16) we have

R2 < I X(w) - x*I, whence

R2<6(x*)+IX(w)-fl
R2/2
for all WECr(wo).

On the other hand, since h(O) = f, we obtain ('1

IX(w) - f I =

<

hl,,#Y(w))y`(w) dt

J0 m2I Y(w)I <_ (m2/m1)1" 11 Y(w)II

IIY(w)II >- (1/2)(m1/m2)1/2R2

= dR2 > p,

therefore q(w) = 0

for w e Cr(wo).

or 0 < s < 1/2 we consider the family XE of surfaces which are defined by XE(w) := h(Y(w) - sq(w))

,how that Xe is an admissible variation of X which is of type II. In HZ (B, R3) and Xe(w) = X (w) for all w c- C since q(w) = 0 for

to show that Xt maps 21-almost all points of I into S. To e w c- I with X (w) e S. If q(w) = 0, then XE(w) = X (w), and

the other hand, if q(w) # 0, we have 11 Y(w)II < p and 2

ml <

ml = 2dRm

2

2

m2/m2

R1 =R m2=R/2.

< 2

m2

2

e S, this estimate yields y3(w) = 0 (see (2')), whence

[Y(w) - cii(w)]3 = 0.

=

2 Rm

7.6 Holder Continuity for Stationary Surfaces

73

Taking the inequalities

IY(w)-Erl(w)I <2IY(w)I
XE=h(Y-Erl)ec'(I',S) provided that 8S = 0. This inclusion holds as well if 8S is nonvoid, since y' (w) >_ a

and -1 < a < 0 implies Y'(w) - ell' (W) = y'(w){l - EA(W)}

a{1 - EA(W)}

a.

Now we define

E-' [h(Y(w) - Erl(w)) - h(Y(w))] 71(E, W)

fore > 0

fore = 0.

-ayk(Y(W)W(W)

Then we have

for 0<e<1/2, and Taylor's formula yields !(E, W) = 'PO(w) + O(e) with

To :_ -h,,k(Y)rlk e H' n

(B, E3)

and V(E, w) -. Wo(w)

a.e. on Base -> 0.

Moreover, the reader readily checks that I V V(E, .)IL2(B) <- const

holds for some constant independent of e E [0, 1/2). Hence the variations {XE}ose- 0, or

SE(Y, rl) < 0,

which implies

J S.(x'o)

1 gjk(Y)DaYiDat1k + 29jk,I(Y)D.yiD.y'q'Jdudv 5 0,

where we have set

74

7. The Boundary Regularity of Minimal Surfaces a

a

D2

ut

= u,

u2 = v,

D1 = au,

av

(summation with respect to Greek indices from 1 to 2, and with respect to Latin indices from 1 to 3). Then it follows that 9;k(Y)D2YJD.yk2(P - II YII)dudv L(WO)

A'(P - II Y II )9;k(Y)(D8YJ)Yk2 II YII -1 S.(WO)

{29;k(Y)(D.yJ)Yk + 9;k,((Y)(D.yOYJYk} dudv

-1 J 2

9;k,z(Y)DaY'Dayky)A(P - II YII)dudv. S(WO)

This is equivalent to 9;k(Y)D«Y'D2YkA(P - II 'II)dudv S.(WO) {[9;k(Y)Yu

Y11

(23)

-

Il

2'(P - Il Yll)

Js,.(wo)

< - 12 J

II

YII] +

[gjk(Y)y]2}dudv

9;k,((Y)DDYJDDYIYIt(P - IIYII)dudv S'(WO)

A'(P - IIYII)9;k(Y)DaYJ

1

Yk

it Yll

Now we set

kp)

J

9;k(Y)DDYJDDYkA(P - II YII )dudv. (w0)

Then, by virtue of the conformality relations (15), we obtain the estimate k II

yk

12

1

+ [ik(y)Y11YIIJ `2IloYll2

where we have set IIVyI12

9;k(Y)D2YJDDYk

Moreover we have II YIIA(P - II YII) <- p2(P - II YII) it Yll A.'(P - 11 YII) < PA'(P - II YD).

7.6 Holder Continuity for Stationary Surfaces

75

Hence the left-hand side of (23) can be estimated from below by

OP) - zPOP); compare (18) and (18') for an analogous computation. The first term on the right-hand side of (23) can be estimated from above by c(n)Kfsr(wo) IVY12m1112IIYllA(P- IYII)dudv


r

IIYII)dudv

S (wo)

where we have set M := c(n)Kmi 312 ,

and where c(n) denotes a constant depending on the space dimension (in our case: n = 3). Analogously, the second term is bounded from above by

c(n)KJ

A'(p- IIYI)IIVYIIIVYIIYl2dudv

r 3 (w0)


< Thus we have derived the following differential inequality

(P) -

ipo,(P) < M[PJ(P) + P20,(P)]

which is equivalent to

-d [p-'O(p)1 <

M

-[P-1i(P)]

with

M:= 2M. Multiplying by e2Mp, we obtain

0
p

,/'

[e2"Mpp-2Y' (P)] + M e

Then by integrating between the limits p and p', p < p', and by applying a partial integration, we infer that 0 < [e2M°P-Z' I'I (P)]o +

Me2Mp

dp[P-1O(P)]dP

P

_ [e2Mpp_2`Y(P)]0 + [Me2MPP-1'Y(P)]P -

2M2e2Mpp-1'Y(P)dp.

fp

7. The Boundary Regularity of Minimal Surfaces

76

Therefore, Me2Mpp-1,(p)]p,

0 < [e2M1 p_2J(P) +

whence e2Mp'

P-20(P) <

+ p'Me2Mp'

e2Mp + pMe 2MP

(P')

2

O(P).

Applying once again the reasoning which led to (20) (that is, choosing 2 = AE, and letting first s -+ + 0 and then p' - dR2 - 0) and setting C(R2) := (1 + dR2M)e2Man2

we arrive at (24)

09-2

IIoYII2dudv S,(wo)n{w 1IY(w)II
< C(R2)(dR2)-2 J

r I V YII2 dudv. s (wo)

Furthermore,

S,.(wo)n{w:IX(w)-fI <2b(x*)} cSr(wo)rl{w: IIY(w)II
g,j(tX + (1 - t)f)(x' - f')dt

Y(w) = 0f,

implies

d-tb(x*) < p.

II Y(w)II < (m2/ml)1121X(w) - fI <

For b(x*) > 0 and p --> d'lb(x*) + 0, we then infer from (24) that ('

d2

b (x*) JS (wo)n{w:IX(w')_fl<2b(x*)}

IIoYII2dudv < C(R2)d-2822

Sr(wo)

J

IIoYII2dudv,

and this inequality can be rewritten in the form (25)

b(x*)-2 J

IVX12dudv.

IVXI2dudv
Sr(w'o)

20

. (.f)

By virtue of

IX(w)-fI<-IX(w)-x*I+If-x*I we obtain Ka(x*)(x*) c K2a(x*)(f),

S,.(wo)

7.6 Holder Continuity for Stationary Surfaces

77

and therefore

R28(x*) 2

IVXI2dudv < C(R2)d J

IVX I2dudv.

J S.(w0)

By virtue of (22), the left-hand side is bounded from below by 27rR2. Thus we obtain ('

R2 <

(26)

1 1/2

J

if 0 < d(x*) < d2R2.

IVX12dudv } s,(-o)

)))

Combining (21) and (26), we obtain from C(R2) < C(R1), Rt = R-,/M2 and d-2 = 4(m2/ml) that t/2

R2 < c

(27)

in case (i): 6(x*) > 0,

IVXI2dudv S.(wo)

if we set

c :=

(237r-t(m2/mt)2C(Rt))t12

= c(R, K, m1, m2).

Case (ii): 6(x*) = 0.

Here we take f = x* as the center of an admissible boundary coordinate system {all, g} for S. Then we obtain (24) for any p e (0, dR2). Setting p' := p mt/m2, it follows that

p-2

IVXI2dudv < C(R2)d-2R22 J

IVX12dudv. S.(wo)

JS.(wo)nKo (x*)

Now let p' -* + 0; then another application of Lemma 1 yields 27r:5 lim sup (p')-2 J P,-+O

IVXI2dudv S.(wo)nK,,(x*)

JSr(WO) I VX I2 dudv R2 A d(

whence we obtain t/2

(27')

R2 < c

IVXI2dudv

in case (ii): 6(x*) = 0.

S.(wo)

Combining (27) and (27'), we arrive at (17).

0

Now we turn to the second step with the aim to prove Theorem 2. 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 '(P, S). Then there exists a constant a e (0, 1) such that the following holds true:

78

7. The Boundary Regularity of Minimal Surfaces

For every d e (0, 1), there exists a constant c > 0 such that (28)

1

SOX I2dudv < crZ

JS.(-o)

holds true for every wo E Zd and for all r > 0. In particular, X is of class Co'a(B V I, R').

We shall use the following simple but quite effective

Lemma 5. Let cp(r), 0 < r < 2R be a nondecreasing and nonnegative function satisfying cp(r) < Ocp(2r)

(29)

for some 0 E (0, 1) and for all r e (0, R]. Then, for 1

a:= 21og0 we have

(p(r) < 2acp(R)(r/R)a

(30)

for all r E (0, R].

Proof. For any r e (0, R] and for any v = 0, 1, 2,..., we have tp(2-vr) <

9cp(2-'r).

Iterating these inequalities, we obtain

(p(2-'r) < 6' p(r) for 0 < r < R. Fix some r e (0, R]. Then there exists some integer v > 0 such that

2'°-' < r/R < 2-`'. Since 6 = 2

and co(r) is nondecreasing, we see that c0(r) <_

(P(2-vR) < 6'(P(R) <_

2-Yagq(R)

<_ 2a(P(R)(r/R)a.

For later use we note a generalization of Lemma 5.

Lemma 6. Let cp(r), 0 < r< 2R, be a nondecreasing and nonnegative function satisfying (31)

(p (r) < 9{cp(2r) + r'}

forsome 0e(0, 1),c> 1,0
0* := max{B, 2`-°(6R` + 1)},

O:= 2109 e**

7.6 Holder Continuity for Stationary Surfaces

79

we have

for all r e (0, R].

cp(r) < 2a{cp(R) + R°-e}(r/R)a

(33)

Proof. Since REO < 1 and 2E-° < 1/2, we infer that

2e-'(OR' + 1) < I and therefore 0 < 0* < 1. Set

9*(r) := 9(r) + ro

e

Then, for any r e (0, R], it follows that cp*(r) < 0cp(2r) + 0r° +

r°-e

= 09(2r) + r'-E(0rE + 1)

< 09(2r) + (2r)°-e2e-°(0re + 1) < 0*{cp(2r) +(2r)°-E} = 0*cp*(2r). Applying Lemma 5, we infer cp*(r) < 2acp*(R)(r/R)a

for all r e (0, R]

and

a := 2109 0*

whence cp(r) < cp(r) + r° E < 2a{cp(R) + R° E} (r/R)a

for 0 < r < R.

11

Proof of Theorem 2. We want to show that the growth estimate (28) is satisfied for any w° e I. Let us first assume that aS is empty. We introduce an admissible boundary coordinate system {°I', g} for S centered at x° := X(w°) with the inverse mapping h = g-t, and we set Y := g(X). Then we have Y n C°(B u 1, R') and Y(wo) = 0, and we can find some number po e (0, 1 - I w° 1) such that

IY(w)I
fB

gk(Y)D.y'Dacpkdudv <

-2 l B

9jk,t(Y)Day'DQykcp`dudv.

(In fact, equality holds true.) Now let r e (0, p0/2], and choose some cut-off function e Q°(B2i(w0)) with (w) _- 1 on B,(w0) and 0 < < 1, 1V l < 2/r. Set T2,

S2r(W0) - S,(wo),

cot:=

y1dudv, J Tz.

y2dudv,

co2:_ Tzr

u)3:=0

80

7. The Boundary Regularity of Minimal Surfaces

where

... stands for messQ I n ... 0 = ((p' (p2 (p3) (pk(w) :_ (yk(w) -

for w e B u 1. Then the test vector

0 is admissible in (34), and we obtain gf

2

SB

9;k.i(Y)DaY'DaYk(y` - w`) 2dudv a dudv.

< -2 J g;k(y)D.y'(yk -

I2

Hence, for any e > 0 and some constant K, (e) > 0, we find the inequality m1

(35)

J

1 J Igjk,1(Y)IID.y'IID.yklIy` 2 B

B

E

fB

B

I1Y-wI121V

where co = (0)1, CO 2, co ). Since

dudv

IIoYI12 < m2I VYI2

we can absorb the term g

f

211VYII2dudv

B

by the first term on the right-hand side, if we choose ml

2m2

Moreover, the absolute value of the second term of the left-hand side of (35) can be bounded from above by

41 f

sB

if we choose r e (0, p1), where pl a (0, po/2) is a sufficiently small number depending on the modulus of continuity of X. Hence there is a number K2 > 0 such that fs'(WO) IVYl2dudv S f (36)

2IDYI2dudv

S2.(wo)


IY-wl2dudv Tz,

holds for all r e (0, pl ).

7.6 Holder Continuity for Stationary Surfaces

81

By Poincare's inequality, there is a constant K3 > 0 such that

Y - wl2dudv < K3rJT2 VYI2dudv

(37)

.

JT2r

is satisfied for 0 < r < P1. Consequently, there is a constant K4 such that IVYI2dudv
IVYI2dudv JS2r(Wo) -Sr(wo)

forallre(0,p1). Now we fill the hole S,.(wo) by adding the term K4SSr(wo) IVYI2dudv to both sides. Then we arrive at

(1 + K4) J

IVYI2dudv < K4 J Sr(wo)

IoYI2dudv S2r(wo)

whence, setting K4

1+K a' we attain IVYI2dudv < 0 fs'(WO)

IVYI2dudv S2r(wo)

for every r e (0, p1). As 0 < 0 < 1, we can apply Lemma 5 to R = p1 and to p(r) := f sr(wo) I V YI2 du dv, thus obtaining

r )2a JIVYI2dudv<22a

IVYI2dudv Pa

Sol(wo)

Sr(wo)

for 0 < r < p1, if we set a:= 1z 2109 0. For

K5 :=

(K5 > 1),

and by virtue of IIVYII2 = IVXI2,

m1IVYI2

-< IIVYII2 <m21VYI2,

we obtain (38)

J

r

IVXI2dudv < K5D(X)(r/p1)2

S (wo)

for all r e (0, p1), and consequently for all r > 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 <<, i1)>

9j&gk

(cf. Section 7.3, proof of Theorem 2, part (i)).

Thus we arrive again at an inequality of the type (36) from where we can proceed as before. This completes the proof of the theorem. Remark 1. A close inspection of the proof of Theorem 2 shows that we would obtain a priori estimates for the a-Holder seminorm in the case that we had bounds on the modulus of continuity of X. Hence only the approach used in the proof of Theorem 1 is indirect. Remark 2. Without any essential change we can replace the class 1(1', S) in the previous reasoning by '(S). In other words, we have analogues to Theorems 1 and 2 for stationary points of Dirichlet's integral in the free boundary class ''(S). A careful inspection shows that the same holds true for stationary points in +(S) and in '(17, S) (cf. Sections 5.1-5.4). For this purpose one has to verify that the variations XL = h(Y - so) that we have previously used will stay in the classes of admissible functions '+(S) and cf(17, S) respectively, i.e., that they do

7.7

83

not violate the topological restrictions that we have imposed on the comparison functions.

7.7 C 1, 1/2 -Regularity

In this section we want to prove C"1'-regularity of a stationary point X of Dirichlet's integral up to its free boundary. As we have seen in Section 7.4, this regularity result is optimal, that is, we can in general not prove X E C1,°(B u 1, 183), I being the free boundary, for some a > 1/2, if the boundary of the support surface S is nonvoid. On the other hand, if as is empty or if XI, does not touch as, then one might be able to achieve higher regularity as we shall see in the next section. As in Section 7.6 we shall restrict our considerations to minimal surfaces with partially free boundaries or, more precisely, to stationary points of Dirichlet's

integral in t(r, S); stationary points with completely free boundaries can be treated in exactly the same way, and perfectly analogous results hold true. Consequently we can use the same notation as in Section 7.6. Our main result will be the following Theorem 1. Let S be an admissible support surface of class C4, and suppose that X (w) is a stationary point of Dirichlet's integral in the class '(r, S). Then X is of class

C1,112(B

u I, 183).

The proof of this result is quite involved; it will be carried out in three steps. In the first step we prove that X E H'(Zd, 183) for any d c- (0, 1), using Nirenberg's

difference quotient technique to derive L2-estimates for 02X. Secondly, using ideas related to those of Section 7.3, it will be shown that X E C'(B U I, 183). In the third part of our investigation we shall see how the boundary regularity can be pushed up to X E C' 112(B u I, 183) by applying an appropriate iteration procedure.

Let us note that, assuming X E C°(B u I, 118'), all regularity results will be proved directly by establishing a priori estimates. Thus the only indirect proof entering into our discussion is that of Theorem 1 of Section 7.6.

Step 1. L2-estimates for V2X up to the free boundary. Let us begin with a few remarks on difference quotients which either are well known (cf. Nirenberg [1], Gilbarg-Trudinger [1]) or can easily be derived. We consider some function Y e HZ(Zdo, 1f8'") with 0 < d° < 1 and m >_ 1, s > 1. For w E Zd and t with It! < d° - d, we define the tangential shift Y by Y(u, v) := Y(u + t, v)

7. The Boundary Regularity of Minimal Surfaces

84

and the tangential difference quotient A, Y by A,Y(u, V) =

I [Y(u + t, v) - Y(u, v)],

that is,

w=u+iv.

[Y(w) - Y(w)],

A,Y(w) =

t Moreover, let D. = au be the tangential derivative with respect to the free boundary I. Then we have: Lemma 1. (i) Let Y e HI (Z,,, 68"`), s >_ 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 << 1

a

if either cp or 0 has compact support in B u I. (ii) Similarly, if Y and E Lq(Zdo, IF "), q

1, then !

fJ Zd

IA,YJ dudv < J

ID.Y - A,YI9dudv = 0.

1im J Zdo

Zd

(iii) Finally, if Y E HZ(Zd,,, ff8'"), then

(V Y), = O° Y"

I Io°Yl2dudv = fv l0 °Yl2dudv, 0 < Itl « 1, Q

for 0
7.7 C'"'"2-Regularity

85

Now we turn to the derivation of L2-estimates for the second derivatives of X. We begin by linearizing the boundary conditions on X. This will be achieved by introducing suitable new coordinates on Q83. Thus let wo be an arbitrary point on 1, and set xo := X(wo). Then we choose an admissible boundary coordinate system {°It, g}, centered at x0, as defined in Section 7.6. Let h = g-1 and Y = g o X, i.e., X = h o Y. Then we can use the discussion at the beginning of Section 7.6; in particular we can employ the formulas (1)-(15) of Section 7.6. By Theorem 1 of Section 7.6, we know that X and Y are continuous on B i 1, and Y(wo) = 0. Hence there is some number p > 0 such that

IY(w)l
for w e 12p(wo),

and, if 8S is nonempty, we have y1(w) > 0-

for w e I2p(wo).

Let r be some number with 0 < r < p which is to be fixed later, and let be some cut-off function of class CC°(B2r(wo)) with 1(w) = 1 on Br(wo), 0 < 11 lV

I < 2/r, and rl(u, v) = rl(u, -v). Now we set

0:= d_,{rl2d,Y}.

X,:=h(Y+so),

0 <E<e0(q),

is an admissible variation of X in '(f, S) of type II (see Definition 2 of Section 5.4) for some sufficiently small Eo (0) > 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 << 1.

Thus Y(w) + eq(w), 0 < E< t2/2, is a convex combination of the three points Y(w), Y(w), and Y_,(w). Since n (w) = 0 for I w - woI

2r, we obtain

21(w)=0, 22(w)=0 iflw-woI>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<2r+21t1

86

7. The Boundary Regularity of Minimal Surfaces

and therefore w, w ± t e S2P(wo)

provided that I tI < p - r.

Hence, for w e 12,+111(w0), the points Y(w), Y(w), Y_,(w) are contained in the convex set

CR:={yell 3:y3=0,Lyj
CR:={yeR3:y3=0,y'>-o,Jy
=

h(CR) ifaS=0 h(CR)

if aS

0.

Thus we obtain

X,(w) = h(Y(w) + so(w)) e S for all w c- 1,

provided that 0 < s < t2/2 and I tI < p - r, and clearly

X,(w)=X(w) forweC=aB-1 since q(w) = 0 on C. Consequently, we have

XX=h(Y+ab)ec'(r,S) for0<s>-0.

Inserting the expression (1) into this inequality, we obtain

f. 9jk(Y)Day'Da{4_,(r124,yk)} dude

-2 fB A-t{n241y`}gjk,j(Y)D,,y'Daykdudv where D, = a , D2 = a , ul = u, u2 = v, and a "partial integration" yields 4,[gjk(y)D.yj]D.(t124,yk)dudv fB 1

2

fB

t124,y`4r[9jk,I(Y)DayiDayk]dudv;

see Lemma 1. Since 41C9jk(Y)Day'] = 9jk(Y)Da4ty' + D8yjJlgjk(Y)

7.7 C`1 -Regularity

87

and

D.(n2d1Yk) = D«n2dtyk + n2Dad:Yk,

we arrive at r

9Jk(Y)Ddy'DA,ydudv JB

<-I (3)

2nD.q j,yk[9;k(Y)Dad,y' + D.Yj df9,k(Y)] dudv B

-

B

- 1/2

dudv. lB

The ellipticity condition for (ggk) yields ml

(4)

JB

JR

Moreover, Lemma 1 implies dr [9Jk, i(Y)Day'D2Yk]

_ (d19Jk,,(Y))D.Y'DaYk +

(5)

9Jk,l(Y)DaYi d,D,Yk.

Furthermore, there is a constant K* > 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 <_ c1D(X)

(9)

holds true for all t with 0 < ItI < p - r. If we let t tend to zero, this inequality yields (10)

I

J s2.(wo)

c1D(X)

7.7 C''"2-Regularity

89

since Y = g o X is of class C3(B, R3), and from (8) and (9) we infer g2IDuYI2IVYI2dudv < c2D(X).

(11) S2r(wo)

Moreover, the conformality relation IID.YII2 = IIDJ112

implies that IDS

YI2

-< (m2/mt) A YI2 ,

whence we obtain ?i2IVYI4dudv
(12) S2r(wo)

taking (11) into account. Moreover, by formula (14) of Section 7.6 we have

4y1 + r,.k(Y)(yjyu + yiyk) = 0

in B,

whence IDv

YI2

< c4(IDu YI2 + IVYI4)

in B.

Combining the last relation with (10) and (12), we arrive at I11V2YI 2dudv

(13)

< cD(X)

whence (14)

IVYI4dudv < c6D(X).

IV2YI2dudv + J S,.(wo)

Moreover, from X = h(Y) we obtain V2X = hy,,(Y)VYVY+ h,(Y)V2Y,

and therefore 102X12
By virtue of (14) it follows that (15)

IVIXI2dudv < c8D(X).

J S,.(wo)

This is the desired estimate of V2X. Before we summarize the results of our investigation, we want to prove the estimate (8) which, so far, has remained open. We shall see how co depends on X, and this will inform us about the dependence of the numbers c1, ... , cg on X.

90

7. The Boundary Regularity of Minimal Surfaces

The estimate (8) will be derived from Theorem 2 of Section 7.6 and from the following calculus inequality:

Lemma 2. Let 0 be an open set in C of finite measure, and define d > 0 by the relation nd2 = meas 0. Suppose also that q e L,(Q) is a function such that Iq(w)Idudv < Qr2a

J

(16)

Qr8,(w0)

holds for some number Q >- 0, for some exponent a > 0 and for all disks Br(wo) in C. Then, for any v e (0, a), there is a number M(a, v) > 0, depending only on a and v, such that (17)

MQDQ(0)d"r2«

Iq(w)I IO(w)I2dudv <

J QC B,(Wo)

holds true for all wo e C, for all r > 0, and for any function 0 e 111(0, R'), m > 1. Proof. As the set C, (Q, l8'") is dense in H2(9Q, R'), it is sufficient to prove (17) for all ¢ e CI(Q, R'), taking Fatou's lemma into account.

Thus let 0eCc'(0,Or),w=u'+iu2,1'= From Green's formula, we infer that O(w) =

-I 271

Iw f.0

I -2(Za -

is satisfied for any w e 0. Set f2,:= 0 n Br(wo); then we obtain f" Iq(w)I I0(w)Id2w 1

<2nJQ,JQ

Iq(w)Ilw - CI-11oO(OId2Cd2w

(18)

12w 1

<

1

2n

Q,

Q

Iq(w)I Iw -

CI2v-2d2Cd2w

1/2

l 1/2

[j0J

Ar

Iq(w)I Iw -

By an inequality of E. Schmidt, we have Iw - t;12"-2d2S < (n/v)d2v;

J A

7.7 C',"2-Regularity

91

the simple proof of this fact is left to the reader. Then, by (16), we obtain (19)

n J S2,

V d2"Qr2a

Iq(w)I Iw -

fo

v

For s > 0 and e C we introduce the function

0(s, 0 := f

n e,cc)

Iq(w)I dew.

By (16), we have

as well as

0< i(s,C)
0(s,0 =\J

Er

where

It follows that d

I q(C + seie)I d0.

0(s, 0 = s fz.

Case 1. Let C E B,(wo). Then we have Iw - (I < 2r for any w e Qr. Accordingly,

l..

f

o

Iw _ CI-2v Iq(w)Id2w

Jo

Jr

s_2vIq(C

d

2r

s-2v 0

=

+ seie)IsdOds

E,

dS

d

2r

(s, C) ds = lira a

2v

ds

>fi(s, C)ds

2r

Jim [S-2vo(S' C)]2, e-+0 Q(2r)2a-2v + 2vQ

S_

2v-1 e-++0

1

2(a - v) Case 2. If C e 0 - Br(wo), then we have

co =w0+FC r wol(C-wo)EBr(wo)

e

(2r)2"-2v = c(a v)Qr2a-2v

92

7. The Boundary Regularity of Minimal Surfaces

Fig.1

Moreover, for all w e Sl it follows by a simple geometric consideration (cf. Fig. 1) that

I(-W1>: 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 <- c(a v)Qr2.-2"

From (18), (19) and (20) we infer that Iq (w)l10(w)ld2w < c*(a,

(21) for

In other words, the function q* := qO satisfies

v)Qd"Ds12(O)r2a-

f n

IVO(C)12d2C.

7.7 C', 112-Regularity

(21')

Iq*(w)I dew < Q*r2a*

93

for all disks B,(wo)

JQB(wo)

where Q* := c*(a, v)Qd°D'12(0),

2a* := 2a - v.

Now let v and y be two positive numbers such that v + µ < a. Then it follows that

06*-a-2-µ>a-(v+µ)>0. v

Hence we can apply the estimate (21') to q*, Q*, a*, µ instead- of q, Q, a, v, and thus we obtain

f

01

Iq*(w)I I0(w)ld2w < c*(a, u)Q*d"D Q

or equivalently Iq(w)I O(w)I2d2w < c*(a, v)c*(a,

p)Qd`1'Dn(0)r2a-(,+p)

Replacing v + µ by v and c*(a, v)c*(a, µ) by M(a, v), we arrive at the desired inequality (17).

Now we come to the proof of formula (8). From Y = g(X) it follows that IVYI <

m2IVxI

whence by Section 7.6, Theorem 2 (and, in particular, Section 7.6, (28)) we obtain

that

f

(22)

IVYI'dudv < Qr2a

Br8,(co)

holds for some constant Q > 0, some a e (0, 1), and for all disks BB&). Therefore, (IVYI2 + I DYI2)dudv < 2Qt2a JS2(wo),go)

for some Q > 0, a e (0, 1), and for all disks B,((o) and all t with ItI < to and 0
IVY(w)12 + V Y(w)I2

,

c(w) := rl(w)dt Y(w)

for w E S2r(wo)

and

q(u, v) := q(u, -v),

O(u, v) := q5(u, -v)

for w = u + iv a B2r(wo) and v < 0.

Applying Lemma 2, we can infer that

In(-B,(co)

Iq(w)I Io(w)I2dudv < 2MQD,a(o)(2r)°i2a

7. The Boundary Regularity of Minimal Surfaces

94

and therefore

In particular, for Co = wo and z = 2r, we have 0 =

I9(w)I I O(w)I2 du dv < 4MQDB2r(wo)(0)(2r)2a JB2(w0)

whence, for reasons of symmetry,

ri2jd'YI2(IVYI2 + IDYI2)dudv <4MQ22ar2 J

fs2

J

S2r(wo)

r(wo)

Moreover IV(I141Y)I2

= Iond,Y+ylVz1

Yl2

< 2rl2IVd,Y12 + 8r-21A, y12.

Setting (23)

co := 25+2«MQ max{1, mi' },

we arrive at formula (8). From (38) in Section 7.6 it follows that Q is of the form

Q = cD(X)

(24)

where c depends on the diffeomorphism g and on the modulus of continuity of X on B u I. Hence also the constants c1, ..., c6 are of the form cD(X) with c depending on g and on the modulus of continuity of X. Let us summarize the results (9)-(15), (22)-(24). Theorem 2. Let S be an admissible support surface of class C3, and suppose that X is a stationary point of Dirichlet's integral in the class '(1, S). Then, for any d e (0, 1), there is a constant c > 0 depending only on d, 1913, D(X), and the modulus

of continuity of X such that

( V2XI2 + VXI4)dudv < c

(25) JZ d

holds true.

Applying Sobolev's embedding theorem (see Gilbarg-Trudinger [1]), we derive the following result from Theorem 2: Theorem 3. Let S be an admissible support surface of class C3, and suppose that X is a stationary point of Dirichlet's integral in '(T', S). Then, for any d c (0, 1) and for any p with 2 < p < oo, there is a constant c > 0 depending only on d, p, 1913, D(X), and the modulus of continuity of X such that (26)

f

I OX I" du dv < c

zd

holds true. Moreover both X. and X have an L2-trace on every compact subinterval of I.

7.7 C','12-Regularity

95

In brief, we have shown that any stationary minimal surface X in 16(F, S) is of class Hz n Hp (Zd, 18') for any d E (0, 1) and any p with 2 < p < oo, and X, X, EL2(I', R') for every 1' c c I.

Step 2. Continuity of the first derivatives at the free boundary. The aim of this step is the proof of the following Theorem 4. Let S be an admissible support surface of class C3. Then any stationary point X of Dirichlet's integral in W(T, S) is of class C1(B u I, R3).

Proof. We choose wp e I, xu = X (wo), p > 0, and a boundary coordinate system {all, g} centered at xo as before, and we set Y = g o X = (y1, y2, y'). Then we have (27)

Ay` + I;k(Y)D8y'D.yk = 0.

Hence, for any 5 _ (q,' (,2,

(ps)

E Cr'(S2p(wo) U I2p(w0), 68'), the equation

SE(Y, ¢) = 0

(28)

is equivalent to g,,(Y)yjrokdu = 0.

(29) '2p(WO)

Case 1. 8S is empty. Then 0 is admissible for (28) if cp3(w) = 0 on I2p(w0). We conclude from (29) that

g;l(Y)y' = 0 a.e. on 12p(wo) (30)

gj2(Y)yvj= 0

a.e. on I2p(wo)

y3 = 0

on 12p(WO)

since Theorem 3 implies that both Y. and Y are of class L2(1', F 3) for every

I' cc I. Case 2. 8S is nonempty. Then 0 is admissible for (28) if 9'(w) = 0 on I2p(wo) and if `Y'I12p(wp) has its support in I(w0) := I2 p(wo) n { y' (w) > Q}. We conclude from (29) that

(31)

9;1(Y)YV = 0

a.e. on I P(wo)

g;2(Y)y,'j = 0

a.e. on I2p(wo)

y3 = 0

on I2,(w0).

Now we claim that, for S e C3 and S e C4, we can find an admissible coordinate system {all, g} for S centered at xo which is of class C2 or C3 respectively, and satisfies

96

(32)

7. The Boundary Regularity of Minimal Surfaces

(9;k(Y1, Y2, 0)) =

9t1(Y1,y2,0)

0

0

0 0

922(Y1, Y2, 0)

0

0

1

for all (y', y2, 0) a BR(O) if 8S = 0, or for all (y', y2, 0) a BR(0) n { y' >_ a} if 8S

0. Note, however, that we lose an order of differentiability if we pass from S to g in case of the particular coordinate system {all, g} with property (32). Let us postpone the construction of this coordinate system; first we want to exploit (32) to derive regularity. The special form of the metric tensor (gJk) simplifies the equations (30) to Yv = 0

yV = 0

(33)

a.e. on I2p(wo)

if 3S = 0,

Y3=0 and (31) takes the special form yV' = 0

a.e. on I 2,(w°)

yv = 0 a.e. on I2p(wo)

(34)

y3

= 0

if 8S

0.

on I2p(wo)

Furthermore, we infer from Theorem 3 that (35)

d Y E Lp(S2p(w°), l) for any p e (1, co),

provided that S is of class C3 which implies h e C2 and Ijk e Co. In case 1, we infer from (33) and (35) by means of classical results from potential theory that Y e Hp (S2r(wo), R3) for any p e (1, co), any wo E I, and any r e (0, p); cf. Morrey [8], Theorem 6.3.7, or Agmon-Douglis-Nirenberg [1, 2], for the pertinent LP -estimates.

Then we obtain y C C1' (S2r(w°), E3) for all fi e (0, 1), taking a Sobolev embedding theorem into account; cf. Gilbarg-Trudinger [1], chapter 7, or Morrey [8], Theorem 3.6.6. If S e C°, then h e C3 and Tk a C1, and consequently !;k(Y)DQy'Dayk e I = 1, 2, 3. On account of (27) and (33), we then obtain (36)

AY E C°'°(S2r(wo), E3)

for any $ E (0, 1).

By classical potential-theoretic results of Korn-Lichtenstein-Schauder, we infer from (33) and (36) that y G 1R3) holds for any fi e (0, 1) and any r e (0, p). A simple proof can be derived from the Korn-Privalov theorem; see Section 7.1, Lemma 6. Since h e C3 and X = h o Y, it follows that

Xe u I, R3). Note that the same result can be derived under the weaker assumption S C C3 if we do not work with the special coordinate system (32) where one derivative is lost. Then we have to use (36) and the more complicated boundary

98

7. The Boundary Regularity of Minimal Surfaces

These remarks complete our discussion in the case that S is empty. Now we turn to case 2, i.e., S 0 0. As before, we have (35), and therefore in particular AY2, 4y3 e L,(S2,(wo))

for anypE(1, co),

and the second and third equations of (34) yield yV = 0

and

y3 = 0 a.e. on

I2p(w°).

By the same reasoning as in case 1 we first obtain y2, y3 E HP(S2,(wo)) for p e (1, oo) and r e (0, p), and then (38)

y2, y3 E C"(S2i(w°))

for any /3 e (0, 1)

and r c- (0, p).

(For this result, we only use S C C3, whence h e C2 and Ijk E Co.) The function y' (w) satisfies d y1 E Lp(S2p(w°)) (39) YV' = 0

a.e. on I'2p(wo),

for any p e (1, oo),

y' = a on

12p(wo) - 12'p(WO)-

It is not at all clear how to exploit (39) (cf., however, Section 7.9). Therefore we shall instead use the conformality relations in complex notation, 9ik(I')YJYW = 0,

(40)

in order to show that y1 e C1(92r(w°)) for some r e (0, p). Our reasoning will be similar as in the proof of Theorem 2 in Section 7.3 (see formulas (16)-(20) of Section 7.3). Since (gk) is a positive definite matrix, there is some y > 0 such that

g11(Y(w))_y for all weS2p(wo). Hence we can rewrite (40) as (41)

In +

91L(Y) yt}2

I911(

)w

= r91L(Y) yL12 _ 9Lmt(Y) yLyM "w L91r(Y) W

911(Y)

where repeated indices L, M are to be summed from 2 to 3. If we introduce the complex-valued function f(w) by (42)

.f(w)

YW +

91L(Y) yL' 911(Y)

w e S2v(wo),

we infer from (41) and (38) as well as from Y E C"(B u I, 683) that (43)

f2EC°',1(S2r(w°)) for0
whence f 2 E C°(S2r(w°))

In addition, we have f e C°(S2r(w°)). By the following lemma it will be seen that f(w) is continuous on 92r(w°).

7.7 C' '12-Regularity

99

Lemma 3. Let f(w) be a complex-valued continuous function on an open connected set Q in C such that f 2(w) has a continuous extension to 6. Suppose also that Q is non-degenerate in the sense that, for every wo e OQ, there exists a S > 0 such that QQ(wo) := 0 n B,,(wo) is connected. Then f(w) can be continuously extended

to 6. Proof. Let wo be an arbitrary point on Q. Then there exists a complex number z such that f 2(w) -* z as w -* wo, w e 0. If z = 0, then I f (w) I2 - 0, and therefore f(w) 0 as w -+ wo. If z 0, then we choose some C 0 0, t' e C, such that z = C2. We pick an e > 0 such that 0 < s < Then there exists a number S > 0 such that S2a(wo) is connected, and that f maps QQ(wo) into the disconnected set BE(/3) u BE(-/3). Since f: Q --> C is continuous, the image f(Q5(wo)) is connected, and therefore already contained in one of the disks B,(f), BE(-/3). Thus lim f(w) W -WO exists and is equal to /3 or -/3. Set

wEQ

f(w) F(w) :

lim f (w)

for

w E aQ .

w-w

Clearly this function is a continuous extension off to i2, and the lemma is proved.

Thus we have found that the function f(w), defined by (42), is continuous on S2r(wo) for some r e (0, p). Since (42')

g(w):=

9t1(w)Yw

is Holder continuous on S2r(wo), we infer that YW = f(w) - g(w)

is continuous on S2r(wo), and therefore Y E C'(S2r(wo), 1183). This implies X E C'(B u I, R'), and Theorem 4 is proved. However, we still have to verify that we can find a coordinate system {all, g} centered at xo which satisfies (32). To this end, we choose a neighbourhood all of

the point xo E all and an orthogonal parameter representation x = t(y', y2), (y', y2) e P, of S n all with xo = t(0, 0). In other words, we have F = 0, where B := lty 12,

$ := ,

:= ty2I2

are the coefficients of the first fundamental form of S. Moreover, set

:_ Ity' A ty2I = 8,- ,, 2 =

d,

and let n :=

(ty1 A t,2)

be the surface normal of S. If 8S = 0, we can assume that the parameter domain

100

7. The Boundary Regularity of Minimal Surfaces

P is given by P = KR where

0 < R < 1.

KR :_ {(Y1,Y2): Iy112 + Iy212 < R2},

If OS 0 0, we can assume that S is part of a larger surface So such that So n °ll is y2), (yl, y2) E KR, and that S n au is given represented on KR in the form x = by x = t(yl, y2), (yl, y2) e P = K. n {y1 >_ o}, o- E [ -1, 0]. We can also suppose that aS n all is represented by ton KR n {y1 = a}. Choosing R e (0, 1) sufficiently small, we can in addition assume that t(y1,

(44)

h(y) := t(y', y2) + Y3n(Y1, y2),

y = (y1, y2, y3) a BR(o),

provides a diffeomorphism of BR(O) = {y e l83 : IYl < R} onto some neighbourhood of xo which will again be denoted by all. Then h maps CR or CR onto S n all if OS is void or nonvoid respectively, where

CC.= {yel3:y3 =0,IYI
CR:={yefl3:y3=0,yt>v,IYI
= Ihy112 = d° - 2y3y + (y3)21ny,12

922

= I hy212 = !J - 2y3.N' + (y3)21ny212

933

=Ihy312=In12=1

912

= 921 = = JIF - 2y3A' + (y3)2

913

= 931 = =

923

= 932 = = + y3 (ny2, n> = 0

n> = 0

n> +

for y e BR(O). Hence we obtain '°(Y1, Y2)

(9gk(Yl, y2, 0)) =

0 0

0

0

1(yl, y2) 0 0

1

for y e CR or CR respectively, and the proof of Theorem 4 is complete.

11

Step 3. Regularity of class C1' 1/2 at the free boundary.

Now we turn to the final part of our discussion. We are going to prove Theorem 1. Our main tool will be

7.7 C1 `2-Regularity

101

Lemma 4. Let f(u) be a complex-valued continuous function of the real variable u on a closed subinterval I' of 1, and set a(u) = Re f(u), b(u) = Im f(u). We suppose that a(u)b(u) = 0 on I', and that there are positive numbers a and c such that a < 1 and (45)

IJ 2(U1) -f2(u2) 1 < C21U1 -

u212a

for all ul,u2e1'.

Then it follows that

If(ul) -f(u2)I -< 2cJul - u2Ia for all u1, u2 E 1'.

(46)

Proof. Let ul, u2 e I', ul # u2, and set fl := f(ul), f2 := f(u2). (i) Let clue - u2Ia <- Ifi + f21. Then we obtain CIUl

- u21alfl - f21 G If1 +f21Ifl -f21 G If12 -,/221 G C2I u1 -

U212a

and consequently If1 - f2I <- clul - u2Ia.

(ii) If 1fl + f2l < cI ul - u2Ia and Re f1 = Im f2 = 0, or else Re f2 = Im fl = 0, then I fl - f2I = If1 + f2 1, and consequently

If1 - f2I < CIUl - u2Ia (iii) 11111 + f21 < cl ul - u2Ia and Im f1 = Im f2 = 0, then either l fl - f2l If, + f2 1, and therefore

Ifl -f2I Ifi + f2 1, whence 1al -+- a21 < 1al - a21

for al := Re f1, a2 := Re f2.

Since a(u) is continuous on I', there is a number uo between ul and u2 such that a(uo) := Re fo = 0, where we have set fo := f(uo). Thus each of the pairs { fl, fo} and {f2, fo } is either in case (i) or in case (ii), and by the previous conclusions we obtain

Ifi - fo1 < clul - uola and Ifi - fol <_ cI u2 - uoI2 whence

Ifi-f2I<-2clul-u2Ia. (iv) If Ifl + f2I < CI ul - u2Ia and Re fl = Re f2 = 0, then we obtain by a reasoning analogous to (iii) that Ifi

-f2I:52cIu1-u2Ia.

Because of a(u)b(u) = 0 on I', we have exhausted all possible cases and the lemma is proved.

Proof of Theorem 1. We choose wo e I, xo := X(wo), p > 0, r e (0, p), and a boundary coordinate system {9l, g} with (32) as before, and set again Y = g o X.

102

7. The Boundary Regularity of Minimal Surfaces

As we have now assumed that S e C4, we have g, h e C3, and therefore I';k e C', 1 (Y) e C1(S2,(w0))

Since we have already treated the case 8S = 0, we can concentrate our attention on the case 8S 0 0 where we have the boundary conditions (34). Let f(w) be the complex-valued function defined by (42). Then, by (32), it follows that

f(w) =

(47)

for all W E 12r(w0).

Furthermore, the equations (32) and (41) imply (48)

f

2 911(Y)(y2)2

-

on 12.(x'0)

911(1')(Yw)2

Since y. = Z(y,', - iy,',) and y2, y3 e C1°fl(S2r(wo)) for any /3 e (0, 1) and Y E C'(SZr(wo), f{B3), we infer that f(u) with u e 1':= I2r(wo) satisfies the assumptions

of Lemma 4 for all a e (0, 1/2). Consequently y' is of class C1""(l2i(W0)) for all a e (0, 1/2). Moreover, the Euler equation

dyl = -1';r'(Y)(Yjyu + yy ) can be written in the form

dyt +ay'' +by' =p+gIVy'J2 with functions a, b, p, q e C°" (S2r(wo)). Then an appropriate modification of potential-theoretic estimates (see Gilbarg-Trudinger [1], Widman [1, 2]) yields y' e C" (S2r(wo)) for all a e (0, 1/2). Next we use the Euler equations dye = -ljk(Y)(Y,',Yu +YvYv)

A y3 = - j; (Y) and the boundary conditions YV = 0,

.

to S2r(W0) (Y,',Y.k

+ Yjy )

y' = 0

in 12,(wo)

for any r e (0, p), as well as Y e l3) to conclude that both y2 and y3 are of class C2 °(S2r(w°, R') for any r e (0, p). By virtue of (48), the function f2 is Lipschitz continuous on I' =12r(WO),

whence Lemma 4 implies that y,' is of class

A repetition of the preceding argument with a = 1/2 yields y' e C""2(S2r(wo)), and consequently Y e C', r"2(S2r(wo), R3) for any r e (0, p).

7.8 Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary In this section we want to consider stationary points of Dirichlet's integral in c(I', S) whose support surface S has no boundary. We shall prove that any such

7.8 Higher Regularity in Case of Support Surfaces with Empty Boundaries

103

surface X is of class Cm"t(BUI, 683), provided that S is an admissible support surface of class Cm,P with m >_ 3 and /3 e (0, 1). Moreover, X will be seen to be real analytic on B u I if S is real analytic whence X can be continued analytically across its free boundary I. Our key tool is the following Proposition 1. Let X be a stationary minimal surface in IK(I', S) and suppose that S is of class C°`, m > 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 <- R2, Ix31 < RI

with 0 < R << 1, the surface S n C(R) is given by (2)

x3 = f(xt, x2),

Ixi I2 + Ix212

<_

R2

where f is a scalar function of class C' or Cm,e if S is of class C' or respectively. Then S has the nonparametric representation t(xt, x2) = (xt, x2,.f(xt, x2)),

(x', x2) e Bx(0),

with the surface normal

(3)

n

_f2

(-k, W'W -

n ,n2,n 3)

( 1

where (4)

ft =f,,

f2:=.f2,

W:=-V1+f2+f.2.

Now we choose some r > 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 << 1. Then we extend X(w) from S,(wo) to B,(wo) by defining the extended surface Z(w) as (11)

Z(w) :=

forw a S,(wo) a(X(w)) - d(X(w))n(X(w)) forw E S,(wo)

(X(w)

It turns out that Z is a weak solution of an equation

106

7 The Boundary Regularity of Minimal Surfaces

AZ = F(w) IVZ12

(12)

in B,(wo)

with some function 'F E L.(Br(wo), 683), i.e., we have (13)

J

r e cwo)

( + IVZ12)dudv = 0

for all rp e HZ n L,,(Br(avo), 683). This is proved by first establishing (12) in S,(wo) and in S*(wo) := Br(avo) - S,(wo), and then multiplying (12) by cp. We integrate

the resulting equation over Sr(wo) n {Im w > e} and S*(wo) n {Im w < -c}, e > 0, and perform a partial integration. The boundary terms on aBr(wo) vanish because of (p = 0, and the remaining boundary terms cancel in the limit if we add the two equations and let s -+ 0; the resulting equation will be (13). The cancelling effect is derived from a weak transversality relation which expresses the fact that X is a stationary point of Dirichlet's integral. Concerning details of the computation, we refer the reader to Jager [1], pp. 808-812.

Then, by a regularity theorem due to Heinz and Tomi [1] (see also the simplified version of Tomi [1]), it follows that Z C C1' (Br-(wo), 683) for some a e (0, 1) and some r' e (0, r), whence X E C',"(Sr, (wo), 683), which was to be proved. Remark 2. Another way to avoid LP -estimates, p > 2, is the approach of Step 1 in Section 7.7. Assuming that S is of class C4, we can estimate the L2-norms of the third derivatives of a stationary point X up to the free boundary I, and this will imply X E C'(B u 1, 683 ). In fact, one can estimate I DsX I L, for any s >- 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,


on 16o

for K = 1, 2,


whence

and consequently
on l6o,

K = 1, 2.

108

7. The Boundary Regularity of Minimal Surfaces

This can be written as//'' X. + M'xx(x1, x2)xu

dwf % +

,II,,

'Yxx(xl

x2) dw f3

on Iaa for K = 1, 2, and the identity x3(u) = Vi(xl(u), x2(u))

for all u E Ibo

yields

-Oixx(x1, x2)xu + xU = 0

on lao

(summation with respect to K from 1 to 2!). Thus we obtain (14)

1

0

OX:(xl, x2)

x.,

0

1

ix(xl, x2)

xU

1

XU

-1' (x1, x2)

1/xz (x l, x2 )

x3

fw + '41i i(Wl, x2)fw Wx2(x', x2)fw

= fw+

0

on I. In matrix notation we may write

A(X)X = l(X, f')

(15)

on lao

with a 3 x 3-matrix A(X), the determinant of which satisfies

det A(X) = 1 +/iz3(x1,x2)+sII22(x1,x2)

0

on

lao

for 0 < 60 << 1. Thus we obtain

X = A-t(X)l(X, f') on lao.

(16)

Let us introduce the function F(w, z) for w c- C and

z=(z1,z2,z3)EC3with lwl0, andlzl-p1 (i.e., x c- B31) by setting

F(w, z) := A"1(z)l(z, f'(w))

(17)

The mapping F: §,,, x Bo -+ C3 is of class C1 (differentiability meant in the "real sense" with respect to w) and holomorphic on Soa x Bp',. Then we can write (16) in the form (18)

d

du

X (u) = flu, X (u))

for all u e lao

if So E (0, po] is sufficiently small. Since X(0) = 0, we obtain (19)

X(u) = fo4 F(t, X(t))dt for all uE100.

7.9 A Different Approach to Boundary Regularity

109

By a standard reasoning this integral equation has not more than one solution in C°(lao, R3) since F(w, z) satisfies a Lipschitz condition with respect to z E B,1, uniformly for all w e S 0. By the same reasoning, the "complex" integral equation Z(w) = J0w F(w, Z(c)))dc°

(20)

has exactly one solution Z(w), W E Sa, in the Banach space d(96) of functions Z: SS -> C3 which are continuous on SS and holomorphic in Sa, provided that 6 E (0, 60] is chosen sufficiently small (cf. the proof of Theorem 3 in Section 7.3, and in particular the footnote; one uses the standard Picard iteration). By the uniqueness principle, we have

Z(u)=X(u) foralluela, whence

ImZ=0 on la. Hence we can apply Schwarz's reflection principle, thus obtaining that Z(w) := Z(w) for w c- Bb with Im w < 0

yields an analytic extension of Z across I,, onto the disk Ba centered at uo = 0. Morever, f = X + iX* is holomorphic in Sa, continuous on Sa, and

Re(f-Z)=0 onlb. Thus we can extend i(f - Z) analytically across 16 by

i{ f(w) - Z(w)} :_ -i{ f(w) - Z(w)}

for w e Bb

with

Im w < 0.

Hence

f(w) := 2Z(W) - f (w) for w e Ba

with

Im w < 0

extends f analytically across Ia. Now f(w) is seen to be a holomorphic function on B5, and X = Ref defines the harmonic extension of X to Ba which, by the principle of analytic continuation, has to be a minimal surface on B5.

7.9 A Different Approach to Boundary Regularity In this section we want to give a different proof of the Holder continuity of VX where X is a stationary point of Dirichlet's integral in W(f, S). This new proof merely requires that S is of class C3. The first step is the same as in Section 7.7 and need not be repeated: one estimates the L2-norms of V2X up to the free boundary. The other two steps are replaced by a new argument: We insert a suitable modification of the test function 0 = d {n2d, Y} into the variational inequality SE(Y, 0) >- 0.

110

7. The Boundary Regularity of Minimal Surfaces

This will lead us to a Morrey condition for V2X which, in turn, implies that VX The essential new feature of this is of class Co' on B u I for some a e (0, approach is that we shall explicitly use the first equation of Section 7.7, (34) which states that y,' = 0 a.e. on I 2p' (w°), and y' = 0 on 2p(w°) - 12p(w°). Throughout this section we shall assume that S is an admissible support surface of class C3 in the sense of Section 7.6, Definition 1. As in Steps 2 and 3 of Section 7.7, we shall use a special boundary coordinate system satisfying (32) of Section 7.7. Note that, therefore, the defining diffeomorphisms g and h of the boundary coordinates are merely of class C2. We also assume that we have the same situation as in Section 7.6, that is: w° e I, x°:= X(w°), {111, g} is an admissible boundary coordinate system centered at x°, h = g-1, Y := g(X), Y(w°) = 0; p > 0 is chosen in such a way that I Y(w)l < R for all w e 92p(w°); in addition, {111, g} is chosen in such a way that (32) of Section 7.7 holds true; we have Z].

a and y3(w) = 0 for all w E 12p(w°),

y1(w)

YV'(w) = 0

a.e. on

I2p(wo) := I2p(wo) - {w: y'(w) = v};

finally, by Step 1 of Section 7.7, Y e H? n H4(S2r(w°), l3) for any r E (0, p), as well as Y e C°' (92r(w°), R3) for all a E (0, 1). Lemma 1. Let q _ ((p1, cp2, (p3) e HZ n L.(S2p(w°), E3) be a test function with cp3 = 0 on 12 p(w°), supp 0 cc S2p(w°) U I2p(w°), and suppose that Xe := h(Y + co),

0 < E < E°(0),

is an admissible type 11-variation of X in W(F, S). Then we have D,,yiD,,gjdudv >_ fB 1

(1)

SB

For

Yj:=D1yj-b',

Yz:=D2Y'-d'

with d' = d2 = 0 and arbitrary constants b', b2, b3, d', we also have

>- fB r(Y)Day'Dykco`dudv.

(2) TB

P roof. Because of (32) in Section 7.7, we infer that also X* := h(Y + e W) with yr = 02, 03), /; := g;k(Y)tpk, 0 = ((pl, 92, 93) is an admissible type IIvariation of X in le(F, S), with 03 = 0 on 12p(w°) and supp ' Y CC S2 p(w°) u 12p(w°).

Hence

SE(Y, T) >0

7.9 A Different Approach to Boundary Regularity

111

and by computations similar to those in the beginning of Section 7.6, we obtain (1).

Secondly, an integration by parts yields the identities yaD,,

dudv = f

dude - fB ( dy')cp'dudv

B

_ -J y2cp'du - J (dy')cp'dudv B

I

(Dvy2)tV2] du - fB

J,

J du - J du A B

('

('

= J dudv = J Day'Dacpjdudv. B

B

Hence (2) is a consequence of (1).

Next we shall prove a generalized version of Poincare's inequality.

Lemma 2. For any y > 0, there is a constant M > 0 with the following property: If wo E IIB, r > 0, T2r := S2r(wo) - Sr(wo), 4 E H2'(T2,) and .fit {w E 12r(wo) -I4(wo): W) = 0} >- yr, then

O'dudv < Mr' IT2, IV

2dudv.

1T2,

Proof. Suppose that r = 1, y > 0, and let ley be the class of functions 0 E H2 '(T), T := T2, with {w E 12(wo) - I1(wo) : Vi(w) = 0} > Y. We claim that there is some number M > 0 such that (3)

f

<M

IV IiI2dudv T

T

is satisfied for all 0 a W.. By a scaling argument we then obtain the assertion of the lemma. Suppose now that there is no M > 0 with (3). Then there is a sequence of functions 0k e WY, k a J, such that

dudv > k IT

JT

Without loss of generality we may assume that

7. The Boundary Regulanty of Minimal Surfaces

112

(4)

fT

0 dudv = 1,

whence (5)

IDOk12dudv < 1/k,

f

k e N.

T

Then, by a well known compactness argument for bounded sequences in Hilbert spaces, there is a subsequence {/i;,} of %1 which converges weakly in HZ(T)

to some /i E HZ(T). Then {c; } converges strongly to >li both in L2(T) and in L2(aT), on account of Rellich's theorem and a result by Morrey (cf. [8], pp. 75-77). As Dirichlet's integral is weakly lower semicontinuous, we infer from (5) that

f. I VOI2dudv = 0. Hence there is a constant c such that >G(w) = c

and, because of (4), we have c

a.e. on T,

0.

On the other hand, we have

f

Ian - cI2dr1 >- faTn[iP.=O} Ion - c12d

1

c'T

=c2. 1(aTn{k/i;,=0})>c2y

forallneN.

As

I0., - cI2d°1 = 0,

lim J n-oo

aT

it follows that c2y = 0, which is impossible since c

0 and y > 0.

Now we want to use the generalized Poincare inequality to establish the basic estimates of the stationary surface X in '(F, S), or rather of its transform Y = 9(X). Lemma 3. Set T2r := S2,(wo) -Sr(wo), and (r) :=

(6)

1

rmin

Dy' 2 du dv,

I Dy' l2 du dv I

.

J/ fT'.' Then, for every 6 e (0, 1), there is a constant c = c(8) > 0 such that the inequality

r

(L1

(7)

lV2YI2dudv <

J S,(w0)

holds true for all r e (0, p).

rl+a +

I02YI2dudv

5r2r

7.9 A Different Approach to Boundary Regularity

113

Proof. Choose a cut-off function n as in Section 7.7, and replace the test function 0 in Section 7.7, (1) by A_,{n2(A,Y - A)},

(8)

= (p', q 2, w3),

where

A=(A',A2,A3):=(0,a,0) is a constant vector with an arbitrary constant a e R. We claim that ¢ is admissible for inequalities (1) and (2) in Lemma 1 provided that ItI << 1. In fact,

Y(w)+rq(w)=11Y(w)+A2Y_1(w)+(1 -Al - A2)Y(w)+yA, Al := Et-2n2(w),

et-2g2'(w),

A2

Y

et-1[g2(w) -i2(w)]

Thus Y (w) + se(w) is a convex combination of the three points Y(w), Y,(w), Y,(w)

which is translated by µA, that is, in direction of the y2-axis, provided that 0 < c < t2. Then, by a repetition of the reasoning used in the beginning of 2 Section 7.7, we infer that X, := h(Y + e¢), 0 < s << 1, is an admissible variation of X in le(T, S) which is of type II. Thus we can insert 0 in (1), whence Day'D.A-,{n2(A,y - A')} dudv >_ J

lj1k(Y)Day'Day1A_,{q2(d,y` - A`)}dudv.

If we multiply this inequality by -1 and perform an "integration by parts" (cf. Section 7.7, Lemma 1), it follows that 1.

A,Day'{n2(A,Day') + 2nDan(A,y' - AJ)} dudv

I;k(Y)DayiDayk{(A tn2)(A,y` - A`) + ni

dudv.

As t tends to zero, we arrive at

fa s

+c

IVy12nIVnIID.Y-AIdudv+c fB

IVYI2n2ID.YIdudv. fB

H ere and in the following, c will denote a canonical constant. Then, by means

of the inequality

2ab < ca2 + s-1b2,

114

7. The Boundary Regularity of Minimal Surfaces

we obtain that c e

1.

E

fB r12IVYl4dudv

fs

+

c

Al2dudv.

s

By choosing E = 1/2, the first term on the right can be absorbed by the left-hand side, and it follows that

JS2(wo)

ID.Y-Al2dudv+c


112IVYl4dudv. .S2.(W0)

From the Euler equation AY` + I;k(Y)D.Y'DDYk = 0

we infer the inequality ID,?YI2 < ID. Y12 + cIVYI4,

and consequently I

r721O2Yl2dudv

(9)

< cr-2 J

Al2dudv + c fs2,(-O) ?12IVYl 4dudv.

I

Tr

Since we have already shown that

0
YeHZnHp(Zd,llV),

for any p e (1, oo), it follows that, for any S e (0, 1), there is a constant c(S) > 0 such that

IVYl4dudv < c(6)r"'

(10) fs2,(-O)

holds for all r e (0, p). Moreover, we have

Al2dudv = fT. JT2r

+ID.y2 - a12 +

Poincare's inequality yields (11)

1VD.y312dudv

ID.y312dudv
J Tzr

Tzr

for all r E (0, p) since y3 vanishes on 12p(w0). Moreover, for

IDuy312)dudv.

7.9 A Different Approach to Boundary Regularity

115

a:= f T2

we infer from Poincare's inequality that J

(12)

al2dudv < cr2 J Tzr

L.

is satisfied for all r E (0, p).

Combining inequalities (9)-(12), we find that IV2YI2dudv (13)

< c(b) {r-2 J (l

I D,,ylI2dudv + J Tzr

IV2Yj2dudv + rt+a Tz.

JJ(

By a similar reasoning, it follows that IV2Y12dudv (14)

JSrwO)

c(8) fr-2JTID,yt12dudv+J

IV2YI2dudv+rr+al rz.

z.

J

holds true if we insert the test function (15)

0:= r12d_,d,Y

in inequality (2) of Lemma 1. We leave it to the reader to check that (15) is an admissible test function for (2), and to carry out the derivation of (14) in detail. Then the desired inequality (7) is a consequence of (13) and (14). Now we are going to prove our main result. Theorem. Let S be an admissible support surface of class C3, and suppose that X

is a stationary point of Dirichlet's integral in '(F, S). Then there exists some a c- (0, 1/2) such that X e Ct"°(B u I, R3).

Proof. We have a.e. on

I2,(wo) := {w E 12r(wo) : yl(w) > v}

Yu = 0 a.e. on

I20r(wo) := {w E 12r(WO): yt(w) = Q}

y,i, = 0

Hence, either

,Yt(Ii(Ko) - Ir(wo)) > r or

,Y1(Iir(t+'o) - Ir(wo)) > 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


n

IZWI(I

l+

clol)d2w

which holds for all 0 e C' (Q, C3) and for all Q c B,(wo) with a piecewise smooth

boundary 852. But this relation is the starting point for the Hartman-Wintner technique; cf. Chapter 8, Section 1. Suppose now that wo e I is a branch point of X, that is, IX (wo)I = 0

and

0

and

Xjwo)I = 0.

Then we have I

I

0

and consequently Yw(wo) = 0.

We claim that there is no r e (0, p) such that Y,,(w) = 0 for all w e S,(wo). In fact, suppose that Yx,(w) - 0 on S,(wo). Then we obtain X,,,(w) = 0 on S,(wo), whence Xw(w) _- 0 on B; but this is impossible for any stationary point of the Dirichlet integral in W(J-, S).

From Yw # 0 on S,(wo) for any r e (0, p) it follows that Zw(w) # 0 on S,(wo),

on account of (1). Then, by virtue of Theorem 1 of Section 8.1, there is some vector P = (pl, p2, p3) # 0 in C3 and some number v e N such that

Z4(w)=P(w- wo)v + o(lw - wol')

as w - wo.

Because of zw(w) _- 0, it follows that p' = 0: P = (0, p2,p3).

Now we consider the function e(w), w e S,(wo) - {wo}, defined by

e(w) := (w - wo)-vf(w),

7.10 Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points

119

where f(w) is defined by (42) in Section 7.7. Since Y(wo) = 0 and

(gjk(°)) =

rso

0

0

0

5o

0

0

0

1

go 0 0,

0,

eo

we infer from formula (41) in Section 7.7 that lim e2(w) exists, and that w-+WO

lim e2(w) w-+WO

ISO

lim (w - wo)-2v(y`2,(w))2

so w-wo

lim (w - woy2v(Y.(w))2 . eo w-WO

Then, by Lemma 3 of Section 7.7, we see that lim e(w) does exist. Set F (f1, f2, f3) where w-+WO

f 1 := lim e(w) = lim (w - wo)-"yw(w), w-wo

w-+WO

f2 := p2

f3 ,

p3

It follows that

Yw(w)=F(w-wo)°+o(Iw-woI') asw-.wo where F e C3 satisfies F

0 and <> = 0, i.e. gk, (0)f

k{J

t = 0.

Because of Xw = h3,(Y)Yw, we obtain the following result:

Theorem 1. Let S be an admissible support surface of class C3 and X be a stationary point of Dirichlet's integral in the class c'(I', S). Assume also that wo e I is a boundary branch point of X. Then there exist an integer v >- 1 and a vector

AcC3with A00and = 0

(6)

such that (7)

Xw(W) = A(W - WO)° + o(I W - WoI")

as w -+ w0

We call v the order of the branch point wo.

From this expansion we can draw the same geometric conclusions as in Section 3.2. To this end we write

A=Z(a-ifl)

with

a,ffEf(83.

Then it follows that

Ia1=1#100,

=0

120

7. The Boundary Regularity of Minimal Surfaces

and

(8)

X,(w) = -a Im(w - wo)' + /i Re(w - wo)" + o(Iw - woI')

as w --> wo, whence Q Xu(w) A X,,(w) = (a A N)IW _ Wol2v + o(IW - wol2") as w -> 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

<

f'(r)dr = f(e) - f(r) r

whence

log- <2m-- loge 1

r

c

for all r E (0, 8]

which yields a contradiction since log 1 --> oo as r -- + 0. Similarly case (ii) leads to a contradiction. r Lemma 2. Let E0 be the singular set of a map X E .9A(Q). Then, for any wo E E0, there is a sequence of positive radii rk, k e rl, tending to zero such that (9)

8

lim

log

k-+oo JC(wo,rk) Or

da =

127rv RV

if wo c- Q

if Wo E 80

where r = 1 w - wo 1, w = wo + re `O, v is the order of the branch point wo defined by the expansion (6), and A = IXJ J2.

Proof. Let us write C(wo, r) _ {w e S2 : Iw - woI = r) as

C(wo, r) = {w = wo + re`°:cpl(r)<(p
for 0
loglX.(w)I Iw -- w0I "d(p.

f(r) W 1(r)

By Lemma 1, there is a sequence rk -- +0 such that rk f'(rk) - 0. Since I XW I = ,/A-/,/2-, we obtain

7. The Boundary Regularity of Minimal Surfaces

124

logIXw(w)I Iw - woI

A(w) - log

= log

- v log r

for w e C(wo, r), whence 8r

logIXW(w)I Iw -woI =- log r

A(w) - v

r'

.

Thus it follows that rN2(rk)

(8

('W2(rk)

-

rkf'(rk)= J

8r

W t(rk)

Jdcp

v p I(rk)

+ rk(p2'(rk) log(IAI + Sk) - rkcpl(rk) log(IAI + t3k )

)

where 16k I and {Sk } are two sequences tending to zero. If wo e 12, we can assume that (p, (r) = 0 and cot (r) = 27r, whence 8

rkf '(rk) =

log

(CJr

J C(WO.rk)

da - 27rv.

Because of rkf'(rk) -* 0, we then obtain

a log,da = 27cv.

lim J k-'co

C(wo,rk)

Sr

If wo e 852, then the smoothness of 812 implies Qp2(rk) - (PI(rk) -+ 7E

and

rk{Icpi(rk)I +I(p2(rk)I) --*0

ask -*cc,

whence a log IA-da = 7rv.

lim iC(wo,rk)

8r

Now we turn to the

Proof of Theorem 1. Let a' = {w,,...,wN} and a" = {w,,...,wM} be the sets of interior branch points and of boundary branch points respectively. We consider N + M sequences {ra )} and {F }, 1 < a< N, 1 < ft < M, of positive numbers tending to zero as j --> oc. Set

12j:={we0:Iw-wal>r,l,Iw-wBl>r`#(JI,1
- fai KdA =

A log A-dudv f0i

t aking I X A X,, I = A into account, and /partial integration yields (10)

-

K dA = J fai

any

(8n log

AI

7.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces

125

where n denotes the exterior normal to aQi. According to Lemma 2, the sequences {r,( )} and

can be chosen in such a way that

(11)

J

log

d.*' -- 21rv(w.)

C(wo,rQ'P) an

as j - oo, and that J

(12)

a log V A $" -4 7rv(W,) an

where v(w,,) and v(0,6) denote the orders of branch points % and wB, respectively, which are defined by the corresponding expansions (6). Moreover, let yk be one of the m closed curves, the union of which is asp, and let (a(o), b(oi)), 0 < v < L, be a parameter representation of yk in terms of its parameter of arc length a which orients 00 in the positive sense with respect

to Q. Then we have a2 + b2 = 1, a(0) = a(L), b(0) = b(L), and n = (b, - a) is the exterior normal to yk with respect to Q. The geodesic curvature xg of the (oriented) curve X o yk can, according to Section 1.3, (46), be computed from the formula

,c91A_ =(ab-ab)+nlog

(13)

From a2 + 62 = 1 we infer that L

(14)

- kb) do = ±2n

(ab 0

where we have the plus-sign if yk is positively oriented with respect to its interior domain while otherwise the minus-sign is to be taken. As 8Q consists of the closed Jordan curves y,, y2, ..., y we can assume that y, forms the outer boundary curve of aQ whereas y2, ..., ym lie in the interior domain of yt. Consequently we have the plus-sign for y, and the minus-sign for y2i ... , y., and we infer from (13) and (14) that

-Jo an log

(15)

,IA-du = 21CEk - J

da

where Ek := 1 for k = 1 and sk := -1 for 2 < k< m. (If there are branch points on yk, the integrals in yk are to be understood as Cauchy principal values.) Adding

(15) from k = 1 to k(-iog = n, we obtain

-J

(16)

a,2

8n

Id r' = 2n(2 - m) xaidXI. J faQ

Thus, letting j tend to infinity, we infer from (10) that

KdA = 2n(2 - m) + 2n Y v(wa) + n Y v(w#)

(17)

fo

a=1

a=i

c9IdX l

J

an

126

7. The Boundary Regularity of Minimal Surfaces

provided that the integral f K dA exists as( principal value 0

J 1 KdA = lim J

(18)

i-*

Q

KdA. Q;

In various instances it is superfluous to assume that the principal value (18) exists. Let us consider some instructive cases. Suppose that X E .PP(Q) is a minimal surface. Then we have K < 0, and we infer that f,2 K dA exists, but it can have the value -oc. If, however, X maps 00 m

topologically onto T = U I where I',, 172, ..., rm are mutually disjoint and =1

regular Jordan curves of class C2, then the geodesic curvature K9 of X100 is bounded by IKBI < K where K denotes the curvature of K. Hence f ao K,,IdX I exists and is finite, and we infer from (10) for j --* oo that f Q I K I dA exists and is finite. Thus Theorem 1 implies the following result.

Theorem 2. Let Q be an m -fold connected, bounded domain in C whose boundary consists of m closed regular disjoint curves y1, ..., ym. Secondly, let X E C°(S2, R 3) n C2(92, l3)

be a minimal surface on 92 which maps the yj topologically onto closed regular and

disjoint Jordan curves Ij, I < j < m, of class C2,", 0 < a < 1, with the curvature K. Then f Q I K I dA and f aQ Kg I dX I = f r Ka ds are finite, and we have

KdA +

(19)

Kgds = 27x(2 - m) + 27r Y v(w) + 7r E v(w) fr

fa

wed"

wed,

where ds = I dX I is the line element of F:= U T, v(w) is the order of a branch i=1 _

point £, a':= S2 n 10, a" := 8S2 n L'o, ED is the set of branch points of X in S2, and Kg is the geodesic curvature of F viewed as curve on the surface X. In particular, equation (19) implies that (20)

2-m+ Y v(w) + 1 E v(w) < 1 J wed'

27t

2 wEo

r

Kds.

Here we have used the fact that the assumption T e C2' implies that X E 9R(SZ), as we have seen in the previous sections of this chapter. We recall that the order of boundary branch points has to be even since X maps 80 topologically onto T.

Remark 2. Because analogous X E C°(52, R3) n C2(0, R3) of

regularity

results

AX = 2H(X)X A X,,, IXu12 = IX,I2,

where H e

<X,,, X0> = 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)<-I

v(w) +

It

2 WEa"

W CO'

f

K ds + h2A(X)

where A(X) = D(X) denotes the area of X which in certain situations can be estimated in terms of the length of I by, say, "isoperimetric" inequalities. Remark 3. Let X e C2 n H2 '(0, !l ) be a minimal surface which is stationary with respect to a boundary configuration (St, S2, ..., Sm) consisting of m regular, sufficiently smooth surfaces S, whose principal curvatures are bounded in absolute value by a constant k > 0, and suppose that Q is an m-fold connected m

bounded domain. Then X is of class 9 (Q) and intersects S := U S; perpeni=t

dicularly. Moreover, the geodesic curvature K9 of the free trace E = XIau can be written as Kg = ±K*

(22)

where K* is the normal curvature of E viewed as curve(s) on S. By virtue of I K* I <- k we then infer that IKgI < k.

Therefore we obtain the Gauss-Bonnet formula (23)

fa

KdA +

K91 dX I = 2n(2 - m) + 27t Y v(w) + 7r Y v(w) WEa'

faQ

WEa"

where a' = Eo n Q, a" = Eo n 8Q, and Eo is the set of branch points wo of X, v(wo) is the order of woe Eo, and we have the estimate (24)

2-m+ Y v(w) + 1 Y v(w)<2 L(E). WEO'

WEO

The length L(E) = f E I dX I of the free trace f can possibly be estimated by other geometric expressions (see Section 6.4).

If we want to state similar formulas for minimal surfaces solving partially free boundary problems, we have to take the angles at the corners of aX into account (see Section 1.4, (12) and (12')). The necessary asymptotic expansions can be found in Chapter 8. Remark 4. For minimal surfaces X solving a thread problem (see chapter 10), the thread .E has a fixed length L(E) and a constant geodesic curvature K9 if we view

f as curve on X. Hence it follows that

Iz K9IdXI = KgL(E).

128

7. The Boundary Regularity of Minimal Surfaces

This observation can be used to draw interesting conclusions from the GaussBonnet formula.

Remark 5. It is not difficult to carry over the Gauss-Bonnet formula (14) of Section 1.4 to minimal surfaces T: M - R3 with branch points which are defined on a compact Riemann surface M with nonempty boundary. Suppose that 8M

consists of m disjoint, regular, smooth Jordan arcs yt, ..., T. which are topologically mapped by . ' onto a system F of disjoint, regular, smooth Jordan arcs I t i ... , r,,,,, and let g be the genus of the orientable surfaces X. Then we have

1.

KdA + J x9ds + 4n(g - 1) + 27cm = 2a Z v(w) + 7r Y v(w) /'

weo'

wed.,

where c' and a' denote the sets of interior and of boundary branch points, and v(w) is the order of any w e a' u a". (See in this context also Section 3.6 and Chapter 11).

7.12 Scholia 1. The first results concerning the boundary behaviour of minimal surfaces are the reflection principles of Schwarz; see Sections 3.4 and 4.8. They insure that a

minimal surface can be extended analytically across any straight part of its boundary, or across any part of its boundary where the surface meets some plane perpendicularly. Schwarz's reasoning is described in Section 3.4, cf. Schwarz [2], vol. I, p 181. (As Schwarz mentions, he learned this reasoning from Weierstrass.) Our discussion in Section 4.8 follows the exposition in Courant [15], pp. 118-119 and pp. 218-219.

2. Another important result, found rather early, is Tsuji's theorem that a minimal surface X e H2 (B, R3) has boundary values X I aB of class H; (8B, I3) if their total variation is finite, i.e., if

IdXI < oo. JOB

(Here we have used the parameter domain B = {w: I w I < 1 }.) The importance of this result, which remained unnoticed for a long time, has been emphasized in the work of Nitsche, see [28]. Tsuji's paper [1] appeared in 1942; it is based on a classical result by F. and M. Riesz [1] from 1916 concerning the boundary values of holomorphic functions. We have presented Tsuji's result in Section 4.7.

3. The result stated as Theorem 3 in Section 7.3 is H. Lewy's celebrated regularity result from 1951; see Lewy [5]. It is the direct generalization of Schwarz's reflection principle guaranteeing that any minimal surface can be extended analytically across an analytic part of its boundary. In Courant's monograph [15], this problem was still quoted as an open question (see [15],

7.12 Scholia

129

p. 118). Lewy succeeded in proving his result without using Tsuji's theorem. Our proof essentially agrees with that of Lewy except that we use the fact that X is

of class C°° on B v y if y is a subarc of 8B which is mapped by X into a real analytic are F of R'. One can, however, avoid using this fact which follows from the preceding results of Section 7.3, see Lewy [5], or Nitsche [28], pp. 297-302. 4. Hildebrandt [1] has in a first paper derived a priori estimates for minimal surfaces assuming them to be smooth up to the boundary. In conjunction with Lewy's result, one then obtains the following : Let X E C°(B, 1183) n HZ(B, 683) be a minimal surface which is bounded by m > 4, u c- (0, 1), and that there is a sequence of real analytic curves T" with

a closed Jordan arc F. Suppose that T e

IT-

(1)

asn- oc.

Assume also that there is a sequence of minimal surfaces X bounded by T,, such that

X - X"lcOca> -' 0

as n -+ Co.

Then X is smooth up to the boundary, i.e., X e Cm""(B, 683).

However, it seems that not every solution of Plateau's problem for F satisfies this approximation condition; it certainly holds true for isolated local minima of Dirichlet's integral; cf. Hildebrandt [1]. By approximating a given smooth curve Tin the sense of (1) by real analytic curves J, and by solving the Plateau problem for each of the approximating curves T,,, the above result yields: Every curve T e

m >: 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
<_-

J

=f

JI

<X,,, no> du

('

I X,I d(p = 1(r),

I XI dcp =

<Xr, no> dcp 5 Jc

J

i.e.,

vl(£) < 1(r). This proof of (3), (4) is not quite correct but it can easily be rectified by the reasoning in Section 6.4. We leave it to the reader to carry out the details. 11

By way of an example Ye showed that the assumption o- > 0 in (2) is necessary if one wants to bound I(E); see Ye [2], p. 101. 14. Now we briefly describe Courant's example of a configuration with a continuous supporting surface S and a rectifiable Jordan arc T with end points on S which bounds infinitely many solutions of the corresponding free boundary problem with a discontinuous and even unrectifiable trace curve; see Courant [15], p. 220. We firstly select a sequence of numbers e > 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<1,

Y

n

( Bn:=S(x,Y,z): z=-En,Ixl<1,Iy- 1n

C", := 1(x , Y, z): Ix I <-1, z = Jl

Cn:={(x,Y,z): l

2E2- 1 \y

En2j, 1 <_

-1-

S3 3,

, - 8n < Z <_ 0 } , J

ri

Ixlz= 1 - 2E Cy-1+E,3,J,-En
2

n

J)

,

134

7. The Boundary Regularity of Minimal Surfaces

A2 214

Z2

/B4\

B3

B2

BI

Fig. 1. Cross section of the sets A,,, B,,, C.1, C. at the levels x = ± 1.

D"1,2

:= the compact region in {x = ± 1 } which is bounded by

{x= see Fig. 1.

Set

S1 := {z = 0} - U A,,, n=1

and define S by 00

S:=S1u U

n=1

(see Fig. 2).

Fig. 2. Cross sections of the surface S at the levels x = 0, ± 1.

Then we observe the following: If F1 denotes the straight segment {z = 0, x = 0, (y - 1I < e'}, then the associated free boundary problem . (f1, S) has at least two solutions, namely the representations of the sets

Al :={z=0,0<x<1,ly-11
Al := {z=0, -1 <x<0, ly- ii <ei}. In fact, there is still another stationary but not minimizing surface bounded by F1 and S, namely the surface describing the compact region in {x = 0} which is bounded by {x = 0} n (F1 u B1 u Ci u C'). Similarly, if we set

F"

1

0 ,Y y-n <En n

7.12 Scholia

135

then we obtain at least two minimizing surfaces in' (rn, S) which are determined by the sets

A? :={z=0,0<x<1,

<En}

Y

n

and

An :={z=0,-1<x<0,

y

< E, n

Now let us lift the curves rn to a height z = En, q >_ 4, and connect the endpoints r,I

Fig. 3. The curve r, lifted.

10, n +

E I, p = (0, n _ El, Eq )

via vertical segments with S, see

Figure 3. Denoting the lifted` curves together with the vertical segments again by rn, it is reasonable to expect the existence of at least two solutions Xn , X E '(rn, S) for the problem 9(I',,, S), provided that q is large enough. In particular, we can expect that the minimal surfaces X,,, Xn "converge" to A and A respectively, if q tends to infinity. At a small height z = E < Ert+1, we connect In and rn+1 with a straight line segment parallel to {z = 0} and omit the corresponding parts of the vertical segments, see Fig. 4.

Y

Fig. 4. Cross section of the boundary configuration

S> at the level x = 0.

This way we obtain a Jordan arc rn.n+1 with endpoints on S. Furthermore, let us assume the validity of the following "bridge principle" : Given any two area-minimizing minimal surfaces X. E'1(I'n, S) and Xi+1 E S), there exists an area-minimizing minimal surface Y, E W(I;,,n+1, S) which converges (in a geometric sense) to the union of XX(B), Xn+1 (B) as E tends to zero.

136

7. The Boundary Regularity of Minimal Surfaces

By means of this heuristic principle we obtain at least four stationary or Xn+1, and X, with S) combining X with minimal surfaces in Xn+, or Xn+1, respectively. Similarly we now define the Jordan arcs 1 n+2' An+l and S bound at which bridge the ditches , An+k Then least 2k+1 area-minimizing minimal surfaces which are stationary in S).

Finally, let I':= F,1, denote the rectifiable Jordan arc which bridges all the ditches and connects the points (0, 0, 0) and (0, 1 + E1, 0). Then it follows that there are infinitely (and even nondenumerably) many stationary minimal surfaces in '(F, S) each of which has a discontinuous and nonrectifiable trace curve. Let us add that the previous reasoning is by no means rigorous as the decisive argument, the bridge-principle, is not established. Thus this example by Courant is merely of heuristic value. 15. Complementary to the existence result for the obstacle problem i(E, C) which we have described in the Scholia 4.11 we want to mention some regularity properties of solutions for 9(E, Q.

The problem Y = Y (E, C) is a special case of a "parametric" obstacle problem which was first treated by Tomi [3], [4], Hildebrandt [12, 13], and Hildebrandt-Kaul [1]. Tomi's results are based on important earlier work by Lewy-Stampacchia [1, 2], whereas Hildebrandt's approach uses the difference quotient technique and some important observations due to Frehse [4]. For "nonparametric" obstacle problems we refer the reader to the treatise of Kinderlehrer-Stampacchia [1] and to the literature quoted there. Here we shall restrict our attention to the two-dimensional parametric case, basically following the papers by Hildebrandt and Hildebrandt-Kaul cited above. Consider the integral

E(X) = lB

+ XX] +
and the class C = C(K,'*) := W* n H2(B, K), where'* stands for '*(F) or '*(I, S) respectively and K c l3 denotes some closed set. Let us also introduce the variational problem

9(E, C) : E -+ min in C. Using the abbreviation e(x, p) = g,j(x) {pipi + PiPi} +
mojpl2 < e(x, p) <_ m1jpI2

holds true for all (x, p) e K x R6 Recall that the existence of a solution of 9(E, C) can be proved under the

mere assumption that K be a closed set. If we want to prove regularity, say Holder continuity, we clearly have to add further assumptions on K. The concept of "quasiregularity" turns out to be of use.

7.12 Scholia

137

Definition 1. We call a set K c 683 quasiregular if it is closed and if there are positive numbers 6°i 81 and d such that for any point x0 e K, there exist a compact convex set K* and a C'-diffeomorphism g of an open neighbourhood of K* which maps K* onto K n Bd(xo), Bd(xo) = {x e R3: Ix - xol < d}, such that the matrix

()T (,g)

ma(y) =

satisfies 60

(7)

(y)

<_ 61112

for all (y, i) e K* x 683. Here a9 denotes the Jacobi matrix of g and

()T

stands

for its transpose.

Remarks. 1. Obviously, each closed convex set in 683 with nonvoid interior is quasiregular. Also, each compact three-dimensional submanifold of 683 with C'-boundary is quasiregular. 2. The preceding Definition 1 and the following Theorem I extend to the case where K denotes some subset of 68N, N >- 3.

3. For our purposes it would be sufficient to assume that g is some biLipschitz homeomorphism.

Theorem 1. Suppose that (6) holds with functions gii e C°(K, 68), g,; = gji, and Q e C°(K, 683). In addition let K c 683 be a quasiregular set such that C(K, 2*) = W* n H2(B, K) is nonempty. Then each solution X of A(E, C) satisfies a Morrey condition of the type (8)

(r)"' DBr(wol(X) < DBR(wQ)(X)

R

in 0 < r < R, for each wo e B1_R(0) and all R e (0, 1) and some constant µ > 0. Hence X is of class Co,"(B p3) Furthermore, if W* = (9*([), then X is also of class C°(B, 683), and for V = (*(T, S) we infer that X E C°(B - I, 683). The idea for proving Holder continuity is to convexify the obstacle K locally by using the definition of quasiregularity, and then to fill in harmonic functions with the right boundary values. Elementary properties of harmonic functions will yield the estimate (8). The reasoning is similar to the argument used in the proof of Theorem 1, Section 7.5; for details we refer the reader to the original paper by Hildebrandt-Kaul [1]. We now give a brief discussion of higher regularity properties of X. First we need the following Definition 2. A set K c R3 is of class CS if K is the closure of an open set in 683, and if, for each boundary point x0 e 8K, there exists a neighbourhood U of xo and a CS-diffeomorphism 0 of 683 onto itself which maps U n K onto

138

7. The Boundary Regularity of Minimal Surfaces

B'(0) = {xE R': IxI < 1, x3 >0}, U n aK onto B°(0) = {x e R3 : IxI < 1, x3 = 0}, and xo onto 0.

We shall also assume that the integrand e(x, p) has the following property

(E): There exist some open set .& c 1B3 with K c .,#and functions Q E C2(,#, 183 ) and G = (gJk)7,k=1,2,3 E C2 (,#, R) with g j, = gk; such that 2

e(x, p) _ Y P.G(x)P8 +
and

molp12<e(x,p)<m,p2 for all (x,p)EK x R6,P=(P1,P2) Theorem 2. Suppose that e(x, p) has property (E) and let K be quasiregular of class C3. Then each solution X E C(K, 1*) of 9(E, C) is of class H, (B', 183) n 683) for all B' cc B and for all s e [1, co) and all a e (0, 1). Remarks. 1. Holder continuity of the first derivatives is still valid for solutions of the elliptic variational problem

f f(u,v,X(u,v),VX(u,v))dudv->min inC=HZ(Q,K)nIe*, n

with a "regular" Lagrangian f : Q x .

x 682" --,. iR, where ff c 18" denotes some open set containing K, and 0 c R2 denotes the domain of definition. For details we refer to Hildebrandt [12], [13]. Assuming the conditions of Theorem 2, Gornik [1] proved that each solu-

tion X E C = C(K, ?*) of 9P(E, C) is in fact of class C" 1(B, R'). Simple examples

show that this result will in general be the best possible one. Gornik's work is

based on fundamental results due to Frehse [1], Gerhardt [1], and BrezisKinderlehrer [1] concerning Cl 1-regularity of solutions of scalar variational inequalities. 16. The first to estimate the total order of branch points of a minimal surface via the Gauss-Bonnet formula was Nitsche [6] who reversed an idea of Sasaki [1]. R. Schneider [1] later established the formula

1 + Y v(w) < 27z Z(F) Wea

for all disk-type minimal surfaces X : B -+ R' which are continuous in B and map

aB monotonically (and hence topologically) onto an arbitrary closed Jordan curve F which has a "generalized" total curvature ((F).

7.12 Scholia

139

The method of Section 7.11 and the generalization of the Nitsche-Sasaki formula is taken from a paper by Heinz-Hildebrandt [2]. 17. Let 0 c l 2 be an open connected domain with smooth boundary and suppose t/i e C2(S2) satisfies maxQ c > 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
I

-0

forallvEKy.

Q

Finally, using ideas of H. Lewy (see, for instance, the proof of Theorem 2 in Section 7.8), Kinderlehrer [6] proved that the "curve of separation" l:= {(xt, x2, x3): x3 = u(x) = O(x), x e 85} possesses a regular analytic parametrization provided that c is a strictly concave, analytic function. Thin obstacle problems were treated by Lewy [6], Nitsche [19], and Giusti [2].

Chapter 8. Singular Boundary Points of Minimal Surfaces

The first section of this chapter will be devoted to the study of minimal surfaces

in the neighbourhood of boundary branch points. The fundamental tool for dealing with this problem is the method of Hartman-Wintner which yields asymptotic expansions for complex-valued solutions f(w) of a differential inequality (1)

Ifw(w)I C clwl-'jf(w)I

on

BR(0)

at the center w = 0 of a disk BR(0) = {w e C: I w I < R}. An appropriate modification of the Hartman-Wintner technique will lead to expansions for vector-valued solutions X(w) of a differential inequality (2)

IAX(w)I < clwl-z{IX(w)I + IVX(w)I} on BR(0)

at w = 0. One of the main features of the Hartman-Wintner reasoning is that, instead of (1) and (2), one treats integral inequalities which can be considered as weak forms of the differential inequalities (1) and (2). This will enable us to deal

with certain singularities at the boundary. In fact, by applying a reflection argument, it will in certain situations be possible to treat boundary singularities as interior singularities of solutions to suitably extended equations. However, to make this artifice valid, it will be indispensable to work with integral versions of (1) and (2) because they require less regularity of their solutions. We refer the reader to Section 7.10 (in particular, Theorems 1 and 2) where we have discussed

the behaviour of minimal surfaces at boundary branch points in detail. We emphazise again that the Hartman-Wintner device is the essential tool in proving the asymptotic relation (7) in Section 7.10. In Section 8.1 we shall describe an extended version of the Hartman-Wintner technique as well as some important generalizations due to Dziuk. In Section 8.2 we shall study the asymptotic behaviour of the gradient of a minimal surface near a corner on the boundary. We shall discuss comers on a Jordan curve as well as corners between curves and supporting surfaces since they occur in partially free boundary problems. The results of Section 8.2 provide the initial regularity indispensable for the methods of Sections 8.3 and 8.4 to work. In these sections a precise discussion of the geometric behaviour of a minimal surface at corners will be given. Section 8.3 deals with the Plateau problem for piecewise smooth contours, whereas in Section 8.4 free boundary problems are investigated.

142

8. Singular Boundary Points of Minimal Surfaces

8.1 The Method of Hartman and Winter, and Asymptotic Expansions at Boundary Branch Points This section deals with the asymptotic behaviour of solutions to certain differential and integral inequalities at interior singularities. In certain situations, boundary singularities can be made into inner singularities by extending a solution, for example, by reflection. First we shall consider complex-valued or even vector-valued solutions f (w) of the differential inequality Ifw(w)I
(1)

in a disk BR(O), where A and c are real constants with 0 < A < 1 and c > 0, and f is of class C'(BR(O), C"), N > 1. As usual we write ag gW =

ag

1

2(g.

8w

ig"),

gw =

--w

1

2(gu + igv)

Secondly, we consider vector-valued solutions X (w) = (X' (w), X z (w), ... , X"(w)) of IJX(w)I <_

(2)

C{wl-{IX(w)I

+ IVX(w)I}

in BR(O), c > 0, 0 < A < 1, which are of class C' on BR(0). If the right-hand side of (2) would not contain X but only VX, (2) could be considered as a special case of (1) by setting f(w) := XW(w).

Both (1) and (2) can be transformed into integral inequalities which require less regularity of their solutions. For instance, let f(w) be a solution of (1) in a domain 0 c C which is of class

C', and let .9 cc Q be an arbitrary subdomain of Q with piecewise smooth boundary a-9. Choose an arbitrary function 0 e C' (Q, C) and apply Green's formula

f

g(w)dw =

a-9

2i f f

awg(w)dudv

to g(w) = O(w)-f(w). Differing from our usual notation, we denote double integrals by two integral signs. The integral S0.9 g(w) dw stands for the complex line integral of the function

g over the boundary 0-9 which is assumed to be positively oriented with respect to -9. Then we obtain

r

('

J

a

O(w) f(w)dw = 2i J

f_

Ca(w) f(w) + O(w) fw(w)] du dv,

and (1) yields

I J(w).f(w)dw < 2 f

[(w)I + cjw1-`IO(w)I]If(w)Idudv. JJ This is the integral inequality associated with (1). (3)

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

143

Similary, we have

JXwdw=2iJJ

[ww+0-X.i,I dude,

and we derive from (2) the inequality O(w) X.(w)dw G 2

(4)

Jai

J

f, {I0w(w)I IXw(w)I

+ clwl-'IO(w)I[IX(w)I + IXw(w)I]}dudv. Here c is one quarter of the constant c in (2) because of dX = 4XwW. In the following we shall work with inequalities (3) and (4) rather than with (1) or (2) respectively. Note that (3) makes sense even for continuous f(w), and

(4) can even be considered for functions X of class C'. Hence we give the following

Definition 1. A mapping f (w) = (f' (w), ..., f N(w)), w e BR(O), is said to satisfy Assumption (Al) on BR(O) if it is of class C°(BR(0) - {0}, CN) and fulfils (3) for every 0 e C'(BR(0), C) and for every 9 cc BR(0) with piecewise smooth boundary a-9.

Similarly X(w) = (X'(w),..., XN(w)), w e BR(0), is said to fulfil Assump-

tion (A2) on BR(0) if it is of class C'(BR(0), ll) and satisfies (4) for every 0 E C'(BR(0), C) and for each -9 cc BR(0) with piecewise smooth boundary.

Then we are going to prove the following two theorems: Theorem 1. Let f(w) satisfy (A 1) on BR(0), and suppose that f(w) # 0 in BR(0), and that there exists a number A' E [0, 1) such that

f(w) = O(Iwl-") as w--*0. Then there is a nonnegative integer v such that the limit lim w-"f(w) exists and is W-0

different from zero. Theorem 2. Let X (w) satisfy (A2) on BR(O) and suppose that there is a nonnegative integer v such that X(w) = o(IwIv)

(5)

Then the limit lim Xw(w)w

as w-+0.

exists. In addition, if X (w) # 0, then there is a first

w-0

nonnegative integer v such that (5) does not hold and that lim XX(w) w-" exists

for p = v - 1 and is different from zero.

We shall prove both theorems simultaneously. The proof of the second theorem differs from the first one in that we have as well to estimate the addi-

144

8. Singular Boundary Points of Minimal Surfaces

tional term involving I X(w) I. We will return to this in detail after completing the proof of Theorem 1. Without loss of generality we may assume that f is a scalar function. Before entering into the proofs, we first mention two interesting corollaries.

Corollary 1. Let f(w) satisfy the assumptions of Theorem 1. Then there exists a nonnegative integer v and a complex number a 0 0 such that

f(w)=aw'+o(IwV°) asw-+0. Corollary 2. Let X (w) satisfy (A2) on BR(0) and suppose that X (O) = 0 but X (w) * 0

on BR(0). Then there exists a nonnegative integer y and a nonzero complex vector A such that

X.(w) = Awµ + o(IwI)

(6)

asw--* 0,

and X(w) = Re{Bwµ+t } + o(IwI1+1)

(7)

as w --* 0,

where B = 2(p + 1)-1A.

Proof of Corollary 2. Equation (6) is nothing but a different formulation of the second statement in Theorem 2. Relation (7) follows by a suitable integration. In fact, X (w) = f 1 IuX (tw) + vX (tw)J dt = f 1 Re[2wXW(tw)] dt o

0

=

f

o

Re[2(Atµwµ+t + t1'o(IwIK+I)]dt = Relµ + 1 Awµ+'1 + O(IWlµtl). J

Now we begin with the proof of Theorem 1. Lemma 1. Let f satisfy (Al) and suppose that there exists some nonnegative integer

µ such that

f(w)=o(Iwj1-1) asw-+0. Then f(w) = 0(1 wIµ) as w ---* 0.

Proof.

0,e<minl IZI,r-IIJ,andput r,E = Br(0) - [BE(0) V BE()].

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

145

Fig. I

Now we test inequality() 3 with the function

1

O(w)

1

= w" w

and the domain

This yields the estimate (8)

fa-9".

w-"(w - )-tf(w)dw < 2c

..IwI-µ-xlw - -llf(w)I dude.

JJ

The result will now follow by letting s tend to zero. To accomplish this it will be necessary to consider the boundary integrals on the left-hand side of (8) separately. Firstly, we have f02 n

s

w-"(w - )-tf(w)dw = i

( + ce`4)-4f( + ee'')dcp,

whence (9)

w-"(w - )-tf(w)dw = i f02

lim J C-0

Iw-SI=E

Furthermore we obtain f(W)

w-"(w - i;)-lf(w)dw

L=2 ICI

where we have used that Iw -

>

I21

rzz JO

Iw(w - )I-t Idwl

If(e`p)I s"-1

dcp,

for w e 8Be(0). Hence

8. Singular Boundary Points of Minimal Surfaces

146

w-"(w - )-1f(w)dw = 0,

lim IJ

(10)

E-ao

IWI-E

into account. taking f(w) = o(Iwlµ Now we conclude from (8) the inequality

w-"(w - )-1f(w)dw

(11) JIWI=

lwl-a-alw

< 2c J J

- I-'If(w)Idudv.

IwISr

Define J1 and J2 by the formulas

J1() :=

Iwl-11w - I-1If(w)l Idwl J Iwl=r

and

J2(0 :J J IwlSr

lwl-µ-zlw

- f I-l lf(w)Idudv;

then (11) implies the inequality 2711f(S)S-' I < J1(S) + 2CJ2(S)

(12)

It follows from (12) that the lemma is proved if we find uniform bounds for J1( ) and J2(c) and all e Bro(0) for some 0 < ro < r. The uniform boundedness of

is obvious since the singularities of the integrand, 0 and , have positive distance from the circle I w I = r. Now we show that

To this end we multiply (12) by for integrate over the disk B,(0); it follows that (13)

ICI-µ-zl

2nJ2(wo) = 21r J J

provides an upper bound wol-1 where wo E B,(0) and

-

Ii(wo) + 12(wo),

141
where

and

+

1t(wo)

f_I

ICI-zl -

12(wo):= 2c J J 141<.

Using the identity )-1

j

+

-WO)-'7

and interchanging the order of integration we obtain I,(wo) <

ff,,,,

.

[1w - I-' + I - wol-171 1-'Iw - woIwl=r

I wI-l' If(w)I Idwl}

dS2

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

Mr' -A

147

Iwl-"lw-wol-'If(w)Ildwl Iwl=r

Mr1-1J1(wo),

=

where we have used the inequality

ft

(*)

CI
{Iw-I-'+-wol-'}Id1d2<Mr1

which will be proved later. Similarly we obtain the estimate

12(wo)<2cJJ I
{JJwl
lw -

)))

2Mr1-zcJ2(wo)

with the same constant M. Finally we infer from (13) the inequality Mr1-z[J1(wo)

2itJ2(wo) <-

+ 2cJ2(wo)]

which implies

2(M-1itrz-1 - c)J2(wo) <_ J1(wo)

(14)

1/(1

If we now choose ro <

GM) J2(wo) <_

z)

then the inequality

(M-1nrz-1 - c)J1(wo)

holds for all wo a Bro(0), and Lemma 1 is proved. Now we have to add a proof of inequality (*). In fact we shall prove a slightly more general result known as

E. Schmidt's inequality (see Vekua [1], p. 39). Suppose that w1, w2 E Br(0), and that a, /i < 2 are positive real constants. Then (15)

M1Iw1-w212-a-B I

w1

ifa+$>2

M2+81IlogIw1-w211 ifa+/3=2

B,ff(O)

M3r2-a-P where R

+

if a + $ < 2

and M1, M2, M3 are constants depending only on a and

148

8. Singular Boundary Points of Minimal Surfaces

Fig. 2

Proof of (15). We replace Br(0) by the larger domain B2r(w1) B,(0). If we put Po = 21 w1 - w21, we have for all e B2r(w1) - BPo(w1) that 21 - w21 >- - wl I which yields I

2r

2"07r

I - W1l-aI -

pt-a-adp

f

JJ

21+P 1w (16)

,

1

W212-a-1

iC1- + fl - 2

2 1+a n log

r

ifa+/3>2, ifa + /3 = 2,

1w1 - w2) 23-a r2-a-6

ifa+/3<2.

Applying the linear transformation * =

w1

Iw2w11

which maps Bo(wl) onto

B2(0), we fcofnclude from the change-of-variables formula that 1

I

1 - w11-aI -

(17) -B

= Iw1 -

I';*I-a

w2I2-a-/3 ffBz(o) S

w2 - w1

Iw2-Wil

dpi

By virtue of a, /3 < 2, the integral on the right-hand side can be estimated by a finite constant M(a, S). Inequality (15) follows by combining the above estimates. Lemma 2. Suppose that f satisfies assumption (Al) and that f(w) = o(I wI"-1) as w - 0 for some nonnegative integer p. Then the limit lim f(w)w-,' exists. W-0

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

149

Proof. Let g(w) := f(w)w-" and Fr(y) =

g(w)(w - )-' dw,

(27Ei)-'

r c- (0, R).

wHr

Then is holomorphic on Br(0), and from inequality (11) we infer for all E Br(0) - {0} the relation

-

awl-"-)Iw

n f fa (o)

I-'If(w)I dudv

I-'dudv,

< ct Jf

whe re we have used Lemma 1. We infer from inequality (15) the estimate

100 - FrO1 <-

czr'-A

for all g e Br(0) - {0}. Again from the boundedness of g(w) we conclude the existence of a sequence tending to zero such that a := lim g(wn) e C, n-oo

whence

la - Fr(0)I <-

c2r'-x,

and since , < 1, we obtain lim Fr(0) = a. r- O

Finally we conclude from IFr(S) - Fr(O)I + IFr(0) - al <

I

c3r'-x

the relation lim g(f) = a.

4-0

-

Proof of Theorem 1. The theorem will be proved if we can find an integer v -1 with the properties f (w) = o(I w I") but f (w) 0 o(I w I"+') as w --+ 0, taking Lemma

2 into account. Let us assume on the contrary that for all nonnegative v the relation f(w) = o(Iwl") holds true. We will then show that f - 0 on BR(0). To accomplish this, we recall inequality (14) with wo = 0: 2(M-tirra-t c)J2(0) <- J,(0),

where

J,(0) =

r J iwrr

Iw1-"-tlf(w)I Idwl,

150

8 Singular Boundary Points of Minimal Surfaces

and

wl'Jf(w)Idudv.

J2(O)=

J wJ=r We select r <

f(ro)

/

ucl zl

and suppose that there exists some o e Br(0) with

CM

0. Clearly there exist numbers 0 < 81 < b2, e > 0, such that B,(go) cc

Br(0) and

2(M-iitrx-1 _

c)51 [1

01 + El-"" < 6,r-v-'.

Therefore there exists some constant cl independent of v such that

(+koI)v+

0
r

This relation, however, cannot hold for all v e Z since I o I + E < r. In conclusion we have shown that f = 0 on Br(0) for some sufficiently small r, and a continuation argument implies f = 0 on BR(O). This completes the proof of Theorem 1.

Next we are going to prove Theorem 2. Lemma 3. Suppose X(w) e C'(B,(O), R') satisfies X(0) = 0 and Xw(w) = o(Iwlji-1) as w -a 0. Then X (w) = o(I w l").

Proof. Fix w e Br(0). Then a simple integration yields 1

w") =

= (18)

w"

O

w"

Xv(tw) dt

µRe(wXw(tw))dt w Jo1

= 2 J o tµ-1 { (tw_ Re(twXw(tw)) } dt

=

2

1

Re(towXW(tow))

P (to x')"

for some to a (0, 1). Consequently, lim

X(w)

W-0 WA

= 0.

The following auxiliary result provides a counterpart to Lemma 1.

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

151

Lemma 4. Let X satisfy Assumption (A2), and suppose that there exists some nonnegative integer y such that Xw(w) = o(Iwlµ-') as w -+0. Then X,(w) = O(1 wl'')

Proof. As in the proof of Lemma 1 we put -9r,E = Br(0) - [BE(0) U Be()] and

Then (4) yields the inequality

I

w-" (w - )-'Xw(w)dw

(19)

2c

IwI-µ-zlw - I-1[IX(w)I + IXW(w)I7dudv,

J J,,

taking 0w = 0 on .9r,E into account. Note that Lemma 3 and the inequality 0< A < 1 imply the boundedness of the integral

f f'r(O) Iwl- "-'Iw - I-'IX(w)Idudv.

J3O

Now we can proceed as in the proof of Lemma 1, i.e., we let s -- 0 and obtain the estimate (20)

J1(S) +

J3(S)1,

where f has to be replaced by Xw in the formulas for J1 and J2 respectively, i.e., J1(S) :=

f

IwI-µ1w - I-1lXw(w)Ildwl, wl=r

and

J2() :

Iwl-'-xlw

f fIwISr

- I-'JX,,(w)I dudv.

Since the boundedness of J1 is obvious for small , we only show the boundedness of J2. To this end we multiply (20) by Ifl-a'I wol'1, wo E Br(0), and integrate over Br(0). Then we obtain

-

(21)

27<J2(w0) < 11(WO) + 12(w0) + 13(w0),

where

I1(wo):=

ff

ICI-11 CI<.

8. Singular Boundary Points of Minimal Surfaces

152

- wol1J2()d1 d2,

12(wo) := 2c JJ 13(wo) := 2c

if

l
CI
As in the proof of Lemma 1 we conclude 11(wo) <- Mr1-2J1(wo), 2Mr1-ACJ2(wo)

12(wo) <-

Similarly we infer from (15) and

(w - ) 1( - wo)-1 = (w - wo)-1[(w - )-1 + ( - wo)-1] the estimate 13(w0) < 2M3c1r211-1

for some constant c1

Finally, the boundedness of J2 follows from (21) and the above estimates if we choose r > 0 suitably small. The assertion of the Lemma follows from relation (20) since the right-hand side of (20) remains bounded as -> 0. Lemma 5. Let X (w) satisfy assumption (A2) and suppose that for some nonnegative integer it we have Xw(w) =

o(Iwlµ-1)

as w -* 0.

Then the limit lira Xw(w)w-" exists. W-0

Proof. We put g(w) :=

Xw(w)w_K and

FF(b) :=

g(w)(w - )-1 dw.

(27ci)-1

w1=r

In relation (19) we let a tend to zero (cf. the proof of Lemma 1) and obtain the inequality Iwl-u-zlw

J J a,lol

holding for all

- I-1CIX(w)I +

IXw(w)I]dudv,

e Br(0) - {0}. Now Lemmata 3 and 4 imply c1

f

aco) J s(0)

- I-ldudv

for all

E Br(O) - {0} and some constant c1. From here on we can proceed exactly as in the proof of Lemma 2.

Proof of Theorem 2. Recall that X (w) satisfies assumption (A2) and that for some

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

153

nonnegative v e Z we have (22)

asw-+0.

X(w) = o(jwI")

We first show that the limit lim Xw(w)w

exists. Since X is supposed to be

w+0

differentiable, this clearly holds when v = 0. On the other hand, if v = 1 we infer from (22) that Xw(w) = o(1)

asw --> 0,

and an application of Lemma 5 implies the existence of lim Xw(w)w-1. W-0

In order to prove the general case v > 1, we shall inductively show that Xw(w) = o(Iwj"-1)

(23)

as w -+ 0

holds for all t c- [1, v]. (The result will then follow by a further application of Lemma 5.) Assume the validity of (23) for some t < v; then there exists some number a c- C such that (24)

Xw(w)w_µ

lim

= a.

W-0

We show that a = 0. To this end, observe that we can write (25)

X(u,0) uµ+1

1

u -

fo 1

tµ

Xu(tu, 0) dt

Xu(tu, 0)X"(t0u, 0) (tu)v dt = (t0U)µ

0

I

X (tou, 0)

it+ 1

(tau)"

C t°dt

j0

for some to c- (0, 1).

On the other hand we infer from (24) that the function g(w) :=

Xw(w)w_µ

is

continuous at w = 0, and again (24) implies

a = lim

n-ao

Xw(U,,, 0)

Un

= lim Xu(un, 0) - 1Xotun, 0) -..

2Un

whence (26)

Re a = lim

X (u 0) for every sequence u,, -+ 0. v 2Un

Now (25), (26), the assumption X(w) = o(Iwl') as w -+ 0, and the continuity of g(w) yield the relation

Rea=0.

154

8. Singular Boundary Points of Minimal Surfaces

Furthermore, we infer from (24) that

a=lim

X.(O' Un) - !X'(0' vn) 2i"v"a

n- r

and in particular, if y is even,

+ Im a = lim n- oc,

2v"

v")

where v,, --> 0.

n

Hence the same argument yields that Im a = 0, provided that u is even. If µ is odd we consider the function Y(w) = wX(w). Then Yw(w) = X (w) + wXw(w), and therefore lim

X(w)

Xw(w) = a. + lim µ+l = lim w_0 w"+1 w-'0

YY(w)

W"

W-0 W

0, whence

Also Y(w) = o(IwI"+2) as w

a = lira Y"(0, yn) "+1Un"+l a 2l

yn)

n

whenever vn --+ 0 with n - co. Thus we obtain that Im a = 0 if we repeat the argument above. This proves the first part of Theorem 2. To establish the second statement we assume on the contrary that, for all nonnegative u a Z, we have

X(w) = o(lwl"). It will then be shown that X. = 0 in BR(O) contradicting the assumption that X # 0 on BR(O). Note that, by the first part of Theorem 2, we obtain the relation for all

Xw(w) = O(IwI")

u,

and in particular Xw(w) =

o(Iwl"-1)

as w -> 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 S
13(0) = 2c

I-x 111(i;)d

ff4l
If

fJ

II<.

z

Y

z

K

S =

1 + i2,

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

155

and

IwI-11w - I-'IXw(w)Ildwl,

J1( ) = I

Iwl-µ- zlw

2() =

- I-1lXw(w)I dudv,

dff1Wj!5r

d3()

=11

wl
IWI-"-'IW - i-'jXw(w)Idudv,

w = u + iv.

As in the proof of Lemma 1, i.e., using the Schmidt's inequality and the identity (28)

(w

-

1 = w-'[(w

lK

- )-' + -'],

we obtain the estimates Mrt-zdl(0),

(29)

Il(0)

(30)

12(0) <

2Mr1-zcJ2(0)

Iw
Again we infer from (28) and (15) that 13(0) = 2c JJ

I-II

dal

I-1 if

Iwlw -

X(w)I dudv

= 2c ffwl
f

CI
<

,J

<

r ftX(tw) 01

f

4cM3r1-a

Iwl

4cM3r1-x

J 01 dt

dt] dudv

L,J

fm
wl '` zI X.(tw) I dudv].

[

fwl
I

Now we put z := tw, z = z1 + iz2i and employ the change-of-variables formula. This yields 13(0) <

4cM3r1-z

f' dt

t.+1-2

ffzj
0

< 4+ ,U

_1

4cM3r1-a

A

if

IzI-"-ZIXw(Z)Idz1dZ2, zI
where we have assumed that It + A >- 2. In the following estimates we let r < 1. Then 13(0) <

wlXw(w))dudv, 4cM3r1-z if

I

wI
8. Singular Boundary Points of Minimal Surfaces

156

or equivalently 13(0) < 4cM3r1-'J2(0).

(31)

The estimates (27), (29), (30), and (31) now imply that

27CJ2(0) < %r0% + 2Mr1-2cJ2(0) + 4cM3r'-zJ2(0), whence we obtain for small r > 0 and some b > 0 independent of p that aJ2 (o)

<- Jt (o),

or, more explicitly,

g ft

Iwl-N-'IXW(w)Ildwl

1w 1-l"" -IIXW(w)ldudv 5 Si' wl=r

wI
for all nonnegative p. If we now assume the existence of some w° such that 0, we are led to a contradiction exactly as in the proof of Theorem 1. We have shown that there exists some finite integer v with

Xw(w°)

X(w) = o(Iwly-1) (32)

X(w)Oo(IwI')

By the first part of Theorem 2 we conclude the existence of the limit lim Xµ,(w)w-y+' = A. W-0

If A = 0, we could infer from Lemma 3 that X(w) = o(Iwl') contradicting (32), and Theorem 2 is proved.

Now we shall consider a further generalization of Theorem 1 which will enable us to treat certain systems of differential inequalities as well. This will be of importance in Sections 8.3 and 8.4. Definition 2. Two complex-valued functions F(w), G(w) are said to satisfy assump-

tion (A3) if they are of class C°,' (Ba, L), Bs = {0 < I wI < S}, and if there are numbers a, /f, v e (0, 1), a + /3 = 1 such that the relations IF(w)I = O(Iwly-a)

(33)

1G(w)I =

O(Iwly-e)'

as w -. 0,

and the inequalities (34)

Iw1e-2aIG(w)I2] IFW(w)I <- c[Iwl-1IF(w)I2 + 1 UGw(w)I <- c[ wIa-2eIF(w)I2 + Iwl-aIG(w)I2]

hold true almost everywhere on Ba = Ba - {0} for some constant c > 0.

Here and in the following we shall work with the concept of "generalized" complex derivatives which are defined analogously to "generalized real" (or

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

157

weak) derivatives, and we refer the interested reader to the monograph of Vekua

[1, 2] for more detailed background information. Note that by a theorem of Rademacher (see e.g. Federer [1]) every Lipschitz-continuous function has a weak derivative which is bounded. Theorem 3. Suppose that F and G satisfy assumption (A3) on B. Then there exists a nonnegative integer m such that the functions fm(w) := w mF(w),

gm(w) := w- `G(w)

satisfy one of the following two conditions (i) or (ii): (i)

f m(w) e C°'"(Ba, C) for all p < min(1, m + a),

fm(0)o0, WIm-e)

Ifw (w)I = O(I

Ig(w)I =

O(IWIm+a-2fl)

gm(w) a Co,"(BB, C) gm(0)

as W

O,

as W

0;

for all it < min(1, m + /f),

0 wIm+fl-2a)

Ifw (w)I = O(I Igw(w)I =

O(IWIm-a)

as w --1 0,

as w --+0.

For the proof of Theorem 3 we shall need the following auxiliary results.

Lemma 6. Suppose that f e

C) satisfies

If(w)I = o(Iwi-t) as w ,0

(35)

and Ifw(w)I = O(IwI') as w --* 0

(36)

with some exponent 2 > -2. Then we have:

feC°,µ(Ba,C) forall p<min(1,1 +2), if2> -1, or

If(w)I = O(I wI-`),

(w -- 0), f o r all e > 0,

if A = -1,

or

If(w)I = O(Iwll+'),

(W-0), f ,i. < -1.

Proof. Since fw(w) e L1(Ba(0), C), we can apply Theorem 1.16 in Vekua [1] which

implies that the sum

f(w) + ! 1 f

a,(o)

w)-1 dal

158

8. Singular Boundary Points of Minimal Surfaces

is holomorphic in Ba(0). Hence it is sufficient to prove that the above alternative holds for the function

9(w) = n f fa,(0) If A >

fw(d)( -

w)-1

1, we conclude for w1, w2 e B,, = B6 - 101 the inequality

19(wl) - 9(w2)1 -

T

j j5

°)fw(S)(wl

-O(w22

0db,dS2 ICI


2

ffado) IW1 - IIW2 Using Holder's inequality, we obtain for each y e (0, 1 + 2) the estimate 19(w1) - 9(w2)I < clwl - w21 lffuo) 1

z

z I2z/(1-p)dSl

(1-p)/2 Y

dS2 (1+p)/2

Y

e,(o)

(Iwl - SIIW2 -

Y

K

Now inequality (15) implies

19(wl) - 9(wz)I <- clwl - wzI

1+(2 -(41(1 +p)))(l +p)/2 = clwl

- w2Ip.

If A < -1, we infer again from (15) that 19(w)I < c

ffBdo)

-

cIw11+x

for some suitable constant c. Finally, if A = - 1, it follows that 19(w)I < c1 + c2 1log1wII

In the discussion about to follow we shall always assume that 0 < a < 1 < < 1. Note that this is without loss of generality since a + /3 = I and because of the symmetry of the following assertions both in a and /3 and with respect to F and G. Observe also that we can (and will, if necessary) decrease the decay exponent v in the relation (33). Lemma 7. Suppose that F and G satisfy assumption (A3) on Ba = B6 - {0} with a < z. Then F e C) for all it e (0, a) and, furthermore, the relations

IFF(w)I=0(IwI-1) I G (w)I =

O(Iw12-2P)

hold true almost everywhere on B6.

Proof. The proof is based on an iteration argument where one has to use Lemma 6 in each step. To start let us assume that v < a, whence for some k° e N u {0}

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

159

we have that 2kov < a < 2ko+tv. Now assume that for some k e N u {0}, k < k°, we have

IF(w)I =

O(Iwj2kv-a),

O(IW121v-F).

IG(w)I =

Then (34) implies IFw(w)I = o(IwI2k

, a-1)>

O(Iw12k+,"-e-1).

IGW(w)I =

From Lemma 6 we infer O(IwI2k."-°)

IF(w)I =

as w --> 0

if k < k°

or

F(w) a C°-" (Ba, C) for all p < 2ko+1v - a

if k = k°.

Also,

IG(w)I = O(1 +

IwI2k.l"-e)

if k < k°.

By virtue of (33) we can start the iteration by putting k = 0. In conclusion we obtain that F E Co,"

for all

it < 2k0+1v - a

and in particular G(w)I = 0(1 + wI2k°`1"-e).

IF(w)I = 0(1), Again we infer from (34) that

IFw(w)I = 0(IwI-1) =

O(IwI1-1),

since 2'- 12V > 2(x, and O(IWIa-21

IGW(w)I =

=

O(IWI'-31)

Finally we infer from Lemma 6 that F a C°-µ(B,, C) for all 0 < u < a.

In the next lemma we improve the regularity of G provided we know that F(0) = 0. Lemma 8. Suppose F and G satisfy (A3) on Ba = B8 - {0} with a < 2, and that F(O) = 0. Then we have G e C°''`(BB, C) for all p a (0, fi) and IFw(w)I =

O(Iwj1-3a),

IGw(w)I = O(Iwj-a)

as w - 0.

Proof. Since F(0) = Owe infer from Lemma 7 that IF(w) I = 0(I w I") for all p < a.

Hence the function f(w) :=

F(w) w

satisfies

If(w)I=0(IwI1-1) By Lemma 7 we have IGw(w)I =

asw-->0, forall0
as w -.4 0, and Lemma 6 yields

8. Singular Boundary Points of Minimal Surfaces

160

IG(w)I = 0(1 +

Iwla-28+1)

if a - 2/3 0 -1,

that is,

if a-2/3=-1

IG(w)I=0(Iwl-E) for all a>0,

(i.e.

a=

13

.

Using inequalities (34) we obtain the system Ifw(w)I <-

(37)

IWI-3,, IG12]

C1[Iwl1If 2 +

IGw(w)I <- c2[1w132If12 + IwI-°IG12]

which holds true for almost all w e B5.

we infer from (37) the

If a - 2/3 _ -1 (or equivalently a = 3, /3 = 3 relations

asw-i0,

Ifw(w)I = O(IWI-1-e)

(38)

IGw(w)I =

for all a>0

O(IWl-113-,)

whence in particular

for ally <

G(w) e C°µ(B5, C)

2

= /3

and (39)

IG(w)I = 0(1),

IG(w)I = 0(IWI-1),

for all e > 0. Inserting (39) into (37) we obtain O(IWI-3a),

Ifw(w)I = O(Iwl-t) = IGG(w)I = O(1) =

O(IW11-3a),

and therefore IFF(w)I = O(1) =

O(1w11-3a),

because of Fw = wfw.

Now we deal with the case a - 2/3 > -1 (or equivalently /3 < 3, a > Inserting the relations IG(w)I = 0(1) and

IG(w)I = 0(Iw1µ-1),

µ
in (37), we obtain Ifw(w)I =

O(IWI-3a)

as w--*0. O(Iwl-a)

Now Lemma 6 implies that IFw(w)I =

O(Iw11-3«),

forally <1-a= fl.

3):

8.1 The Method of Hartman and Wintner and Asymptotic Expansions

161

Finally, we have to treat the case a - 2f < -1 (or /3 > 3 and a < 13): To this end we fix some y < a and select some k° e N u {0} with the property 2k0(p + a) < 1 - a < 2k0+1(p + a). Assume that for some k:!-< k° the relations (

O(1w12k("+a)-a-1)

If(W)I =

k)

IG(w)I =

O(1Wl2k("+a)+a-1)

0,

as w

as w -+ 0,

hold true. Then it follows from (37) that lfw(w)I =

O(IWI2k+1("+a)-a-2)

and O(Iw12k+1("+a)+a-2)

IGw(W)I =

If k < k°, then Lemma 6 applies and we arrive at the relations 0(IW12k+1("+a)-a-1)

If(w)I =

0(lWj2k+1("+a)+a-1);

IG(W)I =

in other words, the validity of (40k) implies the validity of (40k+1). On the other

hand, for k = k° we obtain IWI2ko+1("+a)-a

If(w)I = 0(1 +

(41)

1)

IG(w)l = 0(1).

We can start the induction because (40k) holds with k = 0 taking it < a into account. We insert (41) into (37) and get I fw(w)I = 0(Iwl-'a) as w - 0 and IGw(w)l = O) for all

0(lwl -a) whence we infer by means of Lemma 6 that G E

p<1-a=fl and also

lFw(w)l=0(Iw11-3a)

Next we suppose that both F and G vanish at zero. Then by Lemmata 8 and 6 we conclude that the functions

f(w) := w-1F(w) and g(w) := w-1G(w) fulfil the relations I.f(W)I =

Iw11-3a)

if a 3 O(lwl-E), for all s > 0, if a = 3

{0(1 +

and

Ig(W)I = 0(IwI-1.

Therefore there exists some number A' E (0, 1) such that the mapping h(w) :_ (f(w), g(w))

satisfies the relation Ih(w)i = 0(Iwl-"')

asw-0.

162

8. Singular Boundary Points of Minimal Surfaces

From (34) we easily infer an estimate of the type Ihw(w)I <_ cIwI-'Ih(w)I

holding almost everywhere on B8 with some constants c and , c (0, 1). We are thus in a position to apply Corollary 1 of this section to the function h and obtain the existence of some positive integer m and of a complex vector A E C2 - {0} such that

h(w)=Aw'

(42)

asw-+0

'+o(Iwlm-1)

holds true on B. Now we come to the proof of Theorem 3. Without loss of generality we only consider the case a < i. We distinguish between the following alternatives (which clearly exhaust all possibilities!): (a) F(0) 0 0,

G(0) 0 0,

(/3) F(0) 0 0,

G(0) = 0,

(y) F(0) = 0,

G(O) 0 0,

(S) F(0) = 0,

G(0) = 0.

If (a) or (/3) hold true, then Lemma 5 yields that (i) must be satisfied with m = 0. In view of Lemma 8 we obtain (ii) with m = 0 provided that (y) holds true. Finally, let us assume that F(O) = G(O) = 0. Then (42) is equivalent to F(w) = awm + o(I wlm) G(w) = bwm + o(Iwlm)

as w --+ 0

with complex numbers a, b which are not both equal to zero, and we obtain

f m(w) = a + 0(1)

(43)

gm(w) = b + 0(1)

as w -+ 0.

On the other hand, we easily derive from (34) the inequalities WIm+B-2algm(w)121

(44)

Ifw (w)I < C[IWIm 1Ifm(w)I2 + 1 IWIm-algm(w)I2], C[lWlm+a-2PIfm(W)I2 Igw(w)I < +

and, together with (43), this yields Ifw (w)I =

0(lwlm-1)

and

19w(w)I = O(I

WIm+a-2e)

as w -.0.

But then Lemma 6 can be applied which proves that f m e C°" "(Bb, C) for all µ < 1. Assuming that f m(0) 0 0 we have thus shown that (i) holds true. So let us assume that f m(0) = a = 0 (whence b = gm(0) 0 0). Then clearly If m(w) I = 0(1 wl") as w - 0 for ally < 1, and (44) implies Ifw (w)I =

0(IWIm+#-2a)

and

Igw(w)I =

O(IWIm-a).

Again, by Lemma 6 it follows that gm e C°'"(Bb, C) for all µ < 1, and hence (ii) holds true; thus Theorem 3 is proved.

8.2 A Gradient Estimate at Singularities Corresponding to Corners on the Boundary

163

8.2 A Gradient Estimate at Singularities Corresponding to Corners of the Boundary In this section we consider solutions X = X(u, v) of the Plateau problem 9(T) for a Jordan curve F consisting of two regular pieces T+ and 1'- of class C"" which enclose a positive angle f3 < it at a common point P E T+ n F. We are then interested in the behaviour of X near the corner point P and, in particular, in asymptotic expansions for the gradient VX(u, v) near the point w° e 8B which

corresponds to P. More generally, let X e W(T, S) be a solution to the free boundary problem 9(T, S) and suppose that the configuration <1', S) satisfies some chord-arc condition (see Section 7.5). Then we conclude from Theorem 2 of Section 7.5 that X is globally Holder continuous on the closure of the semi-disk B = {(u, v): u2 + v2 < 1, v > 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 <6,v>0}, 161- :={w=(u,0): 0
1a :={w=(u,0): -6
(i)

(ii) X(Ia)cS,X(0)=0=P1eTnS; (iii) X, I,+ is orthogonal to S along the free trace XI,+.

Then Theorem 2 follows from

Proposition 1. Let X C C2"(B6 - {0}) n C°"°`(BB) be a minimal surface which fulfills assumption (A). Then the gradient OX satisfies (2)

IOX(w)l =

0(Iwla-t)

as w -' 0.

The proof of Proposition 1 rests on a further investigation of solutions X(w) of the differential inequality (3)

IA1(u, v)I < aIVX(u, v)I2

8.2 A Gradient Estimate at Singularities Corresponding to Corners on the Boundary

165

which was already considered in Section 7.2. We recall Proposition 1 of Section 7.2.

Proposition A. There is a continuous function K(t), 0 < t < 1, with the following properties: For any solution k E C2(BR(wo), R") of the differential inequality (3) satisfying

IX(w)I < M,

(4)

w e BR(wo)

for some M with aM < 1, the estimates (5)

and

IVX(wo)I <_ K(aM)

(6)

IVX(wo)I <-

K (aM)

R

R sup

IX(w) - X(wo)I

W E BR(wo)

hold true.

Lemma 1. Let D c Bl(0)be a domain such that D contains the origin. Suppose that k e C2(D, RN) n C°(D, R') satisfies inequality (3). Then there exists some b > 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 <-

x(1/2) E

K(1/2)

IV

(wo) I E

sup I Y(w) - Y(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<
166

8. Singular Boundary Points of Minimal Surfaces

where r, cp denote polar coordinates about the origin. Let X

w = (u, v) e DP,

I

be a mapping of class C°(Dp, R ") n C2 (D, 1N) which satisfies (3)

IdX(w)I :!5; ajVk(w)j'

on D,

and IX(w)I

(8)

c1Iwl'

on Dp

with numbers a, c1 > 0 and 0 < a < 1. For arbitrary fixed 0 e (0, y/2) we put

D,,B:={w=re`°:0<
Lemma 2. Suppose k satisfies Assumption B on D. Then, for every 0 e (0,

2),

there exists a constant c2 = c2(0, a, c1) such that the inequality VX(wo)I <- c2lwola-1

(9)

holds true for all wo e D6,,e and for some b1 e (0, p).

Proof. Let b > 0 denote the number determined in Lemma 1. We take b1 z min(b, p) and put E := i Iwol sin 0. Then BE(wo) c D. n Bp(0) for all wo e Dale and Lemma 1 implies the estimate

IVX(wo)I < const g-1 sup IX(w) - X(wo)I B,(w0)

< const c1E`[Iw0I°` + (Iwol + c)1]
11

The estimate (9) controls the behaviour of the gradient on the segments D6,0. To obtain also some information on the remaining parts Da,o or Db g, we have to make additional assumptions.

Lemma 3. Suppose that X satisfies Assumption B, and let 0 e (0, min

7

41)-

In In addition, assume that X (re`4) = 0 on 0 < r < p and cp = 0 or p = y, respectively.

Then for small b > 0 we obtain the estimate (10) on Da,e or D, ',o respectively.

IVX(wo)I <- const Iwol'-1

8.2 A Gradient Estimate at Singularities Corresponding to Corners on the Boundary

167

Proof. It is sufficient to prove (10) for wo e Da,e. To this end we select some S < min(p, 1) such that

am < 1

(11)

where

M:= sup 1I(W) I Da

and where a denotes the constant in (3). Applying Proposition A, we derive the gradient bound IOX(wo)I < ca-1 sup k(w) - X(wo)I

(12)

BAWD)

holding for some constant c independent of 8 and for all r > 0 satisfying 0 < e < dist(wo, 8D6).

Now we restrict wo further so that I wo I < 2. Put uo = Re wo, RB := 2uo sin 0, w, = (uo, 0) and Ba(wl) := BRB(w,) n { (u, v) : v > 0}. Then we find BE(wo) c BR,(w,)

for all e < dist(wo, ODs)

and

B2Re(wl) C DS

taking the smallness of 0 into account. We define harmonic functions cp(w) _ (ggl(w), ..., (pN(w)) and /(w) by dcp = 0

cp(w) = X(w)

on BZR0(wl),

on OB2R,(w,),

and

di/i = 0 on BiR6(wt),

i(w) = IX(w)I2

on BBzRB(w,)

Consider the function K(w) := <X (w) - W(w), e> +

a aM) {o (w) - IX(w)IZ},

2(1

W e B2+R8(w, ), where e e J N is an arbitrary unit vector. Then )

AK(w) _ -

I AX I -

a

1 - am

{, VX12 + <4X, X> } J

1 am jV X 12 + 1 am JAI, 111


a

1-aM

IVXI2+

2 am 1-am IVXI2=0

for w e B2R0(w,). Furthermore we have K(w) = 0 along BBRB(w,); hence we

168

8. Singular Boundary Points of Minimal Surfaces

conclude from the maximum principle that K(w) > O on ByR,(%,I ), In other words,

<(p(w) - X(w), e) 5

a

cr

2(1 -

2(1 --

aM)IX'(w)12.

Since e is an arbitrary unit vector, this implies the estimate r0 (W) - X(w)15

aaM) W w)

2(1-

1. '(w,)IZ }

in particular (13)

19(w)1:5 Iw(w)I + -..__.`'._,_ I0w)1

2(1 - am)

for »^ E B R(,A,t

On the other hand, we infer from (8) the inequality IX(w)I 5

2R®}a

5c,{1 for all w e BB2R,(w, ), whence IrP(w)I 5

(14)

10(w)I 5

w s B2R,(w, ),

c2(0)Iw0Ia,

cz(e)Iwol2a

5

c2(0)Iw0la

x" E

since Iwo1 < a < 1. Employing the reflection principle for harmonic functions, it is possible to extend (p and 0 harmonically onto the disk B2R.(wl ), taking account

of the fact that (p, ¢ vanish along the line {(u, 0): uo - 2R5 < u < uo + 2R9}.

Denoting the reflected functions again by (p and yt,, we see that (14) continues to hold. The mean value theorem yields the relations

IVOw)I 5

1.

sup Irp(w)I,

W E BR.(wt ),

Re B,y(w,)

IoO(w)I 5 R sup

w E BR,(w, ).

9 BR,(wt)

Together with (14) this implies

Iorp(w)f 5 c3(6)Iw0Ia'' IVV(w)I s c<(B)Iwola-t for all w E BRa(wl).

Finally we conclude from (13) and from the (15)

IX(w)I 5 JP(w) - rp(w,)I

+----al 22(1

5 c5(a, M, O)

Iwora-,

Iw

mean value theorem that

aM)

- w, l

5 c5(a, M, 0)1wola-t2 dist(wo, -OD,),

8.2 A Gradient Estimate at Singularities Corresponding to Corners on the Boundary

169

for all W e Bdist(wo,aD,)(wo). The desired result then follows from (15) and (12) taking e = z dist(wo, aDa).

Lemmata 2 and 3 imply the following

Proposition 2. Suppose that j( satisfies Assumption B and that X (re"') = 0 for 0 < r < p, cp = 0 or cp = y. Then the asymptotic relation IVX(w)l = O(Iwlm-1)

as w -, 0

holds true.

Now we turn to the Proof of Proposition I (and hence of Theorem 2). Since we have assumed that X(0) = P1 = 0, we infer from the Holder continuity the estimate

IX(w)I
IDX(w)I
for all weDa1,B

and some 6, e (0, p). Next we prove (16) on

Da3,e={w=re`co:0
Ihe(z)I <

4

,

i = 1, 2,

z e [0, e).

We extend the functions h1, h2 as even functions to the interval (-e, s) and define

z(w) := x(w) - h1(z(w)) (18)

Y(w):= y(w) - h2(z(w))

for w e D,

2

where we have chosen 62 so as to satisfy z(Da2) c (-s, e). Consider the mapping X(w) := (x(w), y(w)), w e Ds2, which fulfils (19)

X(w) = (0, 0)

on

Ia2

.

Furthermore, since X(w) = (x(w), y(w), z(w)) is harmonic we obtain

170

8. Singular Boundary Points of Minimal Surfaces

dz(w) = -hi(z(w))IDz(w)I2,

w c- D32,

dy(w) = -hz(z(w))IVz(w)I2,

w e Dal,

whence (20)

LX(w)I < c loz(w)I2,

w c- D,,, ,

with a suitable constant c. Relations (17) and (18) imply the estimate

IVx(w)I < 4IVz(w)I + ox(w)I,

weD62.

(21)

IVY(w)I <- 41Vz(w)I + IVY(w)I,

From the conformality condition we conclude IVz(w)I2 < IVx(w)I2 + IVy(w)I2 2IVzI Ivxl + V < I16IVz12 +

+16I IozI2+2Vz1IV

2

I+IV I2

8IozI2+4{IV I2+IVYI2}

on D,,,,

thus (22)

IVz(w)12 <

0IVX(w)I2, w C D62'

Inequality (20) now yields IA (w)I < aIOX(w)I2, w c- Dal, for some constant a. By virtue of relation (19) we are in a position to apply Lemma 3 to the function X, and we obtain the estimate (23)

VX(w)l < clwla-i

on Da3,0

for some number b3 < S2. Finally it follows from (21) and (22) that X itself satisfies (23), i.e. I VX I < c I w I'-' on Da3, 0.

Now we have to verify (23) on the set

Da6,9={w=re`P:0 0 chosen appropriately. Performing a suitable rotation in l assume that S is locally given by z = f(x, Y)

we can

8.2 A Gradient Estimate at Singularities Corresponding to Corners on the Boundary

171

with some differentiable function f defined in a neighbourhood of zero such that f (0, 0) = 01

Vf (0, 0) = 0.

Define

Z(w) .= Z(w) - f(x(w), y(w)), x(w) := x(w) + i(w)fx(x(w), y(w))n(w), y(w) := y(w) + z(w)f,(x(w), y(w))n(w), where

n(w) := [I + f 2(x(w), y(w)) + f2(x(w), y(w))]

t

and w e Db2 with S2 so small that (x(w), y(w)) is contained in a neighbourhood of zero where f is defined. We remark that z(w) = 0 on Ia2 and secondly, because of v) = x (u, v) + i,(u, v)fx(x(u, v), y(u, v))n(u, v) + z(u, v) [ f (x(u, v), y(u, v))n(u, v)]V ,

we have for w e Ib the equality xju, v) = xv(u, v) + {ZV(u, v) - ff(x(u, v), y(u,

- f,(x(u, v), y(u,

v)

v)} f (x(u, v), y(u, v))n(u, v).

Equivalently, for w e Is2, v) =

v) -

v), Ns(X(u, v))>n'(X(u, v)),

where Ns(X(u, v)) = (n1(X(u, v)), n2(X(u, v)), n3(X(u, v)))

denotes the upward unit normal of S at X (u, v). However, X intersects S orthogonally along I+ ; thus 32 x"ju, v) = 0

on Ia2.

Analogously we find y (u, v) = 0 on 1b2 whence the function X(u, v) := (x`(u, v), y(u, v)), (u, v) e Daz, satisfies (24)

.

(u, v) = 0

on I,, .

Furthermore we infer from the definition of x, y, i, from f(0, 0) = 0, Vf(0, 0) = 0, X (O, 0) = 0, as well as from the continuity of X the relation Ixv(w)I2 >_ const{Ix,(w)J2 - E[Iy0(w)12 + IZJw)I2]}

which holds true for w e Da3, b3 = 63(c) < S2, and for arbitrary fixed e > 0. We observe that similar relations hold for x., y,, and y,,.

172

8. Singular Boundary Points of Minimal Surfaces

From the conformality condition we first obtain that I Vz 12 < I Vx I2 and hence on Da,.

IVX(u, v)12 < const IVX(u, v)12

(25)

+ I Vy I2

Similar arguments show that for some b4 < 63 the estimate j4X(u, v) I < const IVX(u, v)12,

(26)

(u, v) e D54,

holds true. We reflect X(u, v) so as to obtain a function X(u, v) given by ,9(u, V) =

X (u, v),

(u, V) a D64

{X (u, -v), (u, -v) e Db4

By virtue of (24) we obtain for each function 0 e C, (Bb,(0) - 1a4, R2) the equalities

VX(u, v) - V(u, v)dudv JB4(O)

=J

VX(u,

v)dudv

Bb4_Dd4

dX Odudv - f"

_ -J Dd4

Xv(u, 0) O(u, 0) du

,

-J

AX 0dudv + J Bb4-D84

r =I

OX (u, -

(u, v)dudv + J

D8(4'

X (u, 0) 0(u, 0) du 14

F(u, V, X(u, V), VX(u, v)) 0(u, v)dudv

JJ aa4

for some function F which grows quadratically in I V X J (compare with inequality

(26)). By construction, the function X(u, v)

is

of class C°(Bs4(0), R2) n

C' (Ba4(0) - Is4), and the preceding discussion shows that it is a weak solution of the two-dimensional system

dX = F(u, v, X, OX) in

B84(0) - Ib4.

Standard regularity theory (see, for instance, Section 7.1, and Morrey [8], Gilbarg-Trudinger [1]) now implies that X is in fact of class C2(Bb4(0) - IB4)

and satisfies (26) classically on all B,,4(0) - I. Finally we apply Lemma 1 to the domain D = B54 - Ib4 and to the function X ; the resulting inequality is

JVX(w°)I < const s-' sup X(w) - X(wo)I R .(WO)

for all w° and s with the property Be(w°) c Ba, - Ia where 65 < 84 is the constant then a suitable determined in Lemma 1. If w° is restricted to lie in Db6,0, 66 := choice of a would bee = 2 l wo 1. Hence ib5,

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

173

IVX(wo)I < const e-1[IwoI' + (Iwol + e)"]

< const IwoI", that is, (23) holds true on Db6,e. Because of (25) we finally arrive at relation (2).

O

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves and Their Asymptotic Behaviour at Corners In the previous section we proved an asymptotic estimate for the gradient of a

minimal surface X at a corner P of a given piecewise smooth boundary arc T+ u 1--. It is the purpose of this section to obtain some more precise information on the asymptotic behaviour of X,,, near the corner P. To give an idea what might happen we start with a simple but characteristic example: Let a e (0, 1) and k e N u {0} be given and define X(u, v) = (x(u, v), y(u, v), z(u, v)),

(u, v) e B = {u2 + v2 < 1, v >_ 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
174

8. Singular Boundary Points of Minimal Surfaces

and

F-={(x,y,z)e 3: z=0,arg(x+iy)=na,0<x2+y2 1 whence X winds around zero k-times. However, there is another possible solution to the Plateau problem determined by T+ and r-, namely the surface X, (u, v) = (xt(u, v), yi(u, v), zt(u, v))

the components of which are defined by Re(w2-a+2k),

xt(u, v) =

yi(u, v) = Im(02

+2k)

zJu, v) = 0,

with w = u + iv E B and w = u - iv. Here the semi-disk B is mapped into the "great" angle (2 - a)n which is formed by r,, r- at zero. Again it is possible that branch points occur and that the surface winds about the origin. In Theorem I of this section we shall show that this behaviour is typical of a minimal surface X which is bounded by two Jordan arcs forming a positive angle an at a corner

P where T+ and F- are tangent to the x,y-plane. Before we can formulate the main theorem of this section, we have to state the basic assumptions describing the geometric situation which is to be considered.

Assumption A. F', F are regular arcs of class C2'",µ E (0, 1), which intersect at the origin, thereby enclosing an angle of it, a E (0, 1). The sets B,, I,, Ia are defined by

Bb :={w=(u,v)eR': jwj <6,v>0},

:0
Ia :={w=(u,0)CIs

(and, as usual, we will identify w = u + iv e C with w = (u, v) e l 2 and Ia with (0, b) c R, etc.).

_

Let X be a minimal surface which is of class C°" (Ba , RI) n C2(BB - {0}, R3)

for some v e (0, 1) and S > 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

2
y>a±(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
It follows that the matrix function T is of class C°([-e, e], R9) since

lim T(t) = T+(0) = Id = T-(0) = lim T(t), and, because of the assumptions on h; , h2 , the matrix T is hence even of class

Co,t/[-c, c], R') (although S±(t) is not even continuous at zero). Next we consider the complex valued function g(w) defined by g(w) := T(x(w)) Xw(w) for all w e BB .

(10)

We claim that g has the reflection property St(0)g(w) = g(w)

(11)

for all w e I6±1

In fact, it follows from (9) that St(0)g(w) = S±(0)T±(x(w))Xw(w)

= St(0)Ot(0)O}(x(w))`Xw(w)

= O}(0)Diag[-1, -1,, l]Of(x(w))`Xw(w) = T±(x(w))S}(x(w))Xw(w)

= T±(x(w))Xw(w) = g(w).

We now reflect g across the u-axis by (12)

{0}

G(w) = {S+

(0)g(..)

if w e B,,

Then we have Lemma 1. The function G is of class C" A (Ba -1g , C3), and there exists some constant c > 0 such that the estimate

178

8. Singular Boundary Points of Minimal Surfaces

IG-(w)I
(13)

holds true almost everywhere on B6 - {0}. Furthermore G(w) satisfies

w e B. - 1

Gi (w) + G2 (w) + G3 (w) = 0,

(14)

IG(w)1 = O(Iw1°-')

(15)

,

as w-*0,

where v denotes the Holder exponent of X. Finally there holds the "jump relation"

lim G(u, v) = S-(0)S+(0) lim G(u, v)

(16)

for alluEla Proof. Since T is Lipschitz continuous and X e CZ(Ba - {O}, R3) we also have {0}, C3), and because of (11) we obtain G e C°,'(Bb - Ib , (C3). ge To establish (13), we remark that almost everywhere on Bb we find GW(w)

9w = S.(0)9w(W) = S+(0)[T'(x(w))Xw(W)]Xw(w)

if w e Ba if w e BB

whence IGw(w)I <- ct I T'(x(w))1 Ix.I IXw1 <- c21Xw(w)IZ

< c31 T-'(x(w))9(w)12 < c4I9(w)I2 < c5IG(w)12

for suitable constants c1, ..., c5. From the conformality condition <XW, Xw> _

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 <-

(20)

c{IwI-IIF2(w)12

IF2W(w)I

Iwl-2aIF3(w)I2}

+

+

Iwle-2aIF3(w)I2}

IWI-aIF3(w)I2}

IF3,.(w)I <- c{Iw1a-2eIF2(w)I2 +

almost everywhere on Bb - {0} for some constant c > 0.

Proof. We conclude from (14) and (17) that 0 = Gi (w) + G2(w) + Gs(w) = z[wa-1F2(w) + w-aF3(w)]2

- I [wa-1F2(w) -

w-,,F3(w)]2

= 2w-1F2(w)F3(w) + F2

+ F1(w)

1(W),

whence (18) follows. From the definition of F(w) = (F1(w), F2 (w), F3(w)) and from

(15) we infer the relations (19). To prove the inequalities (20) we first note that IG(w)I2 = IF1(w)12 +

IwI-2eIF2(w)12

IwI-2aIF3(w)12,

+

whence we obtain from (13) and (17) the inequalities IF1w(w)I <- c[IF1(w)I2 +

IwI-21IF2(w)I2

IwI-IIF2(w)12

IF2w(w)I <- c[IwI°IF1(w)I2 +

IwI-2aIF3(w)I2],

+ +

IwIe-2aIF3(w)I2],

IF3w(w)I < c[IwI'IF1(w)I2 + IwIa-2eIF2(w)I2 + IwI-aIF3(w)12].

On the other hand relation (18) yields the estimate IwI-11wI22-'IF2(w)12

IF1(w)I2 <_

=

IW121a-l)IF2(w)12

= IwI-21IF2(w)I2

+

+

IwI-1IwI1-2aIF3(w)I2

IwI-2aIF3(w)12

+ IwI-2aIF3(w)I2.

Together with the above inequalities we finally obtain (20). This finishes the proof of Lemma 2.

Now we are in a position to apply Theorem 3 of Section 8.1. We can assume

without loss of generality that 0 < a < $ < 1. Lemma 3. There exists a nonnegative integer m such that the functions fm(w) :_ w-mF;(w), i = 1, 2, 3, do not vanish simultaneously at zero and that one of the following conditions holds true: (i)

f2 E C°'"(Ba, C) for ally < min(1, m + a), IfiW(w)I =

O(IwIm-2a)

as w -, 0,

fz (0)

0,

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

If2w(4 = O(I W Im-P)

f3w(W)I = O(I

181

as w --+ 0,

WIm+¢-2/t)

as w -+ 0,

a.e. on Bb - {0}. (ii)

f3 E

C)

for all p < min(1, m + /3),

flw(w)I = O(I

WIm-2a)

f3w(w)l =

0

aSW -+ 0, as W -+ 0,

WIm+Q-2c)

.f2w(w)I = O(I

f3 (0)

asw--+0,

O(IWlm-a)

a.e. on Bh - {0}. In addition, if m > 1, then in both cases f2 (o)f3 (0) = 0 .

(21)

Proof of Lemma 3. From Theorem 3 of Section 8.1 we infer that (i) or (ii) has to hold, except for the assertions concerning f1m. We recall the cases (a), (3), (y) and (8) which occurred in the proof of Theorem 3 in Section 8.1. Let us treat these cases separately.

(a) F2(0) : 0, F3(0) 0 0: Then (i) of Theorem 3 in Section 8.1 holds with m = 0. In particular, IF2(w)I = 0(1) as w-+0, and IF3-,(w)I = O(IwI,-20 ) as w-+ 0. But then Lemma 6 of Section 8.1 implies that O(1

IF3(w)I

ifa-23O -1,

+Iwl°-2p+')

= O(I wl-`) for all e > 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 <(23)

C[IWIm-26If2 (w)I2 +

wlm-2aIf3 (w)I2],

Iwim+6-2aIf3 (w)I2], f2w(w)I <- C[Iwlm-6If2 (w)I2 + C[IWlm+a 261f2 (W)12 + Iwlm

alf3 (w)12],

f3W(w)I

while (18) yields (24)

w[f1 (w)]2 + 2f2(w)f3 (w) = 0

in Ba - {0}.

The relations (22) and (24) imply that as w -+ 0

If2 (w)I, If3 (w)I = 0(1) and

asw -+0.

Ifs (w)I = o(IwI-1)

0(Iwlm-26), and by Lemma 6 of Section 8.1 we find that f, e C°,"(Bo, C) for all p < min(1, m - 2$ + 1). By letting w -+ 0 in relation (24) we conclude (21): ab = f2(0)f3(0) = 0.

Now (231) yields I f w(w)I =

First subcase: a 0 0, b = 0. Then case (i) of Theorem 3, Section 8.1, holds with m > 1, and this implies (i) of Lemma 3 since we have already shown that IfiW(W)I =

Second subcase: a = 0, b

O(Iwlm

26)

asw-, 0.

0. Here case (ii) of Theorem 3, Section 8.1, holds with

m > 1. In particular, If2w(w)I =

asw-,0,

O(IWIm+6-2a)

If3W(w)I = O(Iwlm-a)

as w -, 0.

By virtue of a = f2(0) = 0 and Lemma 6 of Section 8.1 we find If2(W)I=O(1 w1")

for ally <min(l,m+f-2a+1)=1.

Finally we obtain from (231) Iffw(w)I = O(IWIm-26IWI2" +

Thus Lemma 3 is proved.

Iwlm-2a)

=

O(Iwlm-2a)

asw -+0.

8.3 Minimal Surfaces with Piecewise Smooth Boundary Curves

183

Now we finish the proof of Theorem 1. From (10), (12) and (17) we infer wa-1 w-"]F(w)

Xw(w) = T(x(w))*U Diag[1,

=

w"-1 T(x(w))*U

1 w1-2a]F(w).

Diag[w1-"

Let us assume that (i) of Lemma 3 holds true whence in particular

f2 E

C°.µ(B5, C),

0.

f2 (0)

Furthermore Iw1-af{w(w)I

asw-+0,

= 0(lwlm-6)

Iw1-2aJ3w(w)I

O(IWIm+1-a-2fi) = O(Iwlm-R)

=

On the other hand we obtain from (23) that, as w --> 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} z

Y

x Fig. 1

where a e (0,

?). For each k e N u {0} we consider the functions f1(w) = Wa+2k

f2(w) =

f3(w) = - Wa+1+2k,

f4(w) = -

W2-a+2k W1-a+2k

and the associated minimal surfaces X3(u, v) = (xj(u, v), yj(u, v), zj(u, v)),

j E {1, 2, 3, 4}

given by xj(u, v) = Re fj(w),

yj(u, v) = 0,

zj(u, v) = Im f (w),

weB={(u,v)Ef}82:u2+v2<1,v>0},

w=u+iv.

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

187

Then each X;, j = 1, 2, 3, 4, is a minimal surface which maps the interval [ - 1, 0] onto F and [0, 1] into S while X(0, 0) = 0. Also X; meets the surface S ortho-

gonally along its trace X.illo.r], and hence it is a stationary solution of a free boundary problem determined by F and S. We shall prove that any minimal surface with a free boundary behaves "near the corner point" like one of the four solutions constructed above. More precisely, it will be shown that X,(w) = w1O(w)

(1)

as w -* 0,

where y > -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 <_ cIG(w)h

almost everywhere in BR, and we infer from Proposition 1 in Section 8.2 that (18)

IG(w)I S cIwI"

W E R. - {0},

with some constant c, where v denotes the Holder exponent of X.

192

8. Singular Boundary Points of Minimal Surfaces

Next we are going to smoothen the jump of G on the interval 1R by multiplication with a singular matrix function which is related to the eigenvalues of the matrix R1{0)Rs(0). It follows easily that cos 27ta R1(0) Rs(0) =

0

-1

0

sin 27ra

0

-sin 27ta 0 cos 27ca

and Diag[ei2n(a-1) e", e-i2,ra]U*,

Rf(0)Rs(0) = U where U* is the unitary matrix 1

i

U

-i

0

/2

0

11

0

1

The smoothed function F(w) = (F1(w), F2(w), F3(w)) is now defined by (19)

F(w) := Diag[w1-a, w112, wa] U*G(w),

for all w E BR(0) - {0}.

Equation (19) is equivalent to (20) XW(w) = T(X(w))-1U Diag[wa-1, w-112, w-a]F(w) for all w e BR(O) - {0}.

It is easily seen that F is continuous; in particular, we have

lim F(u, v) = lim F(u, v) for all u E 1R V-01

.

V-0-

In fact we find

Lemma 2. The function F(w) is of class C° 1(BR(0) - {0}, C3) and satisfies the relations IF1(w)I = O(Iwl"-a) (21)

IF2(w)I = IF3(w)I =

O(Iwl"-112)

as w --* 0.

O(IwI"-e)

and Q = 1 - a. Furthermore the following differential system holds almost everywhere on BR(0): IF1».I <- c[ wIa-11F112 + (22)

IwI1-3,,IF312]

IF2JI <- c[Iw1111>-2gjF1I2 + IwI(112)-2aIF312]

IF3wl <

c[Iwla-2sIF112

+ IwI 21F312],

where we have dropped the argument w. Moreover, there exist complex valued functions X1, X2' X3 which are Holder continuous on BR(O) such that

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

193

(23) FF(w)Xj(w) + 2Ft(w)F3(w)X2(w) _ [w2a

and X3(0) = 1

Fi(w) + wt 22F32(w)](1 - X3(w)),

for] = 1, 2, 3.

Proof. Relations (21) follow from the definition of F and from (18). The Lipschitz continuity of F on the punctured disk is a consequence of the Lipschitz continuity of G and of the continuity of F at 1j. The conformality condition <Xw, X,,,> = 0,

the definition of G and the relation T(O) = Id imply the existence of Holder continuous functions al(w), a2(w), a3(w) such that al(w)Gi(w) + a2(w)Gz(w) + a3(w)G3(w) = 0

in BR(O) - {0},

and al(0) = a2(0) = a3(0) = 1.

Then (23) follows with X1(w) = a2(w),

X2(w) = z(a1(w) + a3(w)),

X3(w) = 1 +

z(at(w) - a2(w))

From the definition of G we derive IG(w)12 = IwI-21IFt(w)I2 + Iwl-'IF2(w)I2 + Iwl-2aIF3(w)I2,

and inequality (17) together with (19) yields

cjwI'

IF1W(w)I

IG(w)12,

IF2w(w)I <-cIwltl'IG(w)I2, F3W(w)I <- cI wI jG(w)12,

whence F1w(w)I F2w(w)I

IwI-,,IF2I2

c[IwI-eIFtl2 + c[Iwlctn>-2sIFI12

+

+

IwI-112IF212

Iwle-2a1F312],

+

IwIc112)-22IF312],

c[Iwla-21IF112 + IwI-1IF212 + IwI-1IF312].

F3w(w)I 5

On the other hand, we deduce from (23) the inequality IF212 < c[IFII IF31 +

<

Iw121-1IF112 +

Iw11-2aIF312]

c[IwI2a-1IFt12 + Iw11-2a1F312]

These inequalities imply system (22). Relations (211), (213) and (221), (223) are equivalent to (33) and (34) respec-

tively stated in Section 8.1. Hence we infer from Theorem 3 in Section 8.1, similarly as in Lemma 3 of Section 8.3, the following

194

8. Singular Boundary Points of Minimal Surfaces

Lemma 3. There exists a nonnegative integer m such that the functions fm(w) w-mF,(w), j = 1, 2, 3 either satisfy

(i) f,-(O) 0 0, f1 e C°'"(BR, C) for all p < min(l, m + a), and 0(IWIm-1)

Iflw(w)I =

O(IWIm+2a-3/2)

If2w(w)I =

asw-+0,

O(Iwlm+3a-2)

If3w(w)I = or

(ii) f,'(0) = 0 and f3'(0) o 0, f3 e C°-"(BR, C) for every p < min(l, m + fl), and Iflw(w)I =

0(IWIm+fl-2a)

If(w) I =

0(IwIm+1/2-2a)

If3w(w)l = O(Iwlm-a)

almost everywhere on BR.

If m > 1, then in both cases

[fi (0)]2 + 2fi (0)fs (0) = 0.

(24)

Proof. This can be proved like the corresponding result, Lemma 3, in Section 8.3.

Now we can continue with the proof of Theorem 1. Assume that case (i) of Lemma 3 holds true; then we put i

Ow) :=

725

fl (w)

0

-i

+

1

w' -2af3m(w)

1

0

0

+

w1/2-ft (w)

1

0

1

and

O(w) := T-1(X(w))>/i(w),

w e BR .

We claim that 0 e C°'(BR(0), C3) for all p < min(1, m + a). In fact, we have by assumption that f1 is Holder continuous, and by (i) of Lemma 3 it follows that Irwl-2af3

(w)]wI =

I1w1/27f2 (W)hl =

O(IWIm+a-1),

O(IWIm+a-1).

We thus conclude from Lemma 6 in Section 8.1 that w1-2af3 and w1/2-aft are Holder continuous, whence , e C°"" for all p < min(1, m + a). Furthermore, T -1(X (w)) is Holder continuous on BR with the exponent v (since T is Lipschitz continuous); thus also O(w) is Holder continuous on BR . On the other hand, it follows from definition (20) that

8.4 An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem XW(y1) = Wa

(25)

195

W E BR - {0}.

Because of T(0) = Id, we obtain for a < z the relations

x'1(0) = ± f,'"(0)

fi"'(0) 0 0

X3(0) = -

(26)

102(w)I = O(Iwl") for some jc' > 0. Then (25) yields I9w(w)l = O(lwlz')

for some 2' > a - I + m.

If the second alternative of Lemma 3 holds true, we consider instead of Eli the function i given by 01

-t

'Y (w) := = f3 (w)

+

0

1

1_ W2a-t

1(W)

0

wa-1/2fm(w) 2l /

+

1

0

1

and

T-1(X(w))tIi(w).

Then (3(w) is Holder continuous and we have XW(W) = W-a+m (W),

(27)

W E BR(0) - {0},

which together with (25) proves (3) of Theorem 1. Also we find for a < 2 that

cP1(0) _ f3 (0) 0 0,

43(0) _ =Gf (0)

0,

k2(0) = 0,

(w2a-1fi)1(O0)G= since 0 and (wa-u2 f2)(0) 0. The last relation follows because of f j'(0) = 0, relation (24) if m >- 1 or (23) for m = 0, Lemma 3 (ii) and Lemma 6 of Section 8.1.

If a = 2, we obtain relation (3) with

`¢(w) = T -1(X (w))

-1

I

0

0

+ =f1m(w)

0

+ f2 (w)

1

0

1

1

and

X1(0) + 02 (0) + 03 (0) = - 2 [f,2(0) - 2f1(0)f3 (0) + f3 (0)] 2

+ f2 (0) + 2 [f1(0) + 2f1(0)f3(0) + f3 (0)] 2(0)

= 2f1(0)f3 (0) + f2

by (23) and (24).

=0

l

196

8. Singular Boundary Points of Minimal Surfaces

The unit normal of X (w) given by -V(w) _ of (25) or (27) and because of

I,j X AX

Xu A X

(w) satisfies by virtue

(X. A Xj(w) = 2(Im(ywzw), -Im(xwzw), Im(xyw)) the relation Im((P2 (0) d53(0))

lim .,V(w) = 2[I01(0)12 + l 2(0)12 + 1t3(0)l2]-1 1Im((P3(0)dir(0)) w-+o

L Im41(0)02(0)) J

But now relation (15) implies Rs(0)k(0) = 0(0), and this means that

Im0'(0)=0,

'M 02(0)=O, and Reo3(0)=O.

Also, if a < 2, then 02(0) = 0, and we arrive at -*'l (0) = 0,

.A(3(0) = 0,

.N2(0) = ± 1,

whereas, if a = i, we conclude that .

j = 1, 2,

l;(0) _ ±Re -P;(0) [(Re 0t(0))2 + (Re (p2(O))2]-t12,

and

43 (0) = 0.

Finally, we obtain for the tangent vector t(u) =

X.(U, 0)

, u > 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
surfaces with cusps on their traces. In fact, such examples are already well known to us (see, for example Henneberg's surface and Catalan's surface) and have been discussed in Section 3.5. The main part of this chapter is devoted to the study of the free trace of a

stationary surface X within a boundary configuration consisting of a half-plane S and a symmetric curve rwhich has a convex projection onto a plane E orthogonal to aS and which connects the two sides of S. After classifying the possible sets of contact of the free trace E with the boundary DS of the supporting half-plane, we prove a representation theorem for stationary surfaces in le(r, S) which is the key to all further results of this chapter. It essentially states that X can be viewed as a nonparametric surface with respect to the plane E. One of the

main consequences drawn from this representation theorem is a uniqueness theorem stating that there can be at most one stationary minimal surface whose trace is touching aS, and this surface is area minimizing among all surfaces of c'(r, S). Furthermore we shall derive asymptotic expansions of a stationary surface along its free boundary I which will imply that C1.112-regularity is in general the

9.1 An Experiment

199

optimal regularity result. Finally, we shall describe the geometrical shape of the free trace, and we shall exhibit conditions on T which prevent the occurance of branch points.

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

Fig. 1

common with S, except for Pl and P2. We can imagine that r is obtained from a circle by cutting it and pulling its ends slightly apart. Suppose that S is the part {x >__ 0, y = 0} of the x, z-plane and that 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:Iwl<1,Imw>0}, and I denotes the interval

I:= {ueO:Jul
C:=aB-I. Definitions, theorems, etc. concerning surfaces previously defined on the whole disk { I wl < 1 } are then carried over to surfaces defined on the semidisk B by means of a conformal map is w < 1} -> 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 where S is the half plane {x > 0, y = 01,

and ]-is the image of the circular arc C under the mapping (1), that is, the arc {X = X (e"): 0< 0<- n}. The free boundary of.// on S is Neil's parabola (2);,// and S meet at a right angle along this curve.

The orthogonal projection of r onto the x,y-plane is a smooth closed curve. For a later reference we observe that this curve is convex as long as the parameter A remains in the interval 0 < 2 < 20 a 1.014379... (It turns out that 20 is the first positive root of the equation tan(2A) = -22.)

Certain other algebraic singularities of the free boundary are also possible. For the minimal surface represented by the equations x = Re{w2} (3)

y = Re { 2i jW

+-(-04"- 2 dw

111

z = Re

2

-W 2n+1

2n + 1

the free boundary, i.e., the image of I on the plane S = {x >_ 0, y = 0,, is the curve 4x2n+t = (2n + 1)222,

y = 0.

Fig. 4. A boundary configuration
9.2 Examples of Minimal Surfaces with Cusps on the Supporting Surface

205

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

We can state even simpler examples if we do not insist on classical curves as free boundaries. One very simple example is furnished by the minimal surface x = Re{w2 - 18A2w4} (4)

). > 0,

y = Re {iw2 + 18iA2w4},

z = Re{Uw3} which meets the half-plane S orthogonally along the curve

x(u) = u2 - 18,.2u4,

y(u) = 0,

z(u) = 8).u3

which has the expansion z=8.1x3/2+...,

y=0

about the origin. As are r we shall again use the image of the circular arc C, this time under the mapping (4). The orthogonal projection of r onto the x, y-plane is the closed curve x = cos 0 - 18,.2 cos 20,

y=sin0+18).2sin20,

0<0<2n.

2/12 = is, if 0 < 0.117851.... It is not a priori clear at all that the above surfaces are solutions of the minimum problem

This curve is convex if 0 < 18A' < 4, that

(5)

D(X) --+ min

in 1(I', S).

This will, in fact, follow from the uniqueness theorem proved in Section 9.5. In particular Henneberg's surface (1) provides us with a simple example of a solution of the minimum problem (5) which possesses a cusp on its trace.

206

9. Minimal Surfaces with Supporting Half-Planes

9.3 Set-up of the Problem. Properties of Stationary Solutions We will now prepare the mathematical discussion to be carried out in the following sections. We begin by fixing the assumptions on the boundary configuration which are supposed to hold throughout Sections 9.3-9.9. Assumption A. Let S be the half-plane {(x, y, z): x >- 0, y = 0} in W. Moreover,

the curve T is assumed to be a regular arc of class Cl,', 0 < a < 1, with the

endpoints Pt and P2, Pl 0 P2, which issues from Sat right angles and meets S only in its endpoints. Close to Pt, the arc T is supposed to lie in the half-space { y > 0}. Assume also that I' is symmetric with respect to the x-axis, and that the orthogonal projection of T onto the x, y-plane is a closed strictly convex and regular curve y Finally, suppose that the projection of r onto y is one-to-one, except of class for the endpoints Pt and P2 of T which are projected on the same point of y.

Fig. 1. Assumption A.

This assumption is satisfied by the examples discussed in Section 9.2.

Assume that P(s) = (pl(s), p2(s), p3(s)), 0 < s < L, is a parametrization T by the arc length s such that (1)

P(0) = PI = (a, 0, -c),

of

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

where a > 0 and c > 0. Then P3 := P(L/2) is the uniquely determined intersection

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

207

point of T with the x-axis which must be of the form

b > 0.

P(L/2) = P3 = (- b, 0, 0),

(2)

This is illustrated in Fig. 4.

Let us now recall that the definition of stationary minimal surfaces was phrased in such a way that these surfaces are precisely the stationary points of Dirichlet's integral within the class W(F, S). In Chapter 7 we have formulated the following result: Lemma 1. Every stationary minimal surface in c'(T, S) is continuous in the closure B of the parameter domain B.

From the regularity theory of Chapter 7 we can also derive the following result:

Lemma 2. If X is a stationary minimal surface in (e(1, S) and if satisfies Assumption A, then X belongs to C'(B, f183) and satisfies (3)

y(u) = 0,

z,(u) = 0

for u e I.

Thus y(w) and z(w) can be continued analytically across the interval I = { I u I < 11 of the u-axis, and the extended functions are harmonic in the whole disk {w: I wl < 1 }. The set

It := {u e I : x(u) > 0}

is an open subset of R containing the intervals (-1, -1 + 23a) and (1 - 250, 1) for some sufficiently small bo > 0. Hence the set of contact

12={uEI:x(u)=0} is closed in R. In addition, we have (4)

x(u)=0 for u E It.

Proof. The regularity theory of Chapter 7 yields that X is of class C1 on B - { ± 1 }. Since X is stationary in '(F, S), it follows that y(u) = 0

for u e I

holds as well as

forueIl.

9. Minimal Surfaces with Supporting Half-Planes

208

The first equation implies

foruEl. Furthermore, the relations x(u) >- 0

for u e I,

x(u) = 0 for u E 12

imply that

foruel2, whence (0, 0,

z (u)

for isolated points. By virtue of <X,,, X,> = 0, we infer

that 0

for u e 12,

and therefore

foralluel. Thus we have verified (3) and (4), and, in view of the reflection principle, the functions y(w) and z(w) can be continued analytically across the interval I on the u-axis by setting

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

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

for v > 0. The extended functions y(w) and z(w) are harmonic in the disk {W: W1 < 1}.

Since X E C°(B, R') and a > 0, the points X(u) lie in the interior of the half-plane S if u is close to ± 1. Hence there is a number 6° > 0 such that the intervals (-1, -1 + 28°) and (1 - 26°, 1) on the u-axis are contained in I. On the part X (1,) of the free trace, the surface X meets S perpendicularly. Hence we can continue X (w) analytically across I, by a reflection with respect to S, and the extended surface k(w) is a minimal surface on {w: IwI < 1, w 0 12}. Moreover, X (w) is continuous on {w: I w l < 11. Since F issues from S perpendicularly,

the surface X maps the unit circle {w: Iwj = 1} bijectively onto a closed regular curve of class C',a. Then the regularity results stated in Section 7.1 imply that X(w) is of class in the strip {l - 8° < IwI < 1}. Thus X is of class C' on B.

9.4 Classification of the Contact Sets The principal result of this section is the following observation: The free trace of

a stationary minimal surface in '(I', S) either meets the boundary 8S of the halfplane S in a single point, or in a single subinterval, or in no point at all.

9.4 Classification of the Contact Sets

209

More precisely, we shall prove: Theorem. Let X(w) = (x(w), y(w), z(w)) be a stationary minimal surface in c(F, S), and set It := {u e I : x(u) > 0},12 := {ul e : x(u) = 0}, and x0 := min {x(u) : u e 1 }. Then only the following three cases can occur: (I) xo = 0, and 12 consists of a single point uo; (II) xo = 0, and 12 is a closed interval of positive length; (III) xo > 0, that is, 12 is empty, and there is exactly one point uo in I such that xo = x(uo). Consequently, we have x(u) > x0 for u e I with u u0.

Remark. Case I may indeed occur as we see from the examples given in Section 9.2. If we introduce the new supporting surface

SE={(x,y,Z)el3:y=0,x _ -c},

e>0,

for some sufficiently small s > 0 as well as the new coordinates

=x+E,

q=y,

=z,

a surface X(w) = (x(w), y(w), z(w)) of type I is transformed into a surface -(w) _ rl(w), (w)) of type III. Hence also the case III may appear. On the other hand, we shall see in Section 9.6 that minima of Dirichlet's integral are never of type III.

As a first step towards the proof of the Theorem we draw some preliminary information from the maximum principle which is formulated as Lemma 1. The trace X(I) is contained in the strip {(x, y, z): 0 < x < a, y = 0} of the halfplane S whence

0<x0
(1)

Moreover, we have (2)

-b<x(w)
Proof. In fact, if there were some u e I with x (u) >_ a, then there would exist some u* E I such

x(u*) = maxi x(u) >_ a > 0,

since x(±1) = a. Since x(w) is harmonic and nonconstant in B, the lemma of E. Hopf t implies that xv(u*) < 0.

Since u* belongs to I, this contradicts Lemma 2 of the preceding section. 'Cf. Gilbarg-Trudinger [1], p. 33.

210

9. Minimal Surfaces with Supporting Half-Planes

Thus we have proved that x0 := min{x(u): u e I} satisfies (1). Moreover, the x-component p' (s) of the representation P(s) of F satisfies

-b-
-b<x(w)
0<x(u)
The next lemma is the crucial step for the proof of the Theorem. We need the following notations: For each value u c- R we define the open (and possibly empty) subsets B(p), B+(µ), and B-(µ) of B by B(,u):= {w c- B: x(w) :A p)

B+(µ)

{w c- B : x(w) > µ}

B-(µ)

{w c- B : x(w) < µ}.

By virtue of Lemma 1, we obtain

B+(µ) _ 0, B+(µ) = B,

B (µ) = B if u > a, B-(µ) _ 0 if u < -b.

Recall that X(w) provides a topological mapping of the circular arc C onto F. By assumption A there are, for each value µ E (- b, a), exactly two points w1(µ) = e`°1(µ) and w2(µ) = e`B2tµ on C, 0 < B1(µ) < B2(µ) < 7r, with the property

that x(w1(µ)) = x(W2(µ)) = µ

In addition we set

w1(-b) = w2(-b) := i, w1(a) := 1,

w2(a) :_ -1,

01(-b) = 02(-b) := 2 , 01(a) := 0,

02(a) := n.

For y E (-b, a), we define the following open subarcs of C:

C1(µ):= {w=e'B:0<0<01(µ)} C-(µ) :_ {w = ete : B1(µ) < 0 < 02(µ)}

C?(it):= {w=ei°:02(µ) <0 <1t}.

9.4 Classification of the Contact Sets

211

Lemma 2. For each p e (-b, a), the set B-(µ) is simply connected, and the set B+(µ) can have at most two components.

Proof. We proceed as follows: (i) First we fix some it e ( - b, a), and denote by Q the component of B-(µ),

the boundary of which contains the arc Cu). -(If B-(µ) were not connected, there would exist another nonempty component R of B-(µ). Clearly, OR c B u I u {w1(µ), w2(µ)},

x(w)=It for

w E aR n (B u C),

and x(w) < µ for

w e OR n I. (Note that OR n I is void for p < 0.) If OR n 1 were empty, the maximum principle would imply that x(w) _- µ on R, so that, contrary to the facts, x(w) - µ on B. If OR n I is nonempty (this is only possible for µ >- 0), let m = inf{x(u): u e OR n 1}. We claim that m < p. Otherwise, if m = ji, we could obtain a contradiction as before. From x(u) >- 0 we conclude that 0 < m
u:=sup{ueI

OR: x(u) = m}

satisfies - 1 < it < 1. Since m < p, there is a number e > 0 such that

(u-e,u+s)cOR. By the maximum principle and E. Hopf's lemma it follows that x (u) > 0. On the other hand, a right neighbourhood all of u on I must belong to 11. By Lemma 2 of Section 9.3, it follows that x (u) = 0 for u e all, and x is continuous on I. Thus we arrive at the contradictory conclusion x (u) = 0. We have proved that B-(µ) is simply connected for all p c- (-b, a). (ii) Again, we select a value µ e (-b, a). Denote by Q1 and Q2 the two components of B+(µ), the boundary of which contains C1(µ) and C2 (µ) respectively. It is of course possible that Q 1 and Q2 are identical. We assert that B+(µ) cannot have further components. Otherwise, if R were such a nonempty component different from Q1 and Q2, we would have OR c B u I u {w1(µ), w2(µ)}, x(w) = µ for w e OR n (B u C), and x(w) > µ for w e 1 n OR. If OR n I were empty, the maximum principle would lead to a contradiction, as in (i). We may therefore

assume that OR n I is nonvoid. If -b < it < 0, the level set 1(µ) = {w: x(w) = µ} cannot touch I. In fact, there is a strip s, = {w = u + iv; 0 < v < s} abutting on I which is not penetrated by 1(p), so that sE c Qt = Q2. But this is incompatible with the assumption OR n I 0 0. We turn to the case 0 < µ < a. Neighbourhoods in B of the corner points w = ± 1 belong to the components Q1 and Q2. Hence there is a S > 0, such that OR n I c {u: Jul < 1 - 8}. Then there exists a point U e OR n I in which x(u) attains the maximum value m = max{x(u): u c- I n aR}.

As in (i) we conclude from the maximum principle that m > µ. Then there is a o > 0 such that the interval (u - a, u + o) on I belongs to the boundary OR, whence x (u) < 0, again on account of E. Hopf's lemma. On the other hand,

0< µ < m = x(u) implies that the point u belongs to 11 which leads to the contradicting statement x,(u) = 0. Therefore, B+(µ) has no components other than Q1 and Q2.

212

9. Minimal Surfaces with Supporting Half-Planes

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

u'0 := max{u

c-

I : x(u) = x0}.

Clearly we have - i < uo < u'0 < 1. Then one of the following three, mutually exclusive cases must hold: (a)

uo = uo;

(/3)

uo
x(u)=xo for all uE[uo,uo];

(y)

uo < uo ,

x(u) > xo for some u c- (ua, u'0).

We shall show first that case (y) cannot occur. In case (y) we would be able to find two points u,, u2 a [uo, uo], u1 < u2, such that x(u,) = x(u2) = xo and x(u) > xo for u, < u < u2. Set

m = max{x(u): u, < u < u2},

0 < x0 < m < a,

and assume that x(u') = m, u1 < u' < u2. Then

x

0 for u, < u < u2, since u e 1,. Therefore, x(w) can be continued analytically as a harmonic function across the segment u, < u < u2 of the u-axis into the lower half of the w-plane. In a (full) neighbourhood of the point w = u' this function has an expansion

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

,c

O,

v > 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)
y(iv) = z(iv) = 0 for all v E [0, 1].

(5)

Finally, the nonparametric representation z = Z(x, y) of the minimal surface .,11, given by X : B -. R', satisfies

Z(x, y) = -Z(x, - y) for (x, y) E D(xo), and therefore also

Jim Z(x, y)

Jim Z(x, y),

y-+0

y-._0

x

0,

in case II.

Now we come to the proof of Theorem 2. The domain D introduced in the representation theorem is the same for X1 and X2, even if the diffeomorphisms B -+ D given by the first two components differ. Therefore we have the nonparametric representations z = Z, (x, y) and z = Z2 (x, y) respectively with (x, y) e D, for the two surfaces X 1 and X2. The functions Z1(x, y) and

Z2(x, y) have the properties stated in Theorem 1 and satisfy

Z1(x, y) = Z2(x, y) for all (x, y) e y, where y is the projection of F onto the x, y-plane. We will show that Z1 and Z2 coincide in D. For j = 1, 2, we set (6)

a

p': a-ZZ,

a

qq:=ayZi,

W:=

1+pf +q;.

For fixed (x, y) e D and for t e [0, 1] we introduce the notations

9.5 Nonparametric Representation, Uniqueness, and Symmetry of Solutions

215

P(t):=Pt+t(P2-Pt) q(t) W(t)

qt + t(q2 - qt) {l + p2(t) + q2(t)}t/2,

as well as

Pl l At):=(P2-Pt)J P(t) w(t)-Wl +(q2-gi)lw(0- Wt J q(t)

qt

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

f(l) = f'(t)

(7)

Furthermore, a brief calculation yields

f'(t) > 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((?)<-m3(Q)

L,'.

(Z1_Z2)[-(W1 -W)dx+(w-w)dy] 2

1

.

2

Letting S decrease to zero, keeping e fixed, we infer from the boundary conditions (1) and (6) that

216

9. Minimal Surfaces with Supporting Half-Planes

(9)

1(Q) < m3(Q)

c.

(Zt - Z2) [_(W1

P1 W dx + (WI

2)

P2

-W-2) dy

where C, denotes the parts of the circles {x2 + y2 = e2 } and { (x - a)2 + y2 = E3 } which are contained in D u aD. Since the integrand of the right-hand side of (9) is bounded, the line integral tends to zero as t -+ 0 whence 1(Q) = 0 for every compact subset Q of D. It follows that VZ1(x, y) - VZ2 (x, y) in D,

and therefore also

Z1(x,y)-Z2(x,y) inD, on account of (6). Consequently X1 and X2 are conformal representations of the

same nonparametric minimal surface ff, with the same parameter domain B and satisfying the same three-point condition. From this we conclude that X1(w) = X2(w) because a conformal map of B onto itself has to be the identical map if it leaves three points on aB fixed. 0

9.6 Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet's Integral The central result of this section is the following

Theorem 1. Minima of Dirichlet's integral in '*(T, S) are not of type III. In Chapter 4 we have proved that there is always a solution of the minimum problem in '*(r, S). By Theorem 1, this minimum has to be of type I or II. On the other hand, the uniqueness theorem of Section 9.5 states that there is at most one stationary minimal surface in W*(r, S) if surfaces of type III are excluded. Hence Theorem 1 implies the following result : Theorem 2. (i) Stationary minimal surfaces in le(T, S) furnish the absolute minimum

of Dirichlet's integral in '(r, S) if and only if they are of type I or II. (ii) There exists one and only one minimum of Dirichlet's integral in '*(T', S).

Hence the stationary surfaces of type III constructed in Section 9.4 do not minimize Dirichlet's integral within 1(T', S). A proof of Theorem 1 can be based on the following asymptotic expansions for surfaces of type I or III: Theorem 3. Let X (w) = (x(w), y(w), z(w)) be of class I or III. Then w = 0 is a first order branch point of X (w), and we have the expansion

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

217

x(w)=x0+Re{xw2+...} y(w) = Re {ixw2 +...)

(1)

z(w) = Re{µw2"+1 + ...I where K > 0, y is real and

0, and n is an integer > 1.

Proof. We note that Vy(O) = 0, since of Section 9.5. In cases I and III we have

0 for all u e I, and y,(0) = 0 by (5)

x, (u) + zU(u) = yV(u)

(2)

for all u e I.

Combining this identity with yv(O) = 0, we conclude that 0 and 0. Since z,(u) = 0 for all u e 1 in cases I and III, we see that 0x(0) = Vz(0) = 0.

Since x(u) > xo for 0 < Jul < 1, the arguments employed in the proofs of Lemma 2 and the Theorem of Section 9.4 lead, for small w, to an expansion (3)

x(w) = x0 + Re {Kw2 +

... 1,

K > 0.

Hence, w = 0 is a branch point of order one for X.

From the relation z,(u) = 0, u e I, it follows that z(w) can be extended harmonically across the u-axis and that, in view of Section 9.5, (5), an expansion

z(w)=Re{pw'°+...} is obtained in which e is real and 0 and m is an integer >_ 2. Formula (4) of Section 9.5 shows that this integer must be odd so that, near w = 0, z(w) = Re { µw2"+1 + .. .

(4)

Recall now that the vector B

} ,

n > 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 <1,u<0,v>0}. Since y(w) >- 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.

0

This completes the proof of the Theorem.

9.8 Final Results on the Shape of the Trace. Absence of Cusps. Optimal Boundary Regularity An inspection of the foregoing proofs shows that the relations

y(u)>0 for -1
for 0
hold in all three cases I, II, and III. In conjunction with the two expansion theorems of Sections 9.6 and 9.7 we obtain the following result about the shape of the trace of a stationary minimal surface in IK(I', S). This result exactly corresponds to the experimental observations in Section 9.1. Theorem 1. Let X be a stationary minimal surface in W(F, S). In cases I and III, the trace X (u), u e 1, is a real analytic curve which is regular except for the branch point w = 0 of order 1. In case II, X has no branch points on I, and the trace curve X(u), u e 1, is a regular curve of class C1,1J2

z (a)

(b)

(c)

Fig. 1. (a) Case II, z1 < 0 (tongue), (b) Case I (Cusp), (c) Case II, z, > 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<2} C

w=e`B:n 2<0«}.

Let X(w) = (x(w), y(w), z(w)) be the minimal surface )under consideration. Then

z(w) > 0 for w e C+ and z(w) < 0 for w e C-. Denote by Q+ and Q- the two components of the open set Q = {w e B : z(w) # 0} for which C+ c 8Q+ and C- c aQ- respectively. There cannot be further components of Q. In fact, if R were such a component different from Q+ and Q-, then OR e B u I u {i}. Moreover, z(w) = 0 at all boundary points of R in B u {i}. In view of the maximum principle, z(w) cannot vanish everywhere on OR. Hence, there is a point on I where z(w) is different from zero, say positive. Since Q+ is adjacent to C+ and Q- is adjacent to C-, the intersection OR n I must be contained in a compact subinterval of I. Therefore, there is a point u' E I such that

9.9 Proof of the Representation Theorem

223

z(u') = max{z(u): u e OR n I} = max{z(w): w e 8R} > 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<_ 1. "<6< 2

As the angle 0 increases from zero to it, the function y(e`B) decreases from zero

to its minimum value ymin, then increases from ymin to its maximum value Ymax =

ymin+ and finally decreases again to zero. By conformality we have x.'(u) + zu(u) = Yo(u)

on I. Since x (u) -A 0 for u 96 uo, we see that y (u) # 0 for all u e I, with the possible

exception of u = uo. It follows from the maximum principle that the open set { w e B : y(w) : 0} has exactly two components, Q+ and Q-, and that y(w) > 0 Q+, y(w) < 0 in Q-. Applying once more Radb's argument, we infer that in I Vy(w) I > 0 for all w E B. Therefore, the two components Q+ and Q- are separated in B by an analytic arc qf which has points in common with each horizontal line v = Im w = const, 0 < v < 1, considering that y(w) changes signs in B along each such line. We claim that this arc, except for its end points, lies entirely in the domain B-(0), and that it has the end points w = uo on I and w = i on 8B. As a first step we shall show that yt,(1) < 0 and yv(-1) > 0. For this purpose recall that X(w) can be extended to the full disk {w: I w! < 1} in such a way that X(w) is the position vector of a minimal surface defined on {w : 1 - So < I w ! < 1 for a suitable bo > 0. In view of the boundary regularity results stated in Section 7.3, the surface X is of class C" in {w e B : 1 - So < I w! < 11. As the curve X (eie), 0 < 0 < 27r, lies on a convex cylinder, the asymptotic expansion at boundary branch points (cf. Section 8.1) implies that our minimal surface cannot have branch points on the circular arc C. Hence it follows that

9.9 Proof of the Representation Theorem

I X,(e.e)I2 > 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)<0for 0<0<2and y(ei8)>0for2<0
near w = uo an expansion y(w) = Im{ -.1(w - uo)" + ...

},

where n >- 2, and is a real number different from zero. Since the set {w e B: y(w) 0} has exactly two components, we see that n = 2 and 2 > 0; that is, near w = uo,

y(w)=Im{-.1(w-uo)2+...},

(2)

2>0.

The above results imply that the arc a which separates the components Q+ and Q- has as its end points (and only points on 8B) the points w = uo and w = i. Assume that d, except for its end points, is not contained in B-(0). Then

there is a point w, e B on this arc for which x(w,) > 0, y(w,) = 0. From the expansion (2) we see that near u = uo, that is, for small positive values of p, the

arc d has the representation

It then follows from (1) that

x(w) = -xp2 + 0(p3) for w e d in a neighbourhood of w = uo. Therefore, if we traverse the arc .sad from the point w = uo to the point w = w, , we shall encounter a negative minimum for

the function x(w), restricted to sz? Assume that this minimum is attained at the x,y,, = 0 point w2 a B. Since y(w) = 0 on.4, and .sad is a regular arc, we have at w = w2. Thus, there exist numbers p and q, p2 + q2 > 0, satisfying the linear equations Px (w2) +

In fact, p

0 and q

0,

Px,(w2) + gy.(w2) = 0.

0, since IVx(w2)I > 0 and IVy(w2)I > 0. Consider the

harmonic function h(w) = p[x(w) - x(w2)] + q[y(w) - y(w2)] = px(w) + qy(w) + r,

where r = - px(w2). This function vanishes at w = w2, together with its first derivatives. By Rado's lemma (cf. Section 4.10), h(w) must have at least four distinct zeros on the boundary 8B. On the other hand, since pr = -p2x(w2) > 0, the straight line px + qy + r = 0 in the (x, y)-plane passes through the x-axis to the left of the origin and therefore intersects the boundary aD of the slit domain D = D(0) in at most two points. Moreover, the functions x(w), y(w) provide a

226

9. Minimal Surfaces with Supporting Half-Planes

topological mapping of 8B onto aD. Consequently, the function h(w) vanishes on 8B in at most two points. This is a contradiction to the previous statement. We have proved that the arc 4, except for its end points w = u°, w = i, lies entirely in B-(0). This fact will be used in the following way: Let H(w) be a harmonic function in B of class C°(B) such that the open set {w: w e B, H(w) 0 0} consists of exactly four components which are separated in B by four analytic arcs issuing from some point w, e B. Suppose that two end points of these arcs lie on 1, to the left

and to the right of w = u°, and two end points lie on C, to the left and to the right of w = i. Then, regardless of the location of the point w = w,, the null set of the function H(w) in B must contain two points w' and w" in which

x(w')=0, y(w')>0 and x(w")=0, y(w")<0. It can now be shown that the functions x(w), y(w) provide a topological mapping from B to D u aD. We already know that the relation between the boundaries 8B and aD is a topological one and that interior points of B are mapped onto interior points of D. The bijectivity of the mapping follows from the monodromy principle once it has been shown that the Jacobian 8(x, y)/8 (u, v) cannot vanish in B. Assume that 8(x, y)/8(u, v) = 0 at some point wt e B. Then, as before, there exist constants p 0 and q 0 0 satisfying the linear equations gyv(w,) = 0. It follows that the harmonic 0 and function

H(w) = p[x(w) - x(w,)] + q[y(w) - y(w,)] = px(w) + qy(w) + r,

r = -px(w,) - gy(wi), and its first derivatives vanish at w = w,. Rado's lemma implies that H(w) must have at least four different zeros on 8B. On the other hand, any straight line px + qy + r = 0, p 0 0, in the x, y-plane intersects aD in at most four points. The case of four distinct points is only possible for pr < 0. Because of the bijectivity of the relation between 8B and aD we conclude that H(w) possesses exactly four different zeros on 8B if pr < 0. Under these circumstances, the set

{w:weB,H(w)00} consists of exactly four components which are separated in B by four analytic arcs issuing from w = w,. Two end points of these arcs lie on I, to the left and to the right of w = u°, and two end points lie on C, to the left and to the right of w = i. The observation formulated earlier implies that there are two points w', w" e B for which

qy(w') + r = 0,

qy(w") + r = 0,

y(w') > 0,

y(w") < 0.

These relations are incompatible with the inequality q 0 0, and we have proved that the functions x(w), y(w) furnish a topological mapping from B to D u 5D.

9.9 Proof of the Representation Theorem

227

Let w = co(x, y) be the inverse map, and set Z(x, y) = z(cw(x, y)),

(x, y) E D o aD.

The function Z(x, y) provides a nonparametric representation {z = Z(x, y): (x, y) e D u aD} of our minimal surface X = X (w), w c -A Z(x, y) is real analytic in D and on both shores of the open segment 0 < x < a of the x-axis (having of

course different limits lim Z(x, y) and lira Z(x, y)), continuous in D o aD, and y-+o

y-+-o

of class C1 in D o b except at the points (0, 0) and (a, 0). Given that 0

on 1,

and that x (u) A 0, y,(u) 0 0 for u 0 uo, u n I, it also follows from the relation a ay

zux

Z(x y) =

XvYu cu(x.y)=w

that lim a Z(x, y) = 0

y-.±o 8y

for 0 < x < a.

Thus, the proof of the theorem is completed for case I. (ii) We turn now to a discussion of case II, assuming that I2 is a closed interval

u1 < u < u2, where -1 < u1 < u2 < 1. We know that y(u) = y (u) = z (u) = 0 for I u l < i as well as

for-1
x(u)>0, and x(u) =

0

for ul < u < u2.

The functions y(w) and z(w) can be continued as harmonic functions across the diameter I into the disk {w: I w I < 1}. For the function x(w) such a continuation

is possible across the intervals u1 < u < u2, -1 < u < u1 and u2 < u < 1, but the resulting extended function will have isolated singularities at the points w = ul and w = u2. Recall that B-(0) is connected and that B+(0) can have at most two components. From the situation at hand it follows that B+(0) consists of exactly two components and that we have the expansions x(w) = Re{iic1(w - U1)312 + ...},

K1 > 0,

near w = u1 ,

x(w) = Re{/c2(w - u2)312 + ... },

K2 > 0,

near w = u2.

and

The derivation of these expansions is based on the arguments employed for the

proof of Section 9.7, (3), except that we are at the present stage not able to conclude that u 1 = - u2 and K1 = K2

228

9. Minimal Surfaces with Supporting Half-Planes

From here on, we can follow the line of reasoning used in part (i) of the proof. We find that, as u decreases from u, to -1 or increases from u2 to 1 the function x(u) increases strictly from the value zero to the value a, and also that x (u) 54 0 for -1 < u < u, and u2 < u < 1. Since x (u) = 0 on (u,, u2) and since x(w) # const, we also see that xju) 96 0, and therefore x (u) < 0 for u, < u < u2. It can furthermore be proved again that lVx(w)l > 0 for all w e B. As for the function y(w), we see as in (i) that both sets Q+ = { w : w c- B, y(w) > 0}

and Q = {w: w e B, y(w) < 0} are connected, and that yo(u) > 0 near w and yju) < 0 near w = 1. On (-1, u,) u (u2, 1), we have x,2, (u) + z;, (u) = y2 (u), and x2 (u) > 0. Hence, We claim that

0 for -1 < u < u,, and

0 for u2 < u < 1.

0 leads to y (u2) 0 0. The assumption 0, so that w = u, would have to be a branch point of X. However, the

asymptotic expansion of X(w) near a branch point does not allow for terms containing the power (w - U,)112 A similar contradiction arises from the assumption y,(u2) = 0. It follows that the derivative y (u) must vanish somewhere in the interval (u,, u2). Since the set {w: w e B, y(w) 0} has only two components, our standard reasoning shows that there exists exactly one point uo E (u,, U2) such that yL,(uo) = 0. The expansion of y(w) near w = uo is .

y(w)=Re{i2(w-u0)2+...},

A>0.

It follows as in (i) that IDy(w)j > 0 for w e B and that the Jacobian 8(x, y)/8(u, v) cannot vanish in B. The functions x = x(w), y = y(w) provide a mapping between the boundaries 3B and aD. (Here D = D(0) is the slit domain in the (x, y)-plane defined in the statement of the theorem.) This mapping is topological, except on the interval

[u,, u2] of I which corresponds wholly to the point (0, 0) on M. From the non-vanishing of the Jacobian 8(x, y)/0(u, v) in B it follows that x(w), y(w) furnish

a homeomorphism between B and D. A repetition of the further discussion of part (i) leads to the conclusion that the minimal surface X = X(w), w E B, admits a nonparametric representation z = Z(x, y). The function Z(x, y) has the properties stated in the theorem. (iii) Case III can easily be reduced to case I. For the purpose of this reduction,

let x0 = min{x(u): u E I}, and suppose that xo = x(uo). Then, x(u) > xo for u uo, u e I. We choose a new cartesian coordinate system with coordinates , 1, C, defined by relations i = x - xo, n = y, ( = z; see Fig. 1 of Section 9.3. Introduce I'' := T - (xo, 0, 0) and the functions (w):= x(w) - x0,

i1(w):= Y(w),

c(w):= z(w),

and the surface Y(w) = (w), C(w)). Furthermore let S be the half-plane ri, C): >- 0, rl = 0}. Then Y(w) is a stationary minimal surface of type I in g*(Io, S). Applying part (i) of this proof to Y(w), we may deduce the desired properties of X from those of Y by going back to the old coordinates x, y, z. This completes the proof of the representation theorem.

9.10 Scholia

229

9.10 Scholia 1. Remarks about Chapter 9 Except for minor modifications and the second proof of Theorem I in Section 9.6, the results of this chapter and their proofs are taken from the paper [3] of Hildebrandt and Nitsche. There remains the challenging problem to extend the results of this section to non-planar supporting surfaces S and, more generally, to arbitrary configurations . Experimental evidence indicates that it should be possible to prove similar results in the general case. 2. Numerical Solutions So far we have not touched upon the problem of numerical solutions of boundary value problems for minimal surfaces. Both the nonparametric minimal surface

equation div

(1)

Oz

1+Ipzl2

=0

and the parametric equations (2)

4X = 0,

IX.12=IX,,2,

have been treated. In this subsection we want to briefly illustrate how the partially

free boundary problem for (2) with a planar support surface can effectively be solved by means of the finite element method, a comprehensive presentation of which can be found in the treatise of Ciarlet [1]. Before we sketch a numerical approach to partially free problems following the work of Wohlrab [2, 3], we want to give some useful (if rather incomplete) references to the literature concerning the numerical treatment of minimal surfaces. The nonparametric equation (1) was dealt with by Concus [1-4] using a finite difference scheme and solving the resulting finite difference equations by a nonlinear successive overrelaxation method. The finite element method was applied to minimal surfaces by many numerical analysts. We only mention the work of Mittelmann [1-6], Jarausch [1], Wohlrab [2, 3], and Dziuk [9, 10]. Whereas the first three authors used the

variational formulation as a point of departure, Dziuk applied an iteration procedure suggested by the mean curvature flow of surfaces. We also mention the

purely computational work by Wagner [1, 2] and Steinmetz [1]. Very remarkable numerical and graphical work based on methods of geometric measure has recently been carried out by Almgren and his group at Princeton. Now let us outline the main idea of Wohlrab's numerical treatment of the partially free boundary problem with a planar support surface S. To this end we consider a boundary configuration or consisting of the plane S :_ {(x, y, z) e U83 : y = 0} or of the half plane S' := { (x, y, z) e 183: y = 0, x < 0}

respectively, and of a piecewise smooth Jordan arc T which meets S or S' only

9. Minimal Surfaces with Supporting Half-Planes

230

at its endpoints P, and P2. We further assume that P, 0 P2 and that r lifts off from S (or S') under an angle different from zero or it. Moreover, the sets B, C, I and the classes of admissible functions '*(I', S) and 16*(F S') are defined as in Section 9.2. The boundary conditions become

y(u) = 0

(i)

for almost all u e I

or y(u) = 0,

(ii)

x(u) >_ 0

for almost all u e I

in the respective cases le*(I', S) or le*(F, S'). Again we denote by .9*(F, S) or aJ'*(r, S') the problems D(X) --+ min in 2*(T, S),

or in '*(F, S') respectively.

For the following it will be convenient to introduce polar coordinates (r, cp), 7r. Then we have

0 <_ r < 1, 0 :5;

('0t

n

jiXr(r, (p)I2 + Z I X ,

D(X) := 1 J 2

r

Jo

tp)12

rdrdco.

Select M + 1 points Qo, ..., QM, M := 2", n > 1, on T with Qo = P1, QM/2 = P3 and QM = P2 so that the polygon determined by the points Q0..... QM approximates the curve F. Next we define a subset T of 11M+1 by T:=

( , , 0, . .

.,

and for r e T we introduce the finite element space

:_ { f e C°(B, R): f is bilinear in r and 9 on Bk.,,,, 1 < k < M, 1 < m < where Bkm:=

(w=re`s:

Tk-1

<(P
M l)
33

Fig. 1. Discretization of the parameter domain.

M

9.10 Scholia

Also, let { fi: 1 < i < N} with N = 1 + and set

(M + 1)M 2

be a basis of

231

(see Fig. 1),

1,2+i,(M+1)+(2+i),2(M+1)+(2+i),..., 2

(M-II(M+1)+(2+i)}

where 0 < i< M. The discrete analogues of the classes c*(l, S) and '*(F, S') will be denoted by ,*(I S) and *(1', S') respectively; they are defined as follows:

Xn = (xn, y,, zn) e

*(I', S)

(or 1*(I, S') respectively)

if and only if there exists some r e T such that (1)

xn, yn, Zn e

(ii)

Xn(e`T'j)) = Q;

(iii)

X,,

(M )

for 0<j
\ for-IM-1I<m< 2 -l.

ES(orS')

The problems 9*(f, S) (or 9* (F, S')) will be replaced by 9n*(F, S): D(Xn) -* min

in 2*(I', S)

9*(f', S'): D(X.) -+min

in 2*(T, S')

or

respectively. An equivalent problem is obtained as follows. Define the function

d: R' xll

(3)

xR' xT--'l

by

d(x, y, z, t) := 2 {(x, A(t)x> + + },

where A(t) := (aii(t))i
ai,(r) := a(t) + a(t)

and i

n

a'(t):

rarf`(r,(o)arf'(r,(p)drdcp, o

a'(t) for all 0 < i, j < N.

Jo

o

Jo r

f,'(r, q,)a-f'(r,

(o)drdcp

232

9. Minimal Surfaces with Supporting Half-Planes

ME

lu Fig. 2a-L Numerical solutions of partially free boundary value problems.

234

9. Minimal Surfaces with Supporting Half-Planes

The problem *(I', S) (or 91*(r, S')) is then equivalent to minimizing the function d(x, y, z, t) on R" x R" x RN X T with respect to the constraints

N-M<j
(xi,YJ,z')=Q,,

(4)

and

y'=0 for jE.tOu.#M

(5)

(and x' >- 0 for j e d1, u MM if the support surface is S') for

x=(x ,x, ,x M), 0

z=(z ,z, ..,z M ).

Y=(Y0 ,Y,...,y'),

0

1

1

1

Now we fix some initial vector to e T. Then the iteration process for the minimization procedure at step k consists of two parts: (I) First we determine (xk, yk, zk) as minimizers of tk-1) with respect to the constraints (4) and (5). (II) Then we find rk as minimizer of d(xk, Yk, Zk,

).

The first step amounts to finding a minimum of a quadratic function under linear constraints, and it turns out that a conjugate gradient method is suitable for dealing with this problem. At a second step one has to calculate the first and second derivatives of d with respect to r. A straight forward calculation yields V,d(x, y, z, t) = (c1(x, Y, Z, T) - CO(x, Y, Z, T),_, (CM(x, Y, Z, T) - CM-1(x, Y, Z, T)) and

H,d(x, y, z, r) =

bo + b,

- bi

0

-b1

b1 + b2

-b2

0

- b2

b2 + b3

0

...

0

-bM-1 - bM-1

0

bM + bM-1

where H, d denotes the Hessian matrix of d with respect to t and ci(x, Y, z, t) =

Z {xi[ai1(t) - a Ii(t)]xJ + yi[a1(T) - aIi(t)]YJ

1

+ z1[a1 (r) - a2(T)]Z'}, bi = b;(x, y, z, r) =

2

2

{x`a ,(r)xJ + y1a'(t)y + zia1(T)z1 } . IJe _&i

A minimizer tk can now be constructed by Newton's method. Note that the triagonal form of H, leads to a fast determination of the solution. We refer to Wohlrab [2] for a more detailed presentation including a proof of convergence and error estimates. 3. Another Uniqueness Theorem for Minimal Surfaces with a Semifree Boundary As we have mentioned earlier, the uniqueness theorem of Section 9.5 is essentially

9.10 Scholia

235

the only result of this kind that was known for semifree boundary value problems

until recently. Hildebrandt and Sauvigny [1, 2] have now established further uniqueness theorems. We shall present some of the results of [1] for smooth support surfaces S although even piecewise smooth surfaces S are treated in that paper. We shall consider surfaces X(w) = X(u, v) = (x(u, v), y(u, v), z(u, v))

which are parametrized on the semidisk

B={weC:IwI<1,Imw>0} whose boundary consists of the closed semicircle

C={wrC:IwI=1,Imw>_0} and of the open interval

I={u ER:-1
Points of the 3-dimensional Euclidean space R3 will be denoted by P = (x, y, z). We identify the x, y-plane in l with R2, and the points (x, y, 0) of this plane with p = (x, y). Denote by 7L: X83 --> R2 the orthogonal projection of R' on this plane which is given by p = iv(P). With any surface X : B --> R3 in O3 we associate the mapping f : B -+ 182 given by f := it o X, that is, f(u, v) = (x(u, v), y(u, v)).

Fig. 3. The geometrical setting for
236

9. Minimal Surfaces with Supporting Half-Planes

Furthermore, we consider boundary configurations in l

consisting of a supporting surface S and a Jordan arc F with endpoints P, and P, on S. Let Pi = 7r(Pt), P2 = rc(P2) and F = 7r(F) be the projections of P,, P, and I' onto 182.

We assume that P, 0 P2, and that T is represented by a piecewise smooth homeomorphism y: [tl, t2] -+ I'satisfying y(t,) = p, and y(t2) = P2. Moreover, let £o = U(18) be a Jordan arc in l 2 which decomposes R2 - £Q into two disconnected open sets. The parameter representation a: 18 - ff8` of £o be of class C' and satisfy IQ(s)I = 1. We suppose that S is the cylinder £e x 18 above £,,, i.e., 7r(S) = 10. Let p, = U(s, ), p2 = 0-(s2), and denote by £ the subarc £ := c([s 1, s, ] )

between p, and P2. We assume that the interior of the arc I' lies in one of the

two connected components of R2 - £o and that I'u £ is the boundary of a simply connected domain Gin 182. In addition, the preimage 7r-'(p) of any p c- r is supposed to contain exactly one point P e F, except for at most finitely many p E F for which En 7r-'(p) is a whole interval on the vertical axis { p} x R through p. Such an arc F is said to be a generalized graph above F. Denote by i(s) the tangent vector U'(s) of £0 at p = U(s). The exterior normal to £o at p = U(s), s e 18, will be denoted by ,Z(s) = (.r' (s), .Y2(s)), 1. Another representation of the exterior normal of £o at p is given by v(p) := (p)). Then .K(P) := (v(p), 0) is the surface normal to S at any point P e 7r (p) _ {p} x R. For any s e R we introduce the straight line 9(s) := {p e 182: = I,,

the ray

Y(s):={p=U(s)-2',(s):A;>t0}, and the two halfspaces .*'-(s):= {p e 1182: S }, +(s) := {p e 182: > }

Now we can formulate several geometric conditions on the domain G that will be used in the sequel (see Figs. 3-5).

Fig. 4. The condition (C 1).

9.10 Scholia

237

Condition (CI). (i) W(s) n ris empty for all s E Vi4 - [s,, s2]. (ii)

W(s - 0) n ja(r): s1 < r < s} is empty for all s > s2, %(s + 0) n { a(r) : s < r < s2 } is empty for all s < s, .

(iii) For any pair s', s" with s' < s, < s2 < s", at least one of the following two relations holds true : {a(r): s' < -r < s, } c int X -(s")

{a(r): s2 < i < s"} c int $' (s'). Condition (Cl*). Suppose that (i) and (ii) of (Cl) are satisfied as well as one of the following two conditions: (iv) For all i, s with i s1 < s2 < s and o-(-r) E p(s) we have < 0. (v) For all i, s with s s1 < s2 < i and a(t) E s(s) we have <X(t), i(s)> > 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) :=
(s ")>

Clearly cp is continuous on B, harmonic in B and nonconstant. Hence (p assumes its maximum M on 8B, and our preceding reasoning yields that M =
9.10 Scholia

239

Suppose now that s" > 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.

However, E. Hopf's lemma implies that 0) < 0 since p is nonconstant on B, and we have arrived at a contradiction. Thus we can infer that s" = s2. Now essentially the same reasoning, applied to (P(w)


WEB,

yields s' = s1. We only have available conditions (i) and (ii) of (Cl), but since s" = s2 is already established it suffices to show that ( attains its minimum m = at some point u* e I with f(u*, 0) = a (u') if s' < sl. In the same way as before Hopf's lemma implies 0) > 0, whereas (6') once again leads to (p,(u*, 0) = 0. This contradiction can only be avoided if s' = s1. Consequently we have t = f(1). Remark. Although the proof of Proposition 1 is rather simple, the result f (I) = E' is by no means self-evident. In fact, there exist boundary configurations consisting of a cylinder S and of a generalized graph T above the x, y-plane with

endpoints on S which bound solutions X of 91(1', S) such that the relation f(I) = E does not hold for f = 7r o X. To this end, we consider the circle Eo := {(x, y): x2 + y2 = 1 }, the cylinder S = X0 x 18, and the polygon I' consisting of the three segments (see Fig. 1 of Section 7.6)

1 :={0<x<1,y=z=0},

T2:={x,y=0,0
The Jordan curve Tis an improper graph over the polygon Fin R2 which consists of the two segments

).1 ={0<x bounds denumerably many noncongruent minimal surfaces of the type of the disk, each of which is part of a helicoid and a solution of 9(T, S). Apart from the area-minimizing solution, no surface X of this sequence satisfies (rr o X) (I) c E where E denotes the quarter circle

E:=((x,y): x2+y2= 1,x>0,y>0} since (7r o X) (1) = Eo.

The next result is proved essentially in the same way as Proposition 1; hence we shall omit its proof.

240

9. Minimal Surfaces with Supporting Half-Planes

Proposition 2. Let X be a solution of _61A(I, S) which lies on the same side of S as

I', and suppose that the boundary configuration
Proof. By Propositions 1 and 2 it follows that f(OB) c 8G. Since the range f(B) of a harmonic map f : B -- R2 lies in the convex hull of its boundary values, we infer f(B) c G if G is convex, and we even obtain f(B) c G since f is nonconstant. We only have to apply the maximum principle to suitable comparison functions of the kind p(w) :=
We remark that one can easily state variants of conditions (Cl) and (Cl*) which are sufficient to prove the property f(I) = E. It would be of interest to find a condition on
yields a monotonic mapping of C onto F. Now we want to show that under suitable assumptions on
Proof. Consider the continuous function 0: G -> R which is defined by Q(x, y) := s if p(x, y) e 2(s). Because of (C2) we have 0(6) = [s2, s2]. We introduce the continuous function 0 (u, v), w = u + iv e B, by ip := Q o f. We have qi(B) = [s1, s2], a/i(I) = [s1, s2], and >/i(-1, 0) = s1, 0(1, 0) = s2. It remains to be seen that 01 furnishes an increasing map of 1 = [-1, 1] onto the interval [s1, s2]. If this were false, we could find points u1 and u2 such that -1 < u1 < u2 < 1 and t/i(u1) > fi(u2). Then we choose a value s* such that Ou2) < s* < tp(u1) and consider the connected components Z1 and Z2 of {w e B : qi(w) > s* } and {w e B : t/i(w) < s*} which contain u1 and u2, respectively. Claim. Both Z1 and Z2 have a nonempty intersection with the open circular are C.

9.10 Scholia

241

In fact, suppose for instance that Z2 and C were disjoint. Consequently we would obtain that Z2 c Bu I u { -1 }, O(w) < s* on Z2, and [i (w) = s* on aZ2 n B. Set so := infz, 0. Since t/i(w) is nonconstant on I n Z2 we have so < s*. Therefore

the function 'J2 assumes its minimum at some point wo E Z2. If wo E Z2, then po := f(wo) lies on a straight line W*(so) c _'f(so) given by an equation _ for p = (x, y) e 182 where v is a unit vector in 182 such that > for all w E Z2. The function cp: Z2 -+ 18 defined by cp(w) :=
principle, and we obtain that wo = uo e I n Z2. Set _9(w) := for w E Z2. The function cp: Z2 -+ 18 is continuous on Z2, harmonic in Z2, and assumes its minimum on Z2 at some point uo c- I n Z2. Since
cp(uo,0)
Thus we can apply E. Hopf's lemma and arrive at hand, (6) yields (P0(uo, 0) =

0) > 0. On the other

0), A(so)> = ,t(uo)
and we have arrived at a contradiction. Consequently, the intersection Z2 n C is nonempty, and similarly we can prove that Zl reaches the boundary arc C. Hence there exist two points wl = e`°' and w2 = e`O2, 0 < 01 < 02 < it, such that wl E 8Z1 and w2 E 8Z2, and the definition of Zl and Z2 implies that ip(wl) > s* > '(w2). This is impossible since (C2) implies that the function Q(y(t)) is increasing whence Y'(9) := Vi(eie) = S2(f(eie)), 0< 9 < it, is decreasing. Hence we have arrived at a contradiction, and therefore (u, 0) has to be increasing. Now we want to investigate whether solutions of 9(I', S) are embedded. The key to embeddedness is the classical result of H. Kneser and Rado which was already used in Section 4.9.

Proposition 5. Let f e C°(B, 182) be harmonic in B, and suppose that f maps aB onto the boundary 8G of a convex domain G of 182. Then we have: (i) IfJ I aB is monotonic, then f IB maps B diffeomorphically onto G. (ii) If f I aB maps aB topologically onto 0G, then f maps B homeomorphically onto G.

We shall now establish a somewhat more general version of this result.

Proposition 6. Let f e C°(B, 182) be harmonic in B, and suppose that there is a Jordan domain G in 182 such that f(B) c G, and that f I aB maps aB monotonically onto 8G. Then f I B maps B dfeomorphically onto G. If, in addition, f l as furnishes a topological map of 8B onto 3G, then f maps B homeomorphically on G.

242

9. Minimal Surfaces with Supporting Half-Planes

Proof. It suffices to sketch the main ideas. We begin by proving that the Jacobian Jf of the map f(w) = (x(w), y(w)) does not vanish in B. In fact, if Jf(wo) = 0 for some wo e B, then there are numbers a, b e R, a2 + b2 = 1, such that (7)

aVx(wo) + bVy(wo) = 0.

-, 1(p) := + c for p e R2, and consider the {p E R2 :1(p) = 0}. Since po := f(wo) E 2 n G, the set 2 n G straight line 2 is nonempty. The function h(w) := (1 of)(w) =
F(w) = am(w - WO)m + "

We infer that, in a sufficiently small neighbourhood U of wo, the zeros of h lie on m different regular, real analytic arcs which divide U into 2m sectors, and 2m >> 4. Let B+ and B- be the open subsets of B where h(w) is positive or negative

respectively, and Bo := {w E B: h(w) = 0}. By construction, we have f(B0) c . n G. Invoking the maximum principle we see that no component of B+ or Bhas a closure which is completely contained in B. Hence the connected component Z of Bo containing wo has the property that Z n 8B contains at least four consecutive points w1, w2, w3, w4 such that on the subarcs I'1, f 21 Q3, fl4 of 8B

between those points there exist four other points C; e fij, j = 1, ..., 4, satis0, h(C3) > 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


J

1' Zl - WZ 1

vZ2]d.v1

G'

w here v is the exterior normal to 8G', and .e' is the 1-dimensional Hausdorff measure on 8G'. Inserting co = Z1 - Z2, we arrive at

dxdy (16)

G'

(Z 1-Z2)

LWl'aZ,-w21a z2]d.YF1.

faG,

By the reasoning presented in Section 9.5 we find that (17)

[max{ W1, W2}]-31OZ1 - OZ212 <

holds true everywhere on G. Fix some set 0 cc G' cc G, and let

9.10 Scholia

247

m(Q) := supra max { W1 , W2 } .

Then we infer from (16)0 and (17) that I VZI - VZ2 I2 dx dy

m-3 (Q) f12

(18)

4"'(Z1 - Z2) Note that the functions Z; and

[W1

1

W-1

Z1 - WZ 1

Z2] d.e1 .

Z; are uniformly bounded on G, Z; e

C' (G u ±) and av ZZ = 0 on E, and that Z, - Z2 = 0 on r except for finitely many points. Therefore we can choose a sequence {G;,} of domains G,, cc G such

cc G cc

that Q cc Gi cc GZ cc

f

lim o (Z1 - Z2) [ W1- 1

and

de 1 = 0.

Z1 - W2 1 av

.

Z2J

Now it follows from (18) that

fa

IVZ, - VZ2I2dxdy = 0

whence VZ, = VZ2 on Q and therefore also on G since Q was an arbitrary subset of G such that Q cc G. This implies Z, (x, y) =- Z2(x, y) on G because of Z1 = Z2 a.e. on F. U Similarly, from Propositions 3,4 and from Theorem I we derive the following result:

Theorem 3. Suppose that F is convex with respect to G, S E C*, and assume that satisfies (Cl) or (C1*), and (C2). Then (up to conformal equivalence) there is at most one solution of 9(F, S) such that X(B) lies on the same side of S as F. This solution is a graph over the domain G.

Remark. It is not excluded by our reasoning that there could be other solutions of 9(r, S) which penetrate S.

Part IV

Ramifications :

The Thread Problem. The General Plateau Problem

Chapter 10. The Thread Problem

The problem to be studied in this chapter is another generalization of the isoperimetric problem which is related to minimal surfaces. Consider a fixed arc T with endpoints P1 and P2 connected by a movable arc £ of fixed length. One

may conceive F as a thin rigid wire, at the ends of which a thin inextensible thread £ is fastened. Then the thread problem is to determine a minimal surface minimizing area among all surfaces bounded by the boundary configuration


10.1 Experiments and Examples. Mathematical Formulation of the Simplest Thread Problem Imagine N points P1, P2, ..., Pr, in l which are connected by k fixed arcs I'1, ... , I'k and by I movable arcs El , 12, ..., £, in such a way that the resulting configuration <1, £> := consists of n disjoint closed curves C1, C2, ..., C. of finite length. The lengths of the arcs £; are thought to be fixed. Experimentally we can realize the points P1, ... , PN as small holes in a plate or as endpoints of thin rods stuck in a plate. The arcs T are made of thin rigid wires, and the curves £; can be realized by thin and essentially weightless synthetic fibres. Into such a boundary configuration we want to span a surface of minimal area, which can experimentally be achieved by dipping the array into a soap solution and then withdrawing it. This way a soap film will be generated

10.1 Experiments and Examples

253

into account. We shall obtain solutions that are parametrized on a compact connected parameter domain B, the interior B of which consists of at most countably many components. Let us now specify the mathematical setting of the thread problem 1(T, L) that will be solved in the following section.

Notational Convention. In Sections 10.1 and 10.2 we shall, deviating from our usual notation, denote a disk of center zo and radius r by B(zo, r) instead of Br(zo)

An admissible parameter domain for the thread problem is defined to be a compact set B which can be represented in the form

B=[-1,1]UU B,,,

(1)

1
V=1

Here the sets B. with v e N and v < vB denote the closures of mutually disjoint disks B(u,, rv), rv > 0 whose centers u, lie in the open interval {u : -1 < u < 1} on the real axis. Moreover, all disks B, are supposed to be contained in the unit disk B(0, 1).

Introducing the numbers a, and b, by av := uv - rv ,

(2)

by := uv + rv ,

we then have

a,,bve[-1,1].

(3)

Let us denote the set of all admissible parameter domains B by .4. For every B e R, we introduce the two mappings pB and p8 : [-1, 1] -+ aB by

u (4)

Pe (u)

u+i

if rv

-(u-uv)2

ueaBn[-1,1] Iu - uVI < rv.

Fig. 2. (a) A parameter domain, and (b) a corresponding solution to the thread problem consisting of two components.

254

10. The Thread Problem

Let c be a curve mapping a subinterval I' = [a, /3] of 1 = [ -1, 1] into R3, C : I' --). R'.

Then the length of c is given by I(c, I') = sup Y_ lc(ti) - c(ti-,)1

(5)

i=1

where the supremum is to be taken with respect to all possible decompositions

If I' = I we shall write 1(y) := 1(y, I)

For any two intervals I, and I2 in 68 we introduce the set M(1,, I2) of continuous, nondecreasing mappings 0: 11 -+ I2 of

I

onto 12, and we set J1(I)

m(I, I). We observe that the length L of the movable curve I is bounded from below by the distance of its endpoints P, and P2,

IP1 - P21 < L.

(6)

Given a rectifiable Jordan curve T with endpoints P1, P2, and a number L

satisfying 0 < I P, - P21 < L, we are now going to define the set ?(T, L) of admissible surfaces X for the thread problem as follows: Definition 1. The set '1(T, L) consists of the mappings X e C°(B, i83) n H2 (B, with B e -4 which satisfy the following two conditions: (i) l(X o pa) L;

l3)

(ii) there exists some mapping 0 e .11(I), I = [-1, 1], such that 01 oB,1 = id I OB11 and X o pB = y o 0 where y denotes a fixed Lipschitz continuous represen-

tation of T which maps I bijectively onto T.

In other words, a function X is admissible if it is parametrized on some domain B E -4, if it is continuous and has a finite Dirichlet integral, if the length

of the "free part" X o pB is less or equal to L, and if X o pB yields a weakly monotonic parametrization of T. Note that T and I may have one or more interior points in common, that is, E may in part adhere to T. The thread problem 9(F, L) now consists in finding some surface X e '(T, L), defined on some parameter domain B e .4, such that X minimizes the Dirichlet integral (7)

D(X, A) = 1

fi

1VXI2dudv

among all surfaces of IV(T, L).

The solution of this problem will be carried out in two steps. First we shall single out a set B e .4 which can serve as a parameter domain of a solution of 9(T, L); this is the nonstandard part of the construction. We shall obtain such

10.2 Existence of Solutions to the Thread Problem

255

domains B as "minimal elements" with respect to inclusion. In a second step we shall construct a minimizing mapping X parametrized over B. Let us now introduce the following three infima d, d+, and d-: (8)

d = d(I', L) := inf{D(X, B): X E (8(I', L) };

(9)

d+ = d+(r, L) := inf{D(X): X e W(I', L), B = B(0, 1) } , where D(X) := D(X, B(0, 1));

(10)

d- = d- (F, L) := inf{S: b has the approximation property (.cl)} .

The approximation property (cV) is defined as follows: There exists some decreas-

ing sequence of real numbers 2,, > 0 with .1 --.0 and a sequence of surfaces X E cf(r, L + with parameter domains B,, e 24 such that D(X,,, 6 as n

oo.

An obvious consequence of these definitions is the relation

d`
(11)

We shall prove that

d- =d=d+ holds provided that we assume

JP1-P21
10.2 Existence of Solutions to the Thread Problem Consider now the particular case Y(1, L) of the thread problem that was formulated at the end of the previous section. Our main goal is the proof of the following existence result which is formulated as Theorem 1. Suppose that I P, - P2I < L < 1(T). Then we obtain

d-(I, L) = d(F, L) = d+(F, L). Moreover, there exists an admissible parameter domain B and a surface X E W(I', L)

parametrized over B such that

D(X, b) = d(r, L).

256

10. The Thread Problem

This minimizer X is a minimal surface, that is, X is of class C2(B, R') and satisfies the equations

AX = 0 IXuI2 = IX,I2,

<X., Xv> = 0,

in b, and furthermore, the free boundary of X is of maximal length, i.e., l(Xape)=L.

The proof of this theorem is divided into two parts. The first one is concerned with the existence of a minimal parameter domain B e -4. In the second part of

our discussion we will show that such a parameter set B is the domain of a solution X for the thread problem YA(I, L). This will be achieved by establishing whose elements are defined on B the existence of a minimizing sequence

and converge to a solution X of i(T, L). PART 1. Construction of a Minimal Parameter Set B e -4. We begin our discussion with the following Lemma 1. Suppose that X is a surface of class '(T , L) which is defined on B E -4, and let a be an arbitrary positive number. Then there exists some XE e'(T, L + E), parametrized over B(0, 1), such that ID(XE) - D(X, B)I < E.

(Recall that D(XE) denotes the Dirichlet integral with the unit disk B(0, 1) as domain of integration.) Proof. An admissible domain B is of the form given by formula (1) of Section 10.1. Since I(T) < oo and ve

D(X, ]3)

Y D(X, Bv) < co, v=1

we can find a number vo e N such that

Y D(X,

e

v>V0

and

Y l(y, [av, bv]) < e.

v>v0

Set

B':= 1 uB1 uB2

1 = [-1, 1],

and

X 1(w) _

u

X (w)

if w e B1 u B2 u .

y(w)

if w e OB r [-1, 1].

10.2 Existence of Solutions to the Thread Problem

257

Fig. 1. A parameter domain with v. = 2, and the numbers a, e_ d b,..

Then we infer X 1 E W(E, L + e) and 1 D(X, B) - D(X1,

(1)

For each vo with 0 < vo < min{r1, r2, ..., rv0}, there exist numbers c, dv with a, < c, < d, < b, such that pB.(c,) = c. - ivo, pB.(d,) = d - ivo; cf Fig. 1. Now we choose vo so small that also the following conditions are fulfilled: (i) X,(u - ivo) is absolutely continuous with respect to u e U [c,,, dj and V
has square integrable first derivatives; (ii) 1(X1 o

[au, cV]) + 1(X1 o pB., [dv, b,]) 5 2v 0

For some arbitrary number S > 0, we define the set -q by

.Q:={w=u+iv: Jul<1,0b}, .9_:={w=u+iv: w-ivoEB'andv<0}.

v

7W

Fig. 2. The domain 9 = fi(b).

10. The Thread Problem

258

We note that !D is conformally equivalent to the unit disk. Thus, in order to

prove the assertion of the lemma, we shall construct a suitable comparison function X2 defined on 9. This function is defined as follows:

X, (w - i(b + v°)) if w E + if w E !2_ . (w - ivo)

X2(w) := X'

7 0 Sl

7oT2

7 0 S2

7 0 TH

0

\W\di -bi

d 2 b2

2

1

7(u) 7(U)

Fig. 3. The definition of X2-

For 0
u E[c,dv], 1
Xl(u - ivo) y(u)

X2(w) :_

if

y(S,(u)) y(T,(u))

Here Sv is a linear mapping from [av,

u c- [-1, 1] - U (av, bv) v
u e [a,, cv] u c- [dv, by].

onto [av, Bl (c,,)], and Tv is the linear

map from [dv, bj onto [91(dv), bv, where Bl E 0(1) is the transformation I _91 that corresponds to X1. In other words, y o 81 = X 1 a pa.. We infer from the construction that X2 is of class ) n HZ (s). Furthermore, we have C°(_

D(X2, -41 v _) = D(X1, B')

(2)

and

D(X2, j) < 1 2

(2')

f-I vp

IVX2I2dudv

f

I

1

a,

1

ivo)I2du + b

b v=1

c

We, moreover, note that the mapping

X2:82n{Imw<0}->1

IY(t)J2dt. -1

10.2 Existence of Solutions to the Thread Problem

259

is weakly monotonic, and from (ii) we derive the estimate

V2, 0-9 n{Imw>O}) < l(X1 o pB.) + 2 V
{l(X1 o pa,, [av, cv]) + 1(X1 o Pa-, [dv, bv])}

-9 be a conformal mapping of B(0, 1) onto -4, leaving the two points w = ± I fixed. Then X,:= X2 o i is of class W(T, L + 2a), and we infer

I D(XV) - D(X, B)I = ID(X2,

D(X, B)I

<- ID(X2, .-4) - D(X1, B')I + ID(X1, B') - D(X, B)I < S const + E,

taking (1), (2) and (2') into account. Since we can choose b > 0 arbitrarily small, the assertion of Lemma 1 is proved. The following result is an easy consequence of Lemma 1. Proposition 1. The class W(F, L) is nonvoid, and B(0, 1) e R*(I', L).

Proof. Define y(1) +

0 < t:5 7r

[y(- 1) - y(1)]

n

if

7ct<27r

Y(-1 + 2 where y(- 1) = Pt and y(l) = P2. Then y*: 8B -* O1

is Lipschitz continuous,

and a straight-forward computation shows that X*(w) := I wl y* (I

I is of class

_WW1

c9(r, L). Hence '(F, L) is nonempty.

It follows from the definition of d- that there is a sequence of surfaces

X e c'

L+

n

parametrized on domains Be

such that

1

ID(X,,,h.)-d-I < n forallneN. By virtue of Lemma 1, we can choose a sequence of mappings X,* e ? I I', L +

which are parametrized over B(0, 1) and satisfy

ID(X.)-D(X,,,B.)I < 1 , n= 1,2,... n

nI

260

10. The Thread Problem

Thus we infer

n=1,2,..., ID(X*)-d-I
11

In the next lemma we prove the existence of sets B e .J(I', L) which are "minimal" with respect to an ordering of sets defined by inclusion. Lemma 2. Suppose that L < I(F). Then any set of elements B E .4*(f, L) which is totally ordered with respect to inclusion possesses an infimum in L).

Proof. Let {B

be an arbitrary set of elements Ba e.*(r, L) with the index set A which is totally ordered with respect to inclusion, and set

B:=nBQ. aeA

We have to show that B is an element of -4* (T, L). The first step will be to prove B

(i)

0.

In fact, if B were empty, we would have

I:=[-1,1].

I=closlaU (I-Ba)), Then, for any partition

- 1 = tp < tl < t2 < ... < tk = I of I, there exist numbers tj E U (I - BQ) with 0 < j < k and n e N such that aEA

limtj =tj forj=0,1,...,k. n- or

Since the set {Ba }aEA is totally ordered, we infer that, for every n e N, there exists

an index an E A such that tj e I - Ba holds for all j = 0, ..., k. As all domains B are contained in -4*(F, L), n = 1, 2, ... , there exist surfaces Xn e

I \\\

parametrized over B,*,. This implies [k

/

k

/,- I y(t;) - Y(tj 1)I = Y- I XX(t;) - X,,(tj 1)I

j=1

j=1


Since k

lim Z IY(t;) - Al -A = I IY(tj) - Y(tj-1)I,

n-+a; j=1

j=1

I', L + n

10.2 Existence of Solutions to the Thread Problem

261

we arrive at n

I IY(tj) - Y(tj-1)I < L.

j=1

As the partition to, t1, ..., to of I may be chosen arbitrarily, we conclude that 1(F) < L

which contradicts our assumption I(F) > L. Now we turn to the proof of

Bng4*(F,L).

(ii)

We have to find surfaces defined on B whose Dirichlet integrals converge to d-, and whose free boundaries ("threads") exceed L only by an arbitrarily small amount. First of all, every e > 0 there exists some vo E N with 1 < vo < ve such that

E 1(Y,[a.,b,])<_e.

(3)

V>Vo

Fig. 4. The case vo = 2.

For v > vo we define

{w=u+iv: a,,
GE2-v-1}

and choose conformal mappings Tv : B, -* Q,, from B, onto Q, with fixed points av, bv. Here B1i B21 ... denote the components of the domain B (cf. Section 10. 1, (1)).

Then the surfaces X, := Y(Re Tv),

v > vo,

are continuous and havethe Dirichlet integrals (4)

D(XY, BY) = 2

Je2-.-, fl,

6,

IY(u)12dudv = e2-v-1 2

Moreover, Xv o pB is monotonic on [a,, U.

JIY(u)12du. a,

262

10. The Thread Problem

For a e A and v e { 1, 2, ..., vo } there is a uniquely determined v* = v*(v, a) with 1 < v* < vB, such that By c Here B, is the v*-th component of the domain Ba ; cf. Section 10.1, (1). Since {B,, }aeA is totally ordered, we infer from

the definition of B that there is an index ao e A such that the disks B o,,. are mutually disjoint, and that f.a

Ifldt +

(5)

E

YIdt <-

fb,

VO

holds. Here a and by are the numbers associated with B which are defined in formula (2) of Section 10.1, and a°, b° are the corresponding numbers for B

i.e., [a°,b°]:=InBo,v. We have finitely many (at most vo + 1) open intervals .1 c I such that

f-I lyldt- v
(6)

fa,b

lyldtfy IYIdt. J}

For any such interval I, there is a partition

to < ti < t2 < "' < tk,

tj E 5,

such that f.4 Ivl dt

:!5:

1 + JY- IY(tj) - Y(tj-1)1

Passing to a suitable refinement of this partition, we may also assume that there is a subset -f of {1, 2, ..., k} with the following properties: (I) j '' if and only if (tj_1, tj) c (a, for some v > vo; (II) j e if" if and only if [tj_1, tj] c ,0 - B. Moreover we can choose an index ao E A such that the following can be achieved : The values t j e 5 - B are replaced by values t ; e 5 - B all other values tj

remain unaltered and will be called tj ; we have to < t' < Y, IY(tj) - Y(tj-1)I <_

(8)

je 1,

VQ + 1

< tk and

+ I IY(t;) - Y(tj-1)I je .7('

We infer from (3), (7), and (8) that

(9)

1

{'}

Ivldt <- s+ Y f.5

V>VO

3r; + I

J a,b

Iyldt + Y E

{J} ja.X'(J)

Y-

{.r} je -f(J)

IY(tj) - Y(tj-1)I

IY(t;) - Y(t1-1)I .

After these preparations, we proceed as follows: Since B .*O e

is some surface X E'(F, L + e) defined on B,,* such that (10)

D(X,Bo)
*(I', L), there

10.2 Existence of Solutions to the Thread Problem

263

Furthermore, for each v with 1 < v <

v°, there exists a conformal mapping Tv: By -+ B a* v such that X ° T, ° pB_ furnishes a monotonic parametrization of that subarc of F which corresponds to [av, bv]. Then X,' := X o Tv defines a continuous surface defined on By satisfying (11)

B ).

D(X;,,

By virtue of (5) we obtain (12)

1(X;, ° pa , [av, bv]) <- I(X ° p, , [a°,

b°]) + v°

Let us introduce the surface X£ by


rXv(w) XE(w)

vO

y(w)

w E 1 - U B,.

L

v=1

Then we have XE e HI(B, R3) n C°(B, R3), and it follows from (4), (10) and (11)

that (13)

D(XE, B) < D(X, ha ) + Y e2-v-1

l

d-(T,L)+e+c J 1

IYI2dt.

The length of the movable part of the boundary (14)

is estimated by

I(Xeope)< > l(X;,opa,[av,bv])+ E

{f} J

v5v0

l2dt

f-'i

>V0

Ivldt 0

< e + Y l(X o Paso, [a°, b°]) + 3e + v5vO

= 4c + Y_ l(X o paso, [a°, b°]) + vSvO

{5} .le X

{J} je it

I7(ti) - y(ti-1)I

I X(t') - X(tt-i)I

<4e+l(XopBL)<5e+L, on account of (12), (9) and of X e '(F, L + E). The relations (13) and (14) yield B e R*(F, L).

Applying Zorn's lemma we infer from this lemma that the following result holds true:

Proposition 2. The set l*(F, L) possesses minimal elements with respect to inclusion, provided that L < l(r).

264

10. The Thread Problem

PART II. Existence of a Solution of ^F, L). Let B e I*(T, L) be a minimal element the existence of which was established in Proposition 2. We want to prove that B is the parameter domain of some minimizer X.

Lemma 3. If X is a function of class H2'(B(0,1),R3) with a trace a L2(8B(0, 1), l ) on the circle 8B(0, 1) which is of finite total variation f asro, i> Id then the boundary values : 8B(0, 1) -+ R3 actually are continuous.

The proof of this result is an immediate consequence of the CourantLebesgue lemma and has essentially been carried out in part (iii) of the proof of Proposition 3 in Section 4.7. In fact, we even know that is absolutely continuous (see Theorem 1 of Section 4.7). Now we turn to the crucial step in proving Theorem 1 which is to prove

Theorem 2. Let B e -4*(T, L) be a minimal parameter domain with respect to inclusion. Then there exists some X e'(I', L), parametrized over B, such that

D(X, B) = d-(I', L) = d(T, L); thus X is a solution of the minimum problem 1(F, L).

Proof. Since B E .4*(r, L), there is a sequence of surfaces Xn e l I', L +

1),

n c- N, satisfying

D(Xn, B) < d-(f, L) + 1 < M n

for some constant M > 0. Denote by Bn e .19(1) the mappings associated with Xn, i.e., 0nI8Br,r = idI aBr\r

,

X. ° PB = Y ° en

Moreover, let u, r, a, b,, be the numbers corresponding to B, and let B, 1 < v < vB, be the components of B (see Section 10.1, (1) and (2)). Applying suitable conformal reparametrizations, we can achieve that

BU(uv)=u, forneN and 1
v 0 and two sequences {tn}, ay < to < t;, < bv, converging to some point to a [ay, b,,], such that (15)

I9n(tn) - BO(t;,)I >_ eo

with

for all n e N

is satisfied. (Actually, this would hold true for some subsequence of {6n}. However, by renumbering this subsequence we could achieve that (15) is fulfilled.) We want to show that (15) leads to a contradiction. In order to do so, we distinguish

10.2 Existence of Solutions to the Thread Problem

265

the two cases (i) a, < to < b,, and (ii) to = a, or b, Case (i) can be excluded by the discussion given in Chapter 4, where we have proved that the boundary values of a minimizing sequence for the ordinary Plateau problem are equicontinuous. By this reasoning we obtain that the functions y o 0,11,,,d,] are equicontinuous for

every interval [cu, dj c (a, by). The injectivity of y then implies that also the functions 0nI1c,d 1 are equicontinuous which contradicts (15). In fact, there is some no E N such that

holds for n > no and for suitably chosen numbers c and dv. Then it follows from (15) that Y(Bn(tn)) - Y(OO(tn'))I ? c(Fo) > 0

for some fixed number c(eo) > 0 and for all n > no which contradicts the equicontinuity of the sequence y o on Ilc ,d j. Thus case (i) cannot occur. Now we want to exclude case (ii) as well. It suffices to show that to = a, is impossible since the case to = b, can be handled analogously. Thus let us assume that to = a,,. We can choose sequences of numbers 5, r,, and s;, with bn E (0, 1), bn -+ 0, 0 < r,, < Sn, to < sn < u, pg (sn) e 12

a

aq

<

X,(rn> (P) dcp

2irM log 1/bn .

Here r, cp denote polar coordinates around a,,, and the integral on the left-hand side is extended over the (p-interval in [ - 7c, rc] corresponding to the arc in B n aB(av, r,,) which contains (p = 0; cf. Section 4.4, Lemma 1. There is a subsequence of {B,(s;,)} converging to some value uo; renumbering this sequence we may assume that uo as n - oo. By virtue of (15) we have uo > a,, + so. Choose values sn with a,, < s,, <- u and 0,(s,,) = uo, and consider the two closed disks .9t and 92 defined by B

Ca,+uo uo-a,l 2

uo+by b,, - uol

B

2

2

2

Our aim is to define surfaces Yn on .91U-92 such that the surfaces X,*,: B* -- R', given by

(16)

Xw nr )

Xn(w) Y. (W)

for

WEB - B E

-91 U.

2'

W

B*:=(B-B,,)v.91V-92E-4, are of class c'(r, L + A1n) with A,, -- 0 and satisfy

D(X,, B*) -+ d-(T, L)

266

10. The Thread Problem

Fig. 5. The disks -Q 1, .22, and B.

as n -+ x. This, clearly, would contradict the minimality of B, and therefore we would also have ruled out to = a (or by), i.e., case (ii) cannot occur either. Passing to a subsequence and then renumbering, we can achieve that either

s;,<sn or else

sn<s;, forallneN holds true. We only treat the first case; the second one can be dealt with in an analogous way. Consider topological mappings 91

B(av, rn) n B,

Qn: -92 -> By - B(av, r,,)

with Tn(av) = av, Tn(u0) = PB (sn), on(bv) = bv, a (u0) = PB (sn) such that -41 is con-

formally mapped onto B(a, r,) n B, by Tn, and that Qn maps -92 conformally onto

B, - B(a,, rn). Note that K. ° Tn)(u0) = X"(PB (Sn)) = y(en(sn)) -' y(u0) (Xn o Qn)(u0) = Xn(PB(sn)) = y(OO(S.)) = y(u0)-

If we had (Xn o Tn)(u0) = y(uo), we would simply define

10.2 Existence of Solutions to the Thread Problem

267

in-9t a. in-92

jl

and the proof would be complete. As we only know X. o r,(uo) - y(uo) as n -+ co,

we have to adjust the data correctly. The idea is the same as in the proof of Lemma 1: we have to fill in the missing parts of T, thereby slightly changing the Dirichlet integral and the length of the free boundary of Z with y(uo) such X o;: -9i --* R3 a new surface (X o that for w e -9t

Xo

for w e -92

satisfies both

D(Y., -9t u -q2) :! D(X., Bv) + b and 1(Y o p +,,Q,, [a,, bv]) aa

21(y, [0,.(s.), 0,.(s.)])

lim A,,=0, n-.o

'(T, L + .1 -+ 0. This finishes the proof of equicontinuity of the mappings B Ilo ,b i, n e N, for every v e N with 1< v< vB. of Now we can apply the reasoning of Chapter 4 to the sequence

and that the surfaces X,*,: B* -+ l3 defined by (16) are of class

surfaces X E 1 I', L +- which are defined on the minimal parameter domain n

B e R*(1', L) and satisfy

- - < M for all n e N .

(17)

n

From this inequality, together with sup IX,J <_ M'

forallne N

as

and some constant M' independent of n, we obtain that

is a bounded

sequence in HZ (B, 1R3 ).

Passing to a suitable subsequence of X and renumbering it, we can assume tends weakly in H21 (b, R3) to some limit X E HZ(B, t3) that the sequence such that X tends a.e. and also in the L2-sense on every boundary 3B, to the trace of X. By virtue of the equicontinuity result proved above we can assume that the mappings B e .11(I) associated with X tend uniformly on I to some limit 0 e .,1(I) such that the relations O I OB,r = id l aBrr ,

0 PB = y 0 0

268

10. The Thread Problem

hold true for some continuous, weakly monotonic mapping from pB(1) onto F, with the property that and X coincide a.e. on pB (I) - I. Thus we can use S to define X on pB (1) by setting X(w) (w) for w c- pB (I), and we have X °PB =Y00. Moreover, on account of Helly's selection theorem' and of the assumption 1

1(X o pg) < L + -, we can assume that X o pB tends to X o pB everywhere on n

I, and that 1(X o pB)

By Lemma 3 we conclude that X has continuous boundary values on every CBv, and consequently X is continuous on B. Recall that Dirichlet's integral is weakly lower semicontinuous on HZ (B, R') (see formula (7) in Section 4.6), that is, the weak convergence of X to X implies

D(X, B) < lim inf

B).

Then, by (17), we arrive at

D(X, B) < d-(r, L). Consider now mappings H, E C°(B,,, R') n H'(B&, fR') which are harmonic in B, and coincide with X on BBv. Then we also have D(HV, Bv) < D(X,

B X* is of class C°(B, 1R') n H1(B, U8') and satisfies

D(X*, B) < D(X, B),

l(X*oPB)=l(X°PB)
X*oPB =Yo e,

OIae,v=idlaB,,r

Consequently we have X* e c9(F, L), whence

d(T, L) < D(X*, B). Thus we obtain

d-(r, L) < d(r, L) < D(X*, B) < D(X, B) < d-(r, L), and therefore

d(1', L) = d-(1', L) = D(X*, B) = D(X, h)

which implies that X = X* holds, and that X is a solution of P(f, L). ' cf. for instance Natanson [1], p. 250.

10.2 Existence of Solutions to the Thread Problem

269

Theorem 3. Suppose that L < l(1). If X e 16(r, L) satisfies D(X, B) = d(I, L), then X is a minimal surface, that is, the equations

AX = 0 Ixu12 = Ix,12,

<x., x"> = 0

are satisfied in b, and it follows that

l(X opB)=L. (That is, for any solution of 9(F, L), the movable part of the boundary is taut.)

Proof. The minimal-surface property of X can be derived as in Chapter 4, since each of the mappings X IB. solves a Plateau problem with respect to the boundary

curve r,:= X(3BJ. (Here B. denotes the disk-components of the parameter domain B of X.) Thus we only have to prove

l(Xopa)=L. Suppose that this inequality were not true. Then, because of X e 1(I, L), we would have

l(Xope)
l(X o pB) + l(X, B, n 8B(wo, ro)) < L.

Let z be a topological mapping of B, - B(wo, ro) onto B, with z(a,) = a, and z(b,) = b, that maps the interior of B, onto B, - B(wo, ro). We use z to define the comparison map X* e W(T, L) by defining

{X((w))

X*(w) := X(w)

for

weB

WEB-B1.

Then it follows that d(1', L) < D(X*, B), and because of

D(X*, B) = D(X, B) - D(X, B, n B(wo, ro))

= d(T, L) - D(X, B, n B(wo, ro))

we infer that X const, whence X 1B, = coast, as XIBl is harmonic and therefore real analytic. The relation X IB, = const is a contradiction to

X(at)0X(b,). Proposition 3. If I P, - P21 < L then it follows that

d-(r, L) = d+(T, L).

270

10. The Thread Problem

Proof. Case (i). Suppose that L >- 1(T). Then we define the surface Z: Q R' on Q = {u + iv : 1 ul < 1, vl < 8} by setting Z(u + iv) := y(u). It follows that

D(Z, Q) = 6 f

I Y(u)12 du.

Consider a homeomorphism of B(0, 1) onto Q which maps B(0, 1) conformally onto Q. Then X := Z o r is of class '(F, L), and we have d+(F, L) < D(X, B(0, 1)) = D(Z, Q) = 6 J

1

ly(u)l2 du.

As we can make b > 0 arbitrarily small, it follows that d+(r, L) = 0 whence

d(F, L) = d-(F, L) = d+(F, L) = 0. Case (ii). Assume now that 1(F) > L. By Theorem 2, there is some X e le(f, L) such that

d-(F, L)=d(F,L)=D(X,B) where B is the parameter domain of X. For given e > 0 there exists some surface

X,ec'(T, L+E2)nH'(B(0, 1), ER3) with

D(X, B(0, 1)) < D(X, B) + e = d-(F, L) + E, if we take Lemma 1 into account. Consider now the surface X * e'(F, I Pl - P21) n H' (B(0, 1), P3) which was constructed in the proof of Proposition 1. We define the 1-parameter family of surfaces

0<e
X*:=eX*+(1 -E)X,,

Then we infer X* e '(F, LE) where L. is estimated by

Lr<eIP,-P21+(1-&)(L+E2) =SIP1-P21 +L+E2-EL-E3
d+(F,L)
D(X*) = e2D(X*) + e(1 - e)

f dudv + (1 - E)2D(XL) a


10.3 Analyticity of the Movable Boundary

271

for some number K > 0 which does not depend on e with

0<E +0, we arrive at the inequality

d+(I', L) < d-(F, L). On the other hand, we have d-(T, L):5: d(F, L) < d+(T, L) whence

d-(T, L) = d(F, L) = d+(F, L).

11

Theorem 4. If X minimizes the Dirichlet integral D(X, B) in the class W(T, L), then X also furnishes the minimum of the area functional A(X, B) with '(I', L).

Proof. This result can be derived from Morrey's lemma on a-conformal mappings that we have described in Section 4.5. One can proceed in the same way as in the proof of Theorem 4 in Section 4.5.

10.3 Analyticity of the Movable Boundary In this section we want to investigate the regularity of the movable part E of a solution X of the thread problem. Let us begin by considering a special case. We assume that T is a planar curve. By a projection argument it can easily be seen that X has to be contained in the plane E determined by T. In fact, if we assume without loss of generality that E is the plane {z = 0}, and that X(w) = (x(w), y(w), z(w)) is a solution of 9(F, L), then also X*(w) := (x(w), y(w), 0) is a surface of class W(T, L), and we have

D(X*, h)< D(X, B). The equality sign holds if and only if D(z, A) = 0, and D(z, h) vanishes if and only

if z(w) = 0 holds for all w e U B. As X is an absolute minimizer for the thread v=1

problem, there cannot be any surface in c9(F, L) with a Dirichlet integral smaller than D(X, B). Thus we infer that z(w) = 0 on B. Since z u H,',(A), we also have Vp

z(w) = 0 a.e. on B - I. Finally, on B - U By the function z(w) coincides with v=1

the z-component of r so that z(w) vanishes identically on I and therefore on all of B.

Thus, X is in fact a planar surface, and by a classical result of analysis, every

part of the movable curve E not attached to F must be a circular arc, that is, a regular real analytic curve of constant curvature.

272

10. The Thread Problem

It is the aim of this section to show that the same result holds true for any

solution X of 1(T, L) even if r is not a planar curve. As by-product of our investigation we shall also obtain that all free (i.e. nonattached) parts of.E are asymptotic curves of constant geodesic curvature on X, and it can be proved that the curvature is the same for all free parts of E. Clearly we can restrict our discussion of X to any part X IB where B, is an arbitrary disk-component of the parameter domain B of X. Thus we shall assume that X is a solution of a thread problem which is parametrized on a disk, say, the unit disk. For this reason we shall from now on abolish the notation of Sections 10.1 and 10.2 and, instead, return to another notation similar to that used in Chapters 4-9. To be precise, we now denote by B the open disk

B={w=u+iv:Iwj
C+={w=u+iv: Iwl=l,v0}, C-={w=u+ iv: IwI=1,v<0}.

Fig.1

The set '(T, L) of comparison functions X(w), w e B, now consists of all surfaces of class C°(B, 1183) n HZ (B, 683) which map C- in a weakly monotonic way onto

a given rectifiable Jordan arc f, and whose total variation on C+ is equal to a fixed number L, (1)

1(E) := fc. I d X l = L.

Here E denotes the "movable part" X: C+ -+ 683 of the boundary of any X e '(T, L). We assume that (2)

IP1 - P2I < L < 1(T)

where Pl and P2 denote the endpoints of T, and 1(T) stands for the length of the fixed arc T.

10.3 Analyticity of the Movable Boundary

273

Let X e `'(I', L) be a minimizer of thDirichlet integral DB(X) = 2 lB

among all surfaces in c'(r, L). Such a minimizer will now be called a solution of the thread problem Y(F, L). We already know that any such solution has to be a minimal surface. That is, the equations AX = 0 IXu12 = IX,I2

<X.,

,

0

hold true in B. Now we state the main result of this section.

Fig. 2

Theorem 1. Let X e W(I', L) be a minimal surface, that is, X satisfies (3)

AX=0 in B

0 around the as well as the conformality relations. Introducing polar coordinates r, be written as origin by w = reie, these relations can (4)

r2IX,I2 = Xe12,

<Xr, Xe> = 0.

class '(f, L).

Moreover, suppose that X minimizes the Dirichlet integral within the minimal surface across the arc C+, and Then X (w) can be continued analytically as a of even order. it has on C+ no branch points of odd order nor any true branch points R3 is assumed to be an embedding, If, moreover, the boundary mapping X : 8B -> Correspondingly, then X(w) has no false branch points of even order on C+ either. real analytic curve in this case, the free trace X defined by X : C+ -+ R3 is a regular, of constant curvature K 0 0.

behaviour of For the following we recall some results on the boundary values of integral and with boundary minimal surfaces with a finite Dirichlet X(r, 0) = X(ret) DB(X) < oo implies that bounded variation. The assumption

10. The Thread Problem

274

possesses L2-boundary values X(1, 0) on O B which are assumed in the L--sense

as r-+ 1 -0. From fo"dX(1, 0)j < cc we conclude that X(1, 0) depends continuously on 0 (cf. Lemma 3 of Section 10.2). More subtle results have been derived in Section 4.7. For the convenience of the reader, we collect the pertinent statements in the following lemma. Lemma 1. Let X : B --> 683 be a disk-type minimal surfiue, i.e., let (3) and (4) he satisfied, and denote by X*: B -- 683 the adjoint minimal surface to X which, up to an additive constant, is uniquely determined by the equations

Xr=1XB,

(5)

r

1X9=-X*. r

Assume that DB(X) < ce and f,,B IdX I < co. Then we hare: (i) X and X* are of class C°(B, 683) and

jdXj =

DB(X) = DB(X*),

(6)

f'B

IdX*I. f1l

(ii) The boundary values X(1, 0) and X*(1, 0) are absolutely continuous functions of 0, and X9(r, 0), XB (r, 0) tend in the L2-sense to the derivatives X9(1, 0), Xe (l, 0) of the boundary values X(1, 0) and X*(1, 0) respectively as r I - 0. Then, on account of (5), we deduce that also X,(r, 0) and X*(r, 0) converge in L` to boundary values as r -> 1 - 0, and we set X,(1, 0) = lim Xr(r, 0),

r-1-°

X*(1, 0) = lim X*(r, 0).

It follows that a.e. (7)

Xr(1, 0) = XB (1, 0),

(8)

Xr(1, 0)I = X9(1, 0)1,

(iii) If C is an open subarc of 8B, and

X9(1, r)

X*(l, 0),

<Xr(1, 0), X6(l, 0)) = 0. is a test junction of class N2(B, R3) n

L°° (C, 683) with l; = 0 on 8B - C, then ('

(9)

JB rdrd0 =

Jc

<Xr, 5) d0.

(iv) If X # const on B, then X9(1, 0) and X, *(I, 0) vanish at most on a subset of [0, 27r] of one-dimensional measure zero.

Now we turn to the proof of Theorem I which we want to break up into

three parts. In the first one we consider a stationary version of the thread problem; here the existence of a Lagrange multiplier is supposed. Thereafter we prove that

every minimizer in '(I', L) is in fact a solution of the stationary problem by establishing the existence of a Lagrange multiplier, and in the third part we sketch how branch points can be excluded by using the minimum property.

10.3 Analyticity of the Movable Boundary

275

Definition. A minimal surface X : B -). R3 is said to be a stationary solution of the thread problem with respect to some open subarc C of 8B if the following holds: (i) DB(X) < co, f 8B 1 dX I < oo, X (w) * const; (ii) there is a real number 2 :0 such that

(10)

J
dB = 0

f

e

cc

R3)with =0on8B-C.

holds for

Taking the identity (9) into account we arrive at

Ic (<Xr, > + 2IXXI-1 <Xe, e))dO = 0, and (8) yields

IJc <X., >d6 =

Jc

<Xe*, >dO =

-fc <X*, e>dO.

Thus (10) is equivalent to

f, <X* - 2IXBI"'Xe, 9>d6 = 0 for all 1; e C1(B, R3) with

= 0 on BB - C.

DuBois-Reymond's lemma now implies that (11) - and therefore also (10) - is equivalent to the following property of X: There exists a constant vector P E R3 such that (12)

X*=1IXOI-'Xe+P a.e.onC

holds.

We now prove

Theorem 2. Let X : B -+ l3 be a minimal surface which is a stationary solution of the thread problem with respect to the open arc C c 8B. Then, for some P E i83 and some 2 E III, 2 0, equation (12) is satisfied. Moreover, X and its adjoint X* are real-analytic on B u C, and X* intersects the sphere

S={ZEIR3:IZ-PI2=12} orthogonally along its free trace E* defined by X*: C -* IJV. Both X and X* have no boundary branch points of odd order on C. Finally, E = X Ic has a representation i(s), 0 < s < 1, by its arc length s as parameter which is of class CZ and satisfies

I(s)I - 1, Ii(s)I - ICI. x

1

I1I

Thus E represents a regular curve of constant curvature

276

10. The Thread Problem

Proof. As we have noticed, the assumption on X implies that (12) holds for some P E ll and some A E 6&, A 0. Taking the continuity of X*(1, 0) into account, we infer that IX*-P12=A2

(13)

on C.

In other words, the trace E* lies on S. Moreover, equations (12) and (7) yield (14)

X* - P = -21X*1-1-X* a.e. on C.

Therefore the vector X* is normal to S a.e. on C. Thus for almost all w E C the surface X* has a tangent plane which meets S at a right angle. By the reasoning of Section 5.4 (cf. Theorem 1) we conclude that the adjoint surface X* is a critical point of Dirichlet's integral within the boundary configuration
of the arc f* = {X*(w): w e 8B - C} and of the surface S. We can therefore apply Theorem 2' of Section 7.8 to X* and obtain that X* can be continued analytically across C as a minimal surface. (Note that for this regularity theorem

it is not necessary to assume that F* be a Jordan arc which does not meet S except in his two endpoints.) By virtue of (5) we infer that both X and X* are real analytic in B u C, as we have claimed. We furthermore note that, because of (5), X and X* have the same boundary branch points wo e C. Since X + iX* is a nonconstant holomorphic mapping U -+ C3 of some full neighbourhood U of each branch point wo e C, we have the asymptotic formula

asw->wo,

(15)

for some integer v >t 1 and some vector A # 0. Since X* (15')

iXx, we also have

-iA(w-wo)"+O(1w-wo1'+i) asw --+wo.

That is, the order of wo as branch point of X equals its order as branch point of X*. We moreover infer from (15) and (15') that the boundary branch points of X and X* are isolated. In addition, the conformality relations (5) imply
0, = 0.

i point of X on C, we can define the unit tangent If w = ei9 is not a branch vectors T(0) = X9(1, 0) X9(1, d)I '

T*(B)

XB

(1, 0)

I Xe (1, 0)1

of the curves £ and E* at X(w) and X *(w), respectively. Let wo = e`° e C be a branch point of X (and of X*). Then we infer from (15) and (15') that the one-sided limits TT(00) = lim

T(0),

e-ea±O

Tf(00) = lim T*(0) 0-00±0

exist. Moreover, we have (16)

T+(00) = T_(00),

T+(00) = T*(00)

10.3 Analyticity of the Movable Boundary

277

if the order v of the boundary branch point w0 is even, whereas (16')

T(00) =

T-(00)1

T+(00) _

T!(00)

if v is odd.

We note that the limits T+(60), T*(do) are unit vectors. Equation (12), on the other hand, yields that

{X*(1, 0) - P}

T(0)

(17)

holds for all 0 satisfying 0 < 10 - 00 1 < e where s is a sufficiently small number and, moreover, the right-hand side depends continuously on 0 E (00 - E, 00 + E). Therefore, T+(00) = T_(00), and v must be of even order. Hence X and also X* can only have even order branch points on C, as we have claimed. If we define T(00) by T+(00) at a branch point w0 = ei ,, e C of even order, we infer from (16) that T(0) is a continuous function on C with I T(0)l - 1, and (17) holds everywhere on C. Suppose now that C = {e`° : 0, < 0 < 02} and set e2

1=

e2

I Xe*(1, 0)I dd-

I X0(1, 0)I dd = e,

e,

We furthermore introduce o

s = s(0) = J

IXe(1, 0)I dd = J

e

IXe (1, 0)I dd,

0t < 0 < 02 , which is the arc length parameter of E as well as of E*. Since s'(0) = 1Xo(1, 0)1 > 0 has only isolated zeros, the function s(0) can be inverted. Let 0(s), 0:!5; s < 1, be its (continuous) inverse. For 0 < s < 1 we introduce t(s) = T(0(s)),

t*(s) = T*(0(s)),

'(s) = X(1, 0(s)),

'*(s) = X*(1, 0(s)).

So far, we only know that 0(s) is continuously differentiable in s-intervals corresponding to 0-intervals free of branch points. We already know that t(s) and t*(s)

are continuous for 0 < s < 1, and that .' "(s) = t(s), _'*(s) = t*(s) at values of s

which do not correspond to branch points on C. Then a simple argument employing the mean value theorem yields that i(s) and ff*(s) are of class C' for 0 < s < 1, and that (18)

''(s) = t(s),

'*(s) = t*(s)

for 0 < s < 1.

(In these formulas as well as in the following ones, the dot denotes differentiation

with respect to the arc length: regular curves of class C'.

ds .) Thus E and I* are representations of

278

10. The Thread Problem

From (17) and (18) we derive the equation (19)

t(s) = 1 {X*(s) - P}

and (20)

I(s) _

t* (S)

for 0 < s < 1. Thus X(s) is actually of class C2 on (0, 1), II(s)I = IX(s)I = 1/IAI This means, £ represents a regular C2-curve of constant curvature 1/I4 This concludes the proof of Theorem 2. The first of Frenet's equations yields (21)

L(s) = Kn(s),

K= ICI

where n(s) is the principal normal of the curve X(s). On the other hand, differentiating (17) with respect to 0 and employing (7) and (8), we arrive at (22)

I(s) =

.IX,I (1, 0(s)).

Hence n = ± I X, I -' X and thus the normal curvature of I vanishes. Thus as a by-product of our discussion we obtain the following

Corollary 1. Under the assumptions of Theorem I the free trace £ of X is an asymptotic line of the surface X of constant geodesic curvature ±K.

Remark 1. In general, stationary solutions of the thread problem will have boundary branch points of even order. In fact, one can easily construct examples of planar minimal surfaces X*: B -+ R' that satisfy (14) for some nonempty open subarc C of aB and have a branch point wo of second order on C. The adjoint

surface X of -X* will then satisfy (12) or, equivalently, (10). Hence X is a stationary solution of a thread problem with respect to C that has a branch point of second order on C. Next we come to the second part of the proof of Theorem 1. We shall prove

that, for each solution of the real thread problem, there exists a Lagrange multiplier. This is not totally trivial since the applicability of the standard Lagrange multiplier theorem (which requires continuous differentiability of the involved functions) is not clear. The following result provides an appropriate substitute. Lemma 2. Let rp(E, t) and Ii(e, t) be real-valued functions of (E, t) e C-80, 60] x E -to, to],

Eo > 0,

to > 0,

10.3 Analyticity of the Movable Boundary

279

which split in the form (p (c, t) = APO + 4)1(c) + (p2(t),

'(E, t) = 00 + W1(e) + 02(t) Here it is assumed that cpo and 0o are constant, and that WI(0) = 4)2(0) = 01(0) = 02(0) = 0.

We also suppose that 02 is continuous on [-to, to], that the derivatives cpi(0), cp, (0), i (0), 0' (0) exist, and that > 12'(O) = 1. Finally, let the inequality cp(e, t)

cp(O, 0) hold for all (e, t) in [-eo, eo] x Then the relation

to, to] with Vl(e, t) = /o.

(pi (0) + 41(0) = 0

(23)

is satisfied for A = - (p'(0). Proof. The assumptions imply that there is a function q(t), - to < t < to, which satisfies

limrl(t)=rl(0)=0 t-'o

and 1 2(t) = t{1 + ?1(01-

We then choose a number So with 0 < So <

such that In(2t)I <

i

for Iti < So,

2

and infer that

02(- 2t) < -t < t < 02(2t)

for t e (0, S0).

The continuity of 02 now implies the relation

[-8, S] c >/i2([-2S, 28]) for all 5 e (0, So). We also note that lim >[i1(e) = 0 holds. Therefore we can find a number e1 with e

0

0 < e1 < eo such that I >/i1(e) I < bo is satisfied for each e e [-e, e1 ]. Consequently

there exists a real-valued function i(e), -e1 < e < 81, with the properties

2(i(e)) + ¢1(e) = 0,

T(0) = lim i(e) = 0, E-0

2I'1(e)I < t0,

whence also aji(e, T(e)) _ 0o for -e1 < e < e1. From the identities r(e) e

- r(e) - r(0)

i1(e) - l1(0)

1

e

1 + n(T(e))

e

for 0 < I&I < E1 we infer that the function i(e) is differentiable at e = 0, and that (24)

T'(0) = 1im E-0

z(e) 8

_ - i (0)

280

10. The Thread Problem

Moreover, the minimum property for 0 < E < E1

(p(e, r(e)) >_ (p(0, 0)

implies the inequality 0<

(25)

P1(E)

+

(p2(r(E)) E

E

Suppose now that we would have r(E) __ 0 on some interval (0, g'], where 0 < E' < E1. Then we obtain

0<


for 0 < E < E'

on account of (25), and therefore (p' (O) >_ 0. By virtue of (24) we furthermore have

r'(0) = 0 and 0;(0) = 0, whence (26)

0< cpi(0) - 01(0)rP2,(0)

If, on the other hand, there is no s' > 0 such that r(E) 0 on (0, s'], then there exists a sequence of numbers E2, E3, E4, ... tending to zero, with 0 < Ei :!s; s' for i >- 2 and r(e1) 0 0. Set ri = r(e,). We then infer from (25) that

ri E;

Ti

i= 2, 34,

Ei

holds. For i --> oo we once again arrive at the inequality (26) which thus is established. Similarly we can verify the opposite inequality 0 > cpi(0) - 1(0)rp2'(0),

and the Lemma is proved.

In order to apply the previous lemma, we will introduce the class .F(C+) of test functions defined in the following way: _ A function l; is said to be of class .F(C+) if it lies in C1(B, p3), and if there are a point wo e C+ and a number r e (0, 1) such that 5B n Br(wo) is contained in the open arc C+ and that C(w) = 0 for all w e B - Br12(wo). Lemma 3. Suppose that (2) holds and that X is a mapping of class ''(I', L) which satisfies the assumptions of Theorem 1. Then there exists some t' e .F(C+) such that (27)

IX8I_1<X0,

J

CB>d6 = 1.

c+

Proof. It clearly suffices to establish the existence of some t' e .F(C+) for which the integral in (27) is nonzero. To this end, let us suppose that the integral vanishes for all C e .F(C+). Then, by DuBois-Reymond's lemma, there would exist a unit vector e e R3 such that

10.3 Analyticity of the Movable Boundary

281

IX9(l, O)I-'X8(l, 0) = e

for almost all 0 e (0, it). Hence X(C+) would be contained in some straight line

2, and since X : OB --> l3 is a continuous mapping, 2 would have to be the straight line connecting the two points Pt and P2. Applying the reflection principle we could extend X analytically and as a minimal surface across C+. Hence X is real analytic on B u C+ and possesses at most denumerably many isolated branch points on C. Then we infer from the equation X9(1, 0) = I X9(1, 0)Ie

for all 0 E (0, n)

that X(1, 0) yields a strictly monotonic mapping of [0, it] onto the straight segment on 2 with the endpoints P1 and P2, whence we would get

L=J IdX1=IPt-P2I. c

But this contradicts Assumption (2).

Lemma 4. Suppose that (2) holds and that X e ''(F, L) satisfies the assumptions of Theorem 1. Then X is a stationary solution of the thread problem with respect

to the arc C+ ={e`e:0<0<7L}. Proof. By Lemma 3 there is a test function e 3(C+) such that (27) holds. By definition of.F(C+), there exist wo E C+ and r e (0, 1) such that i;(w) vanishes for all w e B - B,12(wo) and that the closed arc y := aB n B,(wo) is contained in C+.

Then C+ - y consists of two non-empty open arcs Ct and C2. We first want to show that X is a stationary solution of the thread problem with respect to C, as well as to C2. Since the reasoning will be the same for both arcs, it suffices to verify the assertion for, say, C,. Firstly, the assumptions of Theorem I imply that IdXI

DB(X) < co,

< oo, and X(w) # const.

faB

Secondly we have to prove that


(28) SB

JCl

holds for some real number .1, 96 0 and for all % E C' (B, O ) that vanish on 3B - C1. Clearly, it suffices to verify (28) for all e C' (B u C lv). We shall, in fact, see that (28) only has to be established for an even smaller class of test functions. For this purpose, we choose some open disk B' with the property that aB n B' = C1, and that 0 := B n B' does not meet the disk B,12(wo). By virtue of some appropriate partition of unity, each element E CC (B u C,, R3) can be written

as the sum = g, + 2 of a function 1 E C. (Q u C1, IR3) and of another function

10. The Thread Problem

282

2 E C(B, 083). We now note that both integrals appearing in (28) vanish separately if is of class C'(B, 083). Thus it remains to prove the following: 0 such that (28) holds for all e C'(Q U C1, p3), There is some number A, This will be achieved by employing Lemma 2. To this end we choose some arbitrary c E C,(Q u C,, R3) which in the sequel is thought to be fixed, and set

XE,,=X+El;+tc,

Itl
IEI <Eo,

for some number go > 0, to > 0. (At present, the subscripts E and t indicate the dependence of the 2-parameter family X,,, on the parameter E and t and do, deviating from the previous way of notation, not stand for partial derivatives.) Let us introduce the functions (P(e, t):= DB(XE,,),

y'(E, t)' L I dX.,,I = fc+

BXE,101 0) d6

go, Eo] x [ - to, to]. Then we have the representations

of (E, t) e

Q(E, t) = (PO + (PI(E) + p02(t),

+'Y'2(t),

'N (E, t) = Wo + 01(c)

where we have set

rPo := DB(X), o

I X8(l, 0) 1 dd, Jc +

zz

rP1(E) := Da(X + ES) - D0(X),

QP2(t) := Dno(X + tC) - D,no(X),

Q0:=BnBr/2(wo), ('

4'1(8) :=

fc

IXe +

dO - J c,

I XBI d0,

q2(t)

(' J 1 Xe +

('

dd - J

Y

1 Xe1 de. Y

We now have once again used: Xe = 6X, etc.) The functions (q1 and 92 are quadratic polynomials, and clearly

0=Q1(0)=rP2(0)=01(0)=02(0) Moreover, the function '2(0's continuous on [-to, to]. We also claim that the derivatives iii (0) and 14 (0) exist. In fact, the formula a2 - b2 = (a + b) (a - b) yields

{IXe+E el -1X91} =fls)+9(E), where 2<Xe, e> 1(E)= IXe+Eel +IXB1'

9(E)=

IXe+Eel + IXBI

Hence we infer that If(E)I s

1g(E)I <

a.e. on C+

10.3 Analyticity of the Movable Boundary

283

and for Isl > 0. By Lebesgue's theorem on dominated convergence the derivatives 0;(0) and /4(0) exist, and (29)

i(0)=

IX0I_1<Xe

Jc ,

e)d0,

(

'(0)=J c

y

IXel-1<X0,S9>dO=1.

Thus the assumptions of Lemma 2 are satisfied, and we obtain (0) = 0,

(Pi(0) + )L1

where , = -;P'(0).

On the other hand, we infer from (29) and from

(P,(e)=eJnrdrd0+ that (28) is true for an arbitrarily chosen e C' (Q u C, , R'), and hence (28) holds for all e C1(B, El) that vanish on aB - C,. Because of the equivalence of relations (10) and (12) we conclude that X* = A1IX01-1X0 + P,

(30)

holds a.e. on C, for some constant vector P, e [l8'. If A, = 0, we would get X* = P,; i.e. X, (1, 0) = 0 a.e. on C,, and this contradicts Lemma 1, (iv). Hence we have indeed A, 0 0, and it is proved that X is a stationary solution of the thread problem with respect to C, (and to C2). By Theorem 2, the mappings X

and X* are real-analytic on B u C, U C2 and have at most isolated branch points.

In order to complete the proof of Lemma 4 we now assume w.l.o.g. that C, = {ei0: 0 < 0 < 0, } for some 0, a (0, tt). Then we introduce the two arcs

Y, ={eie:0<0<20,},

Y2= {ei8:20, <0<m}.

Let us choose two disks B, and B2 with centers outside of B such that aB n B,, Y2 = aB n B2, and that the open sets 0, = B n B, and (12 = B n B. are disjoint. We claim that there is a function , e C, (0, u y,, V8') such that

f

IX01-1<X0, 81>0

= 1.

v,

Otherwise we would have I X01-1X0 = const on y,,

whence by (30) X*(1, 0) = const for 0 < 0 < 201, i.e. X, = XB = 0 on y,. This would be impossible since the branch points of X* on y, are isolated. In addition, we choose an arbitrary function e Q1 (Q2 u Y2, R'). Then we apply the previous reasoning to the 2-parameter family

lel-<eo,

Itl<_t0.

By the same reasoning as before we can establish the existence of a consta vector P E D8' and of a number A e IR, 2 0 0, such that

284

10. The Thread Problem

X* = A1Xel-1X9 + P

(31)

holds on y2, and we also know that X and X* are real-analytic on B U y2. On the other hand, equation (30) is satisfied on C1. Since C1 r-) Y2 = {e`°: ?B1 < 9 < 91

we may infer that A _ Al and P = P1. Thus we have proved that X and X* are real analytic on B u C+, and that (31) is satisfied on all of C+. This in turn yields

rdrd9 + A f,s

for all

< I Xol -1X9, 9> dO = 0 fc

a C1(B, 683) with

= 0 on 8B - C+, and Lemma 4 is proved.

Resuming the results of Theorem 2 and of the Lemmata 2-4, we see that all assertions of Theorem 1 are proved, except for the claim that E is a regular curve. This will be shown in the third and last part of our discussion. We shall proceed

by proving that no minimizer X can have branch points of even order on C. Recall that branch points of odd order were already excluded in Theorem 2; they cannot even occur for stationary solutions of the thread problem. On the other hand, stationary solutions may very well possess branch points of even order, as we have noted in Remark 1. Thus we now really have to employ the minimizing property of X if we wish to exclude branch points of even order. For

this we use arguments that are essentially due to Osserman [12]; important refinements were made by Gulliver [2], Alt [1, 2], Gulliver-Lesley [1], and Gulliver-Osserman-Royden [1]. In what follows we shall sketch the main ideas that lead to the exclusion of true branch points of even order for minimizers X. We shall not discuss the impossibility of false branch points of even order since we have already described the pertinent ideas in Section 5.9. For further information and for filling in all details we refer the reader to the papers quoted before. It will be convenient to choose the parameter domain of any minimizer X as the semi-disk

B = {w = u + iv: lwl < 1,v>0}, and C+, C- will be replaced by

C={w=u+ iv: lwl=1,v - 0} and

I = {u a 68: lul < 1}. We now assume that X : C - P3 yields a monotonic parametrization of r, and X : I -+ P3 describes the free trace of X, i.e., its movable part E of the boundary. It follows from the previous discussion that X can be continued analytically as a minimal surface across I. Let uo be an arbitrary branch point of even order for

10.3 Analyticity of the Movable Boundary

285

X with uo e I. We want to show that the existence of such a branch point contradicts the minimizing property of X.

Without loss of generality we can assume that uo = 0 and that X(0) = 0 because we can always transform u = uo into u = 0 by a conformal self-mapping

of B that keeps the points u = + 1 fixed, and X (O) = 0 can be achieved by a suitable translation of R'. Performing an appropriate rotation of W, we can also accomplish the asymptotic representation x(w) + iy(w) = awm+1 + O(I lWlm+2) Z(w) = 0(I W Im+2)

for the cartesian coordinates x(w), y(w), z(w) of X(w) in the neighbourhood of w = 0, where a denotes some positive constant and m = 2v, v >_ 1, is the order of the branch point w = 0; cf. Section 3.2, formula (6). By a suitable scaling it can also be arranged that x(w) + iy(w) = W,+1 + O(IWlm+2) Z(W) = O(I WIm+2)

holds true for w -+ 0. Because of the power-series expansion of X(w) at w = 0 we may write X(W) + iy(w) = wm+1 + Q(W) (32)

z(w) = 4(w) vka(W) vkO(W) =

O(IWIm+2-k)

for 0 < k < 2

with m = 2v > 0. We will now show that this representation can be simplified even further. Lemma 5. Let X : BR(0) --> l3 be a minimal surface with the representation (32) at

w = 0. Then there exist two neighbourhoods all, YI of 0 in BR(0), a function cp e C2(11) with Vkcp(w)=O(Iwlm+2-k)

for0
and a Ct-d feomorphism F: 0& -+'V of 61l onto P such that the formulas x(w) + iy(w) = Fm+1(w)

(33)

z(w) = cp(F(w))

hold true for w E V.

(Note that we use the complex notation co = F(w) e C; thus wm+1 is the (m + 1)-th power of (o.)

Proof. Define F(w) := w{1 +

w-m-1

(T(w)) 1/(m+1)

286

10. The Thread Problem

on a sufficiently small neighbourhood of w = 0. Because of a(w) = O(Jwj'"+2) this definition is meaningful if we choose the (rn + 1)-th root to be one at w = 0. Moreover, we have lim F(w) = 1. W

Hence 17F(0) exists, and FF(0) = id. Moreover, we have

(D +

I + o(1) as w -* 0,

whence

VF(w) - VF(0)

as w --+ 0,

and this implies F e C'. By the inverse function theorem, there exists a C'-inverse f of F on a neighbourhood ' of the origin; set Q/ := f(f). Since F e C'(07-1 - 101), we see that f e C2(V - {0} ), and it is not difficult to prove that V2F(w) = o(IwL_1)

In order to be able to use the summation convention, we write w = u + iv = u1 + iu2, u' = u, u2 = v. Then the identity fa(F(w)) = ua,

a = 1, 2,

implies

f a(F(w))F1(w) =

in Qt , SY

or

f'(w)F',(f(i))=by in 'V. Moreover, we obtain

feoF.a(f) + fIF,vt(f)fv = 0

in 1 - {0}.

Multiplying this identity by fly we infer

f p, =

-F1"fTaf ff

whence we derive that V2f(w) = o(I01-1)

p

as w-*0, w=F(w).

Now we define cp: 'Y,' -* R by cp(w) = /'(f(')). Then 9 is a well defined function of class C1('Y,) n C2(y/, - {0}) which satisfies (34)

,a = ',Yfa in'V

and (35)

rP.a.6 =V1,YPfafa+0,Yf1arp

in'V-{0}.

10.3 Analyticity of the Movable Boundary

287

The assumptions of the lemma in conjunction with (34) imply that Vq = O(Iwlm+') Thus 02(p(0) exists and is equal to zero. On the other hand, we infer from (35) that O2cp(w) = 0(1wpm) holds. Altogether we arrive at 4) E C2(''), and the lemma is proved. E

Lemma 5 permits the introduction of a new independent variable w = F(w) e Y,- such that X = (x, y, z) can be written as x(w) + iy(w) = w'"+'

(36)

z(')=070

fort'eY

,

where co E C2(7') and V"cp(iv) = O(1 wlm+2-') for 0 < k < 2. (The reader will excuse the sloppy notation X(w) for the transformed surface; actually we should write X(F-'(w)).)

Now we want to describe some local properties of the function q which appears in the representation formula (36). Lemma 6. Let cp be the function that appears in (36), and let w = u' + iu2. Then we obtain (37)

c

2loipl2

on Y'

0

w = u' + iu2.

where c(w) := (m +

(Here, we were even more careless and renamed w as w. Thus the reader should

bear in mind that X (w) actually means the transformed surface X (F-' (iv)). The advantage of our sloppiness is that the following formulas become less cumbersome to read.)

Proof. From (361) we see that every point p c- f- - {0} has a neighbourhood Yl",(p) which is mapped in a regular way onto a neighbourhood 'V2 in the x, y-plane. We write x' = x, x2 = y. On PI-2 the function ip(u', u2) obtains a new representation ii(x', x2), i.e., cp(u', u2) = ifi(xt, x2).

As X is a minimal surface, we infer that

z = /(x', x2) provides a nonparametric representation of this minimal surface. Therefore tli(x', x2) must satisfy the minimal surface equation Da {

01

-- } = 0

l 1 + 0X + q,. )) From cp(w) = O(Re w"', Im w'"), we conclude by a straightforward computation that (37) holds in 1 and therefore also in -1" - {0}.

288

10. The Thread Problem

Now we claim that ,/1 + c-2I Ocp I2 is of class C1('r'). In fact, let A,,:= c-1 cp,,,. Then we see that ).a(w) = O(I wl), whence we can extend ).a(w) in a continuous way to 'V by setting A.a(0) = 0. It follows that ).a(w)1p(w) = O(IwI2), and therefore V(A Ap)(0) = 0. Finally we derive from aa,p

= 9.ap c-1 - c-29," c.p = 0(1)

that 0(22p) = 0(IwI), whence Aa e C1(V). This concludes the proof of (37).

It follows from the representation (36) that selfintersections of X occur at points which are images of points w E 7r with qp(w) = cp(rl'w) where n denotes some primitive (m + 1)-th root of unity, and j # 0 mod(m + 1). Note that cp*(w) := cp(riw) again satisfies (37). Hence the difference O(w) :_ 9(w) - q *(w) is a solution of a linear elliptic differential equation. To be precise, we have Lemma 7. The difference function 0 satisfies (38)

in 'V

0

where aap is of class C'(7I-) and uniformly elliptic on y'', and aap(0) = 6,,#.

Proof. Set Ta(w, q) := qa/ V (P*)

Ta(w,

-

l + c-2(w)Iql2 with Igl2 = gaga, and observe that

Ta(w, V (P)

Ta(w,

= Jo dt =

tV(p* + (1 - t)Orp)dt

Ta 9p(W, tV p* + (I - t)vcp)dt f O.A. (fo,

T hen one sees that the assertion follows for aap(w) :=

fo, Ta,ep(w, tVcp*(w) + (1 - t)O(p(w))dt.

It will be useful to obtain an asymptotic representation for the difference function 0. This can be achieved by the technique of Hartman and Wintner (cf. Section 8.1), which yields the following alternative: Either O(w) - 0 or there exists some integer n >- 1 and some number a e C, a 0 0, such that (39)

rh,,, -

awn-1

+ p(w)

holds with p(w) = o(Iwln-1) as w -+0. Integrating (39), we arrive at O(w) = Re fn a w" } + r(w), )))

(40)

p(w),

o(w) = o(Iwj")

as w -> 0.

10.3 Analyticity of the Movable Boundary

289

Applying once again the reasoning used in the proof of Lemma 5 we obtain the existence of some diffeomorphism T defined on some open disk BR(0) such that

'(w) = Re T"(w),

(41)

and that T(O) = 0 and T'(0) 0 0 hold. Then we derive from the alternative above the following result: Proposition 1. Let X : B u I -+ l83 denote some solution of the thread problem, and suppose that 0 e I is a branch point of X of order m = 2v. Furthermore, let cp, cp*, 0 and T be the mappings which we have defined before. Then there exists some

neighbourhood *0 of the origin 0 in C . R2 such that the following alternative holds true:

(i) Either X1 a can be reparametrized in such a way that it becomes an immersed surface,

_

(ii) or else, there exist two simple CI-arcs yt, Y2: [0, E] -+o n B with y,(0) _ 0, byj,(0)I = 1, yi(0) 0 y2(0), X(y1(t)) = X(y2(t)) for all t e [0, e] and such that the vectors A A Xv(y2(t)) are linearly independent for and all

Proof. Suppose first that O(w) - 0. Then, as in Lemma 5, we can show that (i) holds with Y,o = all. In fact, the system (36) assigns to each w e 'r or to each w e all a unique point X(w) = (x1(w), x2(w), x3(w)), and the surface X may locally be written as x3 = 1/(x1, x2) with 1'!(x1, x2) = gyp('), W = F(w) and 0 e CI since D(p(W) = 0(IWIm+1).

Now we want to settle the case 45(w) # 0 using the expansion (40). We note that n >_ m + 2 since O(w) = O(Iwlm+2) Since m = 2v > 2, we find that n > 4. Define 'ro := F-1(BR(0)) with a sufficiently small number R > 0, and consider the mapping T o F : - o --+ C which is conformal at the origin. Let iC := T o F(w),

and denote by j, 1 < j < 2n, the 2n rays in the c-plane which emanate from = 0 and are defined by Re t;" = 0. The rays 9 correspond to 2n curves yj in 'Y/o via the mapping T o F. Moreover, since n > 4, at least one of the curves y; meets the positive real axis at an angle which is between 0 and it/3. We can assume that y1 (t) is such an are, and we can also assume that t is the parameter of arc length along y1. Then we have 0 = I(F o y1(t)) = cp(F o y1(t)) - cp(ryF o y1(t)) F-1

o (nF o y1(t)), we arrive at X o y1(t) = X o y2(t). Moreover, because of conformality y2(t) hits the positive real axis under an

Setting y2(t) :=

angle which is strictly between 7c/3 and 7c/3 +

m+1

< n. For sufficiently small

E > 0, the mappings y1 and Y2 will map [0, e] into 'Yro r).9. Since co describes the difference of two branches of X and because of (41), it immediately follows that the two surface normals along y1 and Y2 respectively are linearly independent.

290

10. The Thread Problem

Let us now recall the definition of true and false branch points given in Section 5.9.

Definition. The branch point w = 0 of the minimal surface X (w) is called a false branch point if case (i) holds true; otherwise w = 0 is called a true branch point.

Concerning true branch points, we shall prove: Proposition 2. If X : B - X83 is a solution of the thread problem, then there are no true branch points on the interval I = {u E R : J u I < 1 }, which is mapped by X onto the movable boundary I.

Proof. We first recall that X not only minimizes Dirichlet's integral within 1(I, L) but also the area functional AB(X) =

IX,, X A Xvl dude;

f,B

cf. Theorem 4 of Section 10.2.

We may again assume that the true branch point w c- I under consideration is the point w = 0. Since we are now in case (ii) of the alternative in Proposition 1, we can use this fact to construct a comparison surface X* the area of which is equal to the area of X whereas the normals of X* are not continuous. This would yield a contradiction since X* is a minimizer, and therefore X* is of class

C'. _ Choose a neighbourhood )V of 0 in C such that yY in B is diffeomorphic to B. Suppose that the curves y, and y2 first leave *'at yl (26) and y,(26) transversally

to &#' Moreover, let h: 'IV n B -. B be some C'-diffeom_orphism under consideration which, in addition, maps 3'Y' in B onto 8B - 1 and ' in I onto 1. Furthermore we may assume that

hoy1(t)= t for 0 < t < 26, where

0 (a)

hoy2(t)= -t

,

E C denotes some number with I l; = 2.

0

0

(b)

(c)

Fig. 3a-c

10.4 Scholia

291

It is now possible to construct a mapping G : B -* B with the following properties: (I)

G is continuous and one-to-one on q - 0, 2] ;

(II) (III)

GIaB = idlaB;

For E C with Re C > 0 and 0 < t < 1, the following relations are fulfilled :

limG(l(I ±t)+C) =(1 4 C-0

lim G ( l (l 4 (IV)

± t) -

/

(1 - oz;

G is piecewise CI and extends to a C'-diffeomorphism on each edge of the slit [0, 2]

.

We refrain from constructing G explicitly by formulas; Fig. 3 describes the topological action of G. Now we define a comparison function X*: B --+ O by

X*(w) :_

X(w) (X o h -' o G o h) (w)

forwEB - YV

forwe*'.

It is clear that X* e '(I', L) and that AB(X) = AB(X*). Hence X* minimizes DB(X) within W(f', L) and is therefore of class C'. This leads to a contradiction, since any point wo E *' satisfying h(wo) E 10, 2J possesses some neighbourhood

which is mapped onto a surface with two portions intersecting along X(yl). In view of (ii) the normals of the surface near the line of intersection are no longer continuous. The exclusion of false branch points is more difficult. The pertinent reasoning

is sketched in Section 5.9. A detailed discussion can be found in the paper of Gulliver-Osserman-Royden [1], where the theory of branched immersions has been developed. By these remarks we conclude the proof of Theorem 1.

10.4 Scholia 1. The existence of solutions of the thread problem in its simplest form was first proved by H.W. Alt [3]. Except for minor modifications we have presented Alt's existence proof in Section 10.2. Without any changes the proof can be carried over to 2-dimensional surfaces in R', N >_ 2. A different proof has been given by

292

10. The Thread Problem

K. Ecker [1], using methods of geometric measure theory; it even works for the analogue of the thread problem concerning n-dimensional surfaces in l". In the framework of integral currents, Ecker has proved the existence of a minimizer,

the movable boundary of which has prescribed mass. 2. It seems to have been known for a long time that the unattached part of the movable boundary I consists of space curves of constant curvature; cf. van der Mensbrugghe [1], Otto [1]. A satisfactory proof was given by Nitsche [21] under the assumption that the free part of X is known to be regular and smooth; cf. also Nitsche [28], pp. 435-437 and pp. 706-707. 3. The first results concerning the boundary regularity of solutions for the

thread problem were found by Nitsche [23-25]. He proved that the open components of the non-attached part of the movable boundary have a parametrization of class C2' , for some a e (0, 1). Between branch points (the existence of

which was not excluded by Nitsche) these parametrizations turn out to be of class Cx and, therefore, they represent piecewise piecewise regular C`°-curves. The sharper regularity results, presented in Section 10.3, and their proofs are taken from Dierkes-Hildebrandt-Lewy [1]. We have quite closely followed the presentation given in their paper.

By completely different techniques, K. Ecker [1] has established C-regularity of the free part of the movable boundary E in the context of his integral-current solutions; the analyticity is in this case still an open question. It is not known if the thread can have self-intersections; we are tempted to conjecture that this cannot occur. Alt [3] has also proved that the movable arc E must always lift off F in a tangential way whenever it adheres to Fin a subarc of positive length provided that F is supposed to be smooth.

4. As Alt [3] has pointed out, all pieces of the movable boundary I not attached to F have the same constant curvature K. This can easily be proved by the reasoning given in the proofs of Lemmata 2-4 of Section 10.3. 5. In excluding branch points on the free parts of.E we have used arguments of Gulliver-Lesley [1] and of Gulliver-Osserman-Royden [1]. This part of our reasoning is restricted to IV and cannot be carried over to R", n > 4, according to an example by Federer [2].

Chapter It. The General Problem of Plateau

In this chapter we shall present an introduction to the general problem of Plateau that, justifiedly, is often called the Douglas problem. This is the question whether a configuration of several closed curves r may bound multiply connected minimal surfaces that could be of higher genus and even nonorientable. After a brief introduction to the problem we shall exhibit in Section 11.1 several examples of boundary configurations and of minimal surfaces illustrating

the general Plateau problem. In Section 11.2 we shall outline an approach to Teichmiiller theory of compact oriented Riemann surfaces which, in Section 11.3, is extended to oriented Riemann surfaces with boundary. These results are basic

for the variational treatment of the general Plateau problem as we have to deal with variations of the complex structure of two-manifolds. The second basic tool is a compactness theorem of Mumford which will be combined with Courant's

condition of cohesion to obtain nondegenerate solutions of the variational problem connected with the general Plateau problem. This variational problem will be discussed in Section 11.5. We shall outline the crucial ideas for solving this problem and state some of the principal existence results due to Douglas, Courant and Shiffman. Finally, in Section 11.6, we shall tackle the general Plateau problem in combination with an obstacle problem in order to obtain a sufficient condition for the solvability of the general Douglas-Plateau problem that can be formulated in geometric-topological terms.

11.1 The General Problem of Plateau. Formulation and Examples In Chapter 4 we discussed the classical problem of Plateau as it was solved by

Douglas and Radb, and we presented the solution found by Courant and, independently, by Tonelli. In the restricted sense formulated in Definition 1 of Section 4.2, Plateau's problem consists in finding a disk-type minimal surface spanning a prescribed closed Jordan curve. This is to say, for a given closed Jordan curve T we are to find a harmonic and conformal mapping of the disk B = {w e C w I < 11 into l which maps the boundary of B topologically onto F.

11.1 The General Problem of Plateau. Formulation and Examples

295

Fig. 5. The catenoid (below), and four of its relatives bounded by 4, 5, and 6 closed curves respectively constructed by Wohlgemuth.

296

11. The General Problem of Plateau

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

(b)

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

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

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

11.1 The General Problem of Plateau. Formulation and Examples

297

Fig. 10a-c. Two-sided minimal surface of higher genus bounded by a single closed curve. (a) A Chen-Gackstatter surface of genus one. (b) A Chen-Gackstatter surface of genus two. (c) An Ennepertype surface of genus two with a Y-handle.

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

As we already pointed out, this is neither the most general nor the most natural way to formulate Plateau's problem, but merely the simplest and most convenient one as we do not run into the problem that parameter domains of the same topological type may be of different conformal type. By Riemann's mapping theorem all simply connected bounded domains in the plane are conformally equivalent. Nevertheless, boundary configurations consisting of several closed curves may bound multiply connected minimal surfaces as we see in the classical case of minimal surfaces of revolution, the catenoids. Moreover, soap film experiments show that certain configurations may bound minimal surfaces of higher topological structure, say, nonorientable ones, or minimal surfaces that are not of genus zero. Figures 1-9 depict several such contours as well as minimal surfaces spanning them. In certain cases it is not difficult to see that topologically more

complicated minimal surfaces may have smaller area than disk-type surfaces bounded by the same contour. The first to state Plateau's problem in a general form was Jesse Douglas who attacked this question in a series of profound and pioneering papers. Hence many

authors speak of the general problem of Douglas (or of the Douglas problem) instead of what Douglas himself called the

298

1 1. The General Problem of Plateau

General problem of Plateau. Given in 683 a configuration F =
of prescribed Euler characteristic, orientable or not, that span the contour F. Also this problem has many generalizations. We might replace 683 by 68", N >_ 2, or by any N-dimensional Riemannian manifold, say, by the N-sphere S" or by the N-dimensional hyperbolic space H". Moreover, instead of minimal surfaces (i.e., surfaces of vanishing mean curvature) we might look for surfaces of prescribed mean curvature, and instead of fixed boundary configurations we could investigate other boundary arrays leading to free or partially free boundary problems for minimal surfaces as were considered in Chapters 5-7 and 9. In all such cases the phenomenon of solutions of higher topological type may occur. In order to tackle solutions of this kind it is necessary to have some hold on their

parameter domains which will be Riemann surfaces with boundaries (nonoriented surfaces can be treated by passing to the double cover). For this purpose, some results from the theory of Riemann surfaces will be needed, particularly results concerning the so-called Teichmuller theory. In Sections 11.2 and 11.3 we shall outline a new approach to Teichmuller theory due to Fischer and Tromba which avoids function-theoretic tools and yields a very natural geometric method to obtain the results needed for treating the general Plateau problem. We do not plan to give an exhaustive description of all results obtained for the general Plateau problem. Rather the reader should take this chapter as a first

introduction to the problem and as a way to become acquainted with some of the methods to treat boundary value problems for minimal surfaces of higher topological type and of more complicated boundaries. As an example of what can be done we shall discuss in Section 11.6 a few existence results of Tomi and Tromba. Some of the existence results of Douglas, Courant and Shiffman will be formulated in Section 11.5 without proof. References to the literature and to more complete discussions of the general Plateau problem will be provided in Section 11.5 as well as in the Scholia Section 11.7.

In Douglas's approach to the general Plateau problem the main feature is a sufficient (but not necessary) condition ensuring the existence of a solution. This condition requires that the greatest lower bound of area (or of Dirichlet's integral) for nondegenerate admissible surfaces is less than the greatest lower bound for surfaces of "lower type". Here a surface is called of lower type if at least one of the following two possibilities occurs: (i) It has a smaller Euler characteristic than the prescribed one; (ii) It is degenerate in the sense that it consists of two of more disconnected surfaces whose total Euler characteristic is less than or equal to the prescribed characteristic and which are bounded by "subconfigurations" of F = whose union is the prescribed configuration F. Some of the examples pictured in Figs. 1-9 illustrate Douglas's sufficient condition. We shall see why for the general Plateau problem extra conditions have to be posed in order to guarantee the existence of solutions. On account of the

11.2 A Geometric Approach to Teichmuller Theory of Oriented Surfaces

j

299

enclosure theorems presented in Chapter 6 it is in any case clear that the general Plateau problem is not always solvable. However, there is still a big discrepancy between necessary and sufficient conditions for solvability. In general no conditions are known that are both necessary and sufficient.

11.2 A Geometric Approach to Teichmuller Theory of Oriented Surfaces

jc

In this section we shall outline a geometric approach to Teichmuller theory for compact oriented surfaces without boundary and of genus greater than one. Here a surface M is meant to be a two-dimensional manifold of class C'.

It is well-known that there is a collection of coordinate charts {Gj, co} of open sets Gj c M satisfying U Gj = M, and of bijective mappings (pj: Gj -+ C such that 9j o cpk' is holomorphic. Such a collection of charts {Gj, co} called a complex structure; we denote complex structure on M by c. When we think of M having a fixed complex structure c we shall designate this by writing

the pair (M, c). However, a given M may have many complex structures. For example, let f : M - M be a C°-diffeomorphism. Then we can construct a new complex structure f *c, the pull-back of c by f, considering the coordinate pairs { f -' (Gj), cpj o f }. The maps ((pj o f) o ((pk o f)-' are also holomorphic, and hence

f *c is indeed a complex structure, and f : (M, f *c) --> (M, c) is a holomorphic

mapping. It seems natural to identify (M, c) with (M, f *c) and to consider equivalence classes of complex structures on a fixed manifold M under this equivalence relations. To be precise, let le = '(M) be the space of all complex structures on M, and denote by.9 = -9 (M) the space of all orientation preserving C°-diffeomorphisms of M to itself. Then consider the space FP(M) := `e(M)l_9(M)

of equivalence classes of complex structure on M as described above. The space R(M) is called the Riemann space of moduli of M. The structure of 9P = %/9 is

even today not completely understood. Riemann conjectured that R(M) is a (69 - 6)-dimensional space in case that the genus p of M is greater than one, as we have assumed. To investigate the structure of 9P, Teichmuller introduced a somewhat larger space

= .T (M) defined by

where '/-qo is the quotient space of le with respect to the subgroup -90 of diffeomorphisms f E 2 which are homotopic to the identity map on M. 20 can also be characterized as the set of diffeomorphisms which are homotopic to id through diffeomorphisms. One calls (M) the Teichmuller space of M. Then we have

_P(M) = J(M)/T

11. The General Problem of Plateau

300

where r:=

/-9a

is the modular group of M. Nielsen [1, 2] proved that r is a discrete group. (Actually, r turned out to be isomorphic to the group of outer automorphisms of M.) One may attempt to understand the moduli space R(M) by first investigating .°l(M) and then studying the action of F on 9-(M). In the following we shall not employ any of Teichmuller's ideas. Rather we shall outline a description of Teichmuller spaces based on work by Fischer and Tromba [1]. It is somewhat difficult to investigate the space W of complex structures on M directly. We shall come to understand and the action of -9 on it through a somewhat circuitous route.

Definition 1. An almost complex structure J on M is a C°°-section of the (1, 1)tensor bundle T,1(M) over M such that J2 = -1,1 the identity map. More colloquially, for each p e M, J(p) is a linear map of the tangent space TPM into itself such that J2(p) = -1(p), where 1(p) is the identity on TPM and where the mapping p -- J(p) is of class C. We say that J is orientation preserving if, for each p and for non-zero XP e TPM, the pair (XP, J(p)XP) is a positively oriented basis for TPM. We shall denote by d (M) the space of all almost complex structures. It is not difficult to see that with any given structure c e ' we can associate a unique J E .W. This can be done as follows. Let (cpj, G;) be a coordinate chart e c. Define J,(p): TPM -- TPM by Jj(P) = drpf

i

0

10]

dco

1

where [0

Q] is the canonical almost complex structure on l 2.

We have J;(P) = Jk(P)

if dcp; 1[°

Oldrp,=d(Pk1L0

]dk

or if -1

dpi [0

]d'=[?

o] ,

where cli is a complex analytic mapping. Write i = x + iy, x = x(u, v), y = y(u, v).

11.2 A Geometric Approach to Teichmiiller Theory of Onented Surfaces

Then dpi is represented by the matrix x° [Y.

XV]

301

which, by the Cauchy-Riemann

YV

equations, is of the form

xu] = [b

Y,. [xu

Also, a

b

[b

a

t_

1

a]

[a b

a2 + b2 - b a]

Thus we have a

1

a2+b2[b

a][11

0][-b a]-[0

0]

and consequently J;(p) = Jk(p).

By this computaton we see that each c e' induces a J E d if we define J(p) = J;(p) for any coordinate chart (G;, gyp;) E c.

We shall see that the converse is also true, i.e., given any J e sad, there is a unique c e W which induces J in the above described manner. This converse is a much deeper fact which is only true in the two-dimensional case. The diffeomorphism group.9 acts on .4 in a natural way. For f E -9 define (f *J) (p) := dfp 'Jf(p) dfp. It is clear that f *J E .say if J e 4, and one easily sees that for the correspondence c - J we have f *c --> f *J. Thus the map which sends c to J is -9-equivariant. We shall now introduce another space of objects into the picture, the space of all C°'-Riemannian metrics.,# on M. Definition 2. The space A of Riemann C°°-metrics on M is the space of positive C°-sections of the (0, 2) tensor bundle To (M).

This is to say, if g e A#, then for each p e M the metric g(p) is a positive definite symmetric bilinear form on 7 M, g(p) : TM x TM R, so that p -+ g(p) is of class C. It is easy to see that A is a Frechet manifold since A is open in the linear space S2 of all symmetric two-tensors on M (S2 is defined in the same way as Af except that positive definiteness is not required). The diffeomorphism group 2 acts on A' as follows: (f*g)(p)(V, W) = g(f((p))(dffV, dffW), where dfp: T,M --> Tf(p)M

denotes the tangent mapping defined by f. Let Y be the space of all positive C°°-functions M, and let Al/9 denote the quotient space of .off with respect to 1; that is, we identify two metrics gl and

302

11. The General Problem of Plateau

g2 if gI(p) = p(p)g2(p) for all p, where b is a strictly positive C'-function on M. In brief, X/J/' is the set of classes of pointwise conformally equivalent metrics g. Theorem 1. There is a bijective equivalence (in fact, a diffeomorphism) between .rah

and .11/°.

Proof. Given g E A there is a standard way to construct a new and unique nondegenerate, alternating, antisymmetric bilinear form µg(p): T,M x TpM such that, if Vp, WP is an oriented basis for TpM, then pg(p)(Vp, W,) > 0. This bilinear form pg is called the volume element determined by g and by the orientation of M. It is defined by the equation Vp)

Pg(P) (Vp Wp) :=

C1

p, ,

g(Vp, Wp)

Vp) Ow" fVp)

if (V,, Wp) is an oriented basis for TM, and by pg(p)(Vp, Wp) :_ -µg(p) (WW, Vp) if not.

Since g is also nondegenerate, we can for each p e M transform µg(p) into a linear map 'P(g)(p): TpM - TpM via the rule

g(p)[ (g)(p)Vp, Wpb = µg(P)(Vp, Wp).

Let J(p) = fi(g)(p). One then checks that J2(p) = -1(p) and that J is an almost complex structure on M. Let us define the map d5: /t -> 4 by g -* 0(g). It is not bijective since we easily see that 0(p - g) = 0(g) for p a 9. However, one can check that 45 passes to a bijective map from M191' to .sad.

In each coordinate chart (G, (p), the metric g has a local representation g2,(w)duadufl for w = (u', u2) E cp(G). It is an obvious and natural question to ask: Given g, is there an orientation-preserving coordinate mapping p which makes g locally look as nice as possible? (w) 6.0, 6a6 = 0 if a /3 In this case "as nice as possible" means that gap(w) and SQ,, = 1 if a = /3, and where 2 E Y. The answer is yes and is a classical result (see Chapter 1). Such a qp is called a conformal coordinate system for g.

It is elementary to check that, given any g E .11, the set of conformal local coordinates gives a complex structure c(g) for M. One can further see that, as before, c(p g) = c(g) for p c- Y. Thus we obtain a map 4 -->' which is indeed the inverse of the map f --> sat which we defined earlier. Thus we have sketched a proof of the following result.

Theorem 2. For compact two-dimensional manifolds M without boundary there exists a bijective correspondence between IV and .sad and, moreover, between W and .4'/Y. This correspondence is 2 -equivariant.

11.2 A Geometric Approach to Teichmuller Theory of Oriented Surfaces

303

The quotient space is still a bit too cumbersome to work with. For this reason we shall introduce another model for With any metric g on M we associate its scalar curvature R(g). We view R:.A --> C'(M) as mapping from # into a space C°°(M) of C°°-functions on M. In conformal coordinates u1, u2 we obtain gap and we then define

R(g) := - A `d log A a 2f

where df =

a 2f

au2 + 8v

(u,

v) = (ut, u2). According to formula (32) of Section 1.3

we have R(g) = 2K where K is the Gauss curvature of a two-dimensional surface

in l3 whose first fundamental form is just g. Therefore we define the Gauss curvature K(g) of a metric g on M by K(g)

log ! = R(g). 2

Definition 3. Set . _1 := K-'(-1), i.e., the set _,#_1 is to consist of all those C°°-metrics whose Gauss curvature is the constant function -1.

The next proposition is well known. We shall present a proof of a more general version later in this section. Proposition 1. Let M be a compact surface without boundary of genus greater than one. Then, given any g e .11, there exists a unique A e 9 with K(Ag) = -1.

This establishes the bijective correspondence between &-, and /#/9, and hence W. In Fischer-Tromba [1] the manifold properties of these spaces are investigated, and it is shown that (as manifolds) they are diffeomorphic.

Recall that the group of diffeomorphisms of M also acts on .#. The map K: .,# --> C`°(M) has the property that K(f *g) = K(g) o f. As a consequence of this basic fact it follows that f *g e ,&_1 if g c- '#_I Thus _Q acts on ._1, and the correspondence between _#_1 and 4' is -equivariant. Hence we have found as the quintessence of our discussion: .

In order to understand the quotient spaces le/.02 and W190, it suffices to understand the quotient spaces

and

We now proceed to study M_1/_Q0. Let us think of &_I as an infinite dimensional surface in the linear space S2. Let g e '..,. Since 3 acts on .#_1, it would be useful to known what the

"tangent space" is to the orbit Cg(-q) of -9 through g. So let f, -s < t < s, be a smooth one-parameter family of diffeomorphisms of M. A "tangent vector"' to

'We use quotation marks here because we are working in the category of C°-maps and tensors and thus have no implicit function theorem. In Fischer-Tromba [1] it is shown that this presents no serious difficulties, and that formal tangent vectors are indeed tangent vectors.

304

1 1. The General Problem of Plateau

Og(_9) at g is given by d

(1)

dt

Here V :=

df I

dt =0

(.r*g)Ir=o = Ld .q

.

is a vector field on M, and the left-hand side of (1) is the

definition of the Lie derivative of the metric g with respect to V which is denoted by Lvg. Thus, to paraphrase, all tangent vectors to (''X/) at g e . 4' are of the form Lvg. Locally Lvg has the following form: c7

(Lvg)ap = V y utt + gyp

V'' Oua +

Since we are interested in the quotient space .M_, /t4 we wish to collapse all such orbits Oa(f). As a result we are not too much interested in the subspace of the tangent space Tg.,#_, to .K_, at g consisting of all symmetric tensors of the form Lvg, but in the complement of this subspace if it exists. We have the following basic result, the proof of which is given in Fischer-Tromba [1]. Theorem 3. Let h e T.-W-1 be a symmetric two-tensor. Then h can he uniquely, for some unique V, expressed as a direct sum

h = hTT + Lvg, where hTT is a symmetric two-tensor on M with the property that, in a conformal coordinate system with respect to the metric g and with local coordinates designated as u + iv, hTT has the local representation

hTT = adu2 - adv2 - 2bdudv, where a + ib is a holomorphic function.

Corollary. Every h e Ta..K_, can be expressed uniquely as a direct sum

h=

Lvg,

where (w)dw2 is a holomorphic quadratic differential on M with respect to the complex structure induced by g.

Proof of the corollary. Note that

adu2- adu2 - 2bdudv = Re{(a + ib)(du + idv)2} = Re{c;(w)dw2}. Now w g(w) is holomorphic by the theorem, and (w)dw2 is a complex valued two-tensor over M which in any complex (or conformal) coordinate system is holomorphic. Thus c(w)dw2 is, by definition, a holomorphic quadratic differential on M. O

11.2 A Geometric Approach to Teichmuller Theory of Oriented Surfaces

305

Therefore, formally speaking, the tangent space to the quotient space ought to be the set of real parts of holomorphic quadratic differentials. At this point some more historical remarks are in order. As we mentioned earlier, Riemann had conjectured that the space V/9 is of dimension 6g - 6 if y > 1, y = genus(M). Teichmuller later observed that, as a consequence of

the Riemann-Roch theorem, the dimension of the space of holomorphic quadratic differentials is as well 6g - 6. This led him to connect, via quasi conformal mappings, the space 16/-90 with these differentials. However, the

route that Teichmuller chose at that time was more complicated than the approach which we have taken here. Let us proceed with our discussion. Infinitesimally (on the level of tangent spaces) we have that the TeichmUller space (or the Riemann space of moduli) is represented naturally as a finite dimensional space consisting of the real parts of holomorphic quadratic differentials.

This is only one step away from actually putting a manifold structure on M_,1_9 or on N_1/-90. As it turns out, one can put a natural manifold structure on #_,/-90 but not on and we shall shortly see the reason why. We want to push down the tangent space onto #_, to (at least locally) produce a submanifold 9 of M_, which is a candidate for a coordinate chart for t_,/_9 or _,/-90. This will be carried out as follows. Consider the affine space determined by the set of symmetric two tensors of the form g + for "small" . If is small enough then

g+ is a Riemannian metric. By Proposition 1 there exists a unique 1(l;) such that the Gauss curvature -1. Clearly, since K(g) = -1, we have A(O) = 1. Moreover, one can verify that the mapping -* () is C-smooth, and that its derivative at 0 is zero. This implies that the mapping (2)

--*

Re(c(w)dw2)]

as a map of the holomorphic quadratic differentials into .#_, has a derivative at 0 e TM-, which is the identity map on all tensors of the form dw2). An application of the implicit function theorem now implies that, locally, the image of this map is a submanifold 9 of ' #_1.2 We thus have found a candidate for a coordinate chart for . lf_, /.9 or ,#_,/.90, namely the slice Y. We want to collapse all orbits onto Y, and we must

merely check that each point of . corresponds to only one orbit of .9 or 20. Precisely at this step the distinction between -9 and .90 enters the theory. Generally, points of Y may represent more than one orbit of _Q, but each point of S" represents only one orbit of the group -90.

24_, is again a Frechet space; however .So consists of C°-metrics, and by the implicit function theorem, it is a submanifold of 4'_1, consisting of metrics of Sobolev smoothness class H', for any

s>2.

306

11. The General Problem of Plateau

The following theorem by Palais and Ebin (see Ebin [1], or Tromb4 ETH-Lecture Notes [20]) shows that the action of .9 on .1l_1 (in fact on, fl ) proper. Theorem 4 (Ebin-Palais Lemma). Let g e

be a sequence of metrics converyit

to some metric g e ', and f e -9 be a sequence of diffeomorphism such tlt f,,* g,, -' 9 a .# (where --+ means convergence in any H'-topology, s > ? dim M Then f has a convergent subsequence

The next result shows that the action of.90 on ,#_1 is free. This is not tri for the action of on . &. Theorem 5. -90 acts freely on . #_1 -

Proof (a sketch). To act freely means that if f *g = g and f e .90, then f = i< Assume that f id. Then, if c(g) denotes the complex structure associated to it follows that f *c(g) = c(g), i.e., that f is a holomorphic self-mapping. Since is not the identity, the fixed points of f must be isolated, and in fact be nor degenerate. This implies that the Lefschetz fixed point index A(f) is positiv4 However, since A(f) is a homotopy invariant, we have A(f) = A(id), and this i equal to the Euler-characteristic of M, namely (2 - 2 genus M) < 0. This give us the desired contradiction, and hence f must be the identity. Using the Theorems 4 and 5, standard arguments (see Fischer-Tromba [ 1 ] allow us to conclude the following: In a neighbourhood of g we have a bijective correspondence between points i ,&-,/go (and hence of T/_90) and points in 9' (which is a nice C°°-submanifold o

One checks by elementary techniques that the pieces .9' constitute a co ordinate atlas for .&_1/90. We summarize this as Theorem 6. Let M be a compact oriented surface without boundary and with genu g > 1. Then the spaces #-1/-90 and W/90 have the natural structure of a simpl_ connected, finite dimensional C°°-manifold of dimension 6g - 6. The tangent space

at a point [g] E'#_i/go can be naturally identified with Re(l;(w)dw2), the rea parts of quadratic differentials which are holomorphic with respect to the complex structure on M induced by the metric g. (Remark. The dimension 6g - 6 follow:

from the Riemann-Roch theorem.) In a less elementary manner one can show (see Fischer-Tromba [4]): Theorem 7. The C°°-manifold M_1l_9o is d ffeomorphic to Euclidean R68-6 space.

As a consequence of Theorem 7 it follows that the principal -90-bundle

(iv, #-t, #-t/9o) with the natural projection map iv: _#_1 -+.#_1/!20 ha;

11 3 Symmetric Riemann Surfaces and Their Teichmuller Spaces

smooth sections. This implies that the Teichmuller space embedded as a (6y - 6)-dimensional submanifold I into ir; I --, /-90 is onto, and that at every g e I we have

307

can be in such a way that

T9E 1

Ts900) = T9-#-t (A particularly natural and beautiful embedding using harmonic mappings was given by Earle and Eells [1], but, of course, there are many possible such embeddings.) Equality (3) expresses the fact that I is transverse to the orbits of ADO, and this will be of fundamental importance in the following approach to Plateau's problem. (3)

11.3 Symmetric Riemann Surfaces and Their Teichmuller Spaces In attempting to obtain higher-genus solutions of Plateau's problem via the calculus of variations, the natural domains of definition for our mappings are not closed oriented Riemann surfaces but oriented Riemann surfaces with boundaries. In this section we shall indicate how to construct the Teichmiiller space for surfaces with k boundary components C,,..., Ck, each diffeomorphic to the unit circle. Let M be an oriented, two-dimensional C°°-manifold with

BM=Ctu...uCk which is not the disk or the annulus. We wish to determine the structure of W/90 where % is the space of all complex coordinate atlasses for M, and !20 is the space

of diffeomorphisms which fix the boundary (any f e -9o maps each C; to itself and is isotopic to the identity). The trick to handle manifolds with boundary is

(a)

(b)

Fig. 1. (a) A Riemann surface M with boundary. (b) Its Schottky double.

308

11. The General Problem of Plateau

to reduce the problem to manifolds without boundary by means of a construction due to Schottky. Given a manifold (M, c) with a fixed complex structure c, we can consider an exact duplicate of it, say, (AI, c). The manifold M again has k boundary components Ct, ... , Ck and the same coordinate atlas which we now denote by

a. Moreover, for each point p e M there is a "symmetric point" P e M. We construct the double 2M of M by forming the disjoint union M u M and identifying each point p E C; with its symmetric point p e C3, 1 < j < k.

Then we have to check that 2M has a complex structure induced by the complex structure c of M. For points away from the curves of transition CC we define the complex coordinate mappings as follows. If po e 2M and po aM, then po e M u A%I. Suppose that po e M. Let (GPO, (p) e c be a coordinate neighbourhood of po disjoint from M. Define our new coordinate mapping Y': GPO -, C by 1'(p) = cp(p), the complex conjugate of cp(p). If po E M, there is a coordinate neighbourhood (GPO, cp) E (M, c') disjoint from M. In this case define Y': GPO -, c by I'(p) = cp(p). If po e C;, then there is a complex coordinate system cp e c which

takes a neighbourhood GPO of po in M to the upper half B' of the unit disk in the complex plane with cp(GPO n C;) = (- 1, 1) and 4(po) = 0. Similarly we can consider the map gyp: GPO - C, (GPO, 0) the exact duplicate of (GPO, (p). Define a new coordinate neighbourhood GPO of Po in 2M by the disjoint union GPO = GPO u Gp,,

with identifications made along aM as described above. Define a new coordinate mapping : GPO -+ C by setting q1(p) = cp(p) if p e GPO and O(p) _ O (p) if p e 600, Doing this for all po e M we obtain a complex coordinate system 2c for 2M.

It is, however, important to note that 2c is not just an arbitrary complex structure for 2M, but also a symmetric complex structure. By this we mean that there is a map S of (2M, 2c) into itself which is an antiholomorphic (locally cS = 0) diffeomorphism of 2M with S2 = identity. We just set S(p) = p, S being

aw the symmetry for (2M, 2c). Since we assumed at the outset that M be not the disk

or the annulus, the genus of 2M is greater than one. We shall now consider only compact oriented surfaces 2M without boundary with a Cx-involution S, S2 = id, such that the fixed point set of S consists of k disjoint curves C, u ... u Ck. Denote by `es the set of all complex structures on 2M with the property that, for each c e %s, the involution S is an antiholomorphic map of (2M, 2c) to itself. Definition 1. The Teichmuller space (2M) is defined to be the quotient space Ws/go, where do are those C°°-diffeomorphisms of 2M to itself which are homotopic to the identity, S-symmetric and map each half of 2M to itself.

We now want to follow the construction in Section 11.2 to show that the Teichmuller space has the structure of a C°°-smooth finite dimensional manifold

11.3 Symmetric Riemann Surfaces and Their Teichmuller Spaces

309

of dimension - 3x(M), x(M) the Euler-characteristic of M. To start we do not consider all metrics on 2M but only those metrics for which S is an isometry. Let

as call these symmetric metrics, and denote them by .mss. The condition of symmetry immediately implies that the curves of transition are geodesic. In fact, this property characterizes those metrics on M which are the restriction of

symmetric metrics on 2M. That is, if g is a metric on M such that OM are geodesics, then g can be extended to a symmetric metric on 2M. Then, as before, we obtain an identification between 's and the space #S of all those C°-symmetric metrics whose curvature is the constant -1. Moreover, we can identify ks-t/.9o with 16s/-9o and where Ys is the set of C°-symmetric positive functions on M. To achieve this identification we shall need the symmetric version of Theorem 3 of Section 11.2. 1

Theorem 1. Let M be a smooth closed surface of genus greater than zero, endowed with a smooth metric g. Then there exists a metric G of constant scalar curvature conformal to g. The metric G is uniquely determined and depends smoothly on g if we require that the Gauss curvature K(G) satisfies K(G) = -1 in case of a negative Euler characteristic x(M), and that the volume of (M, G) equals that of (M, g) if X(M) = 0. If (M, g) is symmetric with respect to the isometry S, then also (M, G) is symmetric with respect to S.

Proof. By the Gauss-Bonnet formula we have

I

(1)

K(h)dyh = 27rx(M) < 0

M

for any metric h on M. Hence the curvature K(G) of the theorem must indeed be negative or zero depending on whether x(M) is negative or zero, since K(G) is supposed to be constant. Writing G = evg we obtain in conformal coordinates x", xO that 9"e = AS"f ,

K(g)

1 d log A,, 2A

Gap = ev;tL"e ,

K(G) = -e-vd log(Aev), 2A,

whence

K(G) = e-v[K(g) - 2._dv]

.

Requiring that K(G) _ -1 if x(M) < 0 and K(G) = 0 if x(M) = 0 we obtain (2)

-49v + 2ev = -2K(g)

and (2')

-dsv = -2K(g)

310

11. The General Problem of Plateau

respectively, where d9 denotes the Laplace-Beltrami operator of (M, g). Let us

first turn to (2'). As the kernel of dg consists of the constants, equation (1) expresses the orthogonality of K(g) to this kernel in L2(µg). Hence (2') is uniquely

solvable under the normalization condition v dµ9 = 0. M fm

If, therefore, g is symmetric with respect to S, then v = v o S since v o S is also a solution to (2'). By a simple scaling argument we see that the condition of equal volumes guarantees uniqueness as well. We now turn to the nonlinear equation (2). The uniqueness of G follows immediately from (2) by means of the maximum principle for elliptic equations. We remark that (2) is the Euler equation of the functional 1(v) = E(v, g) + 2 J

{e° + K(g)v}dµ9 M

where E(v, g) = fm g(Vv, Vv) dµ9, and Vv denotes the gradient of v with respect i Dirichlet's functional evaluated on the pair (v, g). to g. This is, in fact, We shall solve (2) by minimizing I in an appropriate set of functions, Y. We observe that, if a solution v of (2) attains its maximal value in a point p e M, then

necessarily -d9v(p) > 0, and hence

0< evla) < -K(g)(p), v(p) < logimin K(g)l =: K. We therefore choose

'':= fu EH2(M):u(p)< 1 +ic}. Since the first eigenvalue of -dg on M is zero, with the corresponding eigenspace consisting of the constant functions, we have the estimate

2E(u, g) ? 2

(3)

JM

u 2 dµg

for all u e HZ (M) with f u dµg = 0, where .12 is the second eigenvalue of - d9. It follows that the expression

E(u, g) +

(4)

AI

J

u dug

,

A:= fm dµg ,

is a norm on H2 (M).

Let us now show that I is bounded from below on Y, and that any minimizing sequence is bounded in the norm (4). Let u e'.7Y', and decompose u

in the form u = uo + m with fu uodµg = 0 and m = 1 J A

udµ9. Using (1) and M

11.3 Symmetric Riemann Surfaces and Their Teichmiiller Spaces

311

(3) we obtain

1(u) >- E(u, g) + 2 fm K(g)uodµg + 2m J I K(g)dµg M

> E(u, 9) - 12 IM

2

4A' IM K (g)2 dg + 4mX(M)

E(u, g) + 4nmX(M) - 4,.2-1

K (g)2 dµIn JM

case that m > 0 we have Iml = m < 1 + K and therefore 1(u) ?

JK(g)2dRg.

I E(u, g) - 4z(1 + K)IX(M)I -

If, however, m < 0, we obtain 1(u) ? 2 E(u, 9) + 4711 X(M)I I mI

4.1z 1

JK(g)2d. µg

We conclude immediately from the last two inequalities that I is bounded from

below on -f and that any minimizing sequence in A' is bounded in H'(M). Standard compactness and lower semicontinuity arguments then give the existence of a minimizer v e .%' (see Morrey [8], and also Chapter 4). Let us show that v < K.

For arbitrary w e A' we obtain 1(v) < 1((1 - t)v + tw) = 1(v + t(w - v)). Then, using the fact that dt 1(v + t(w - v))

0, we clearly obtain the varia=o

tional inequality

1.

{g(Vv, V(w - v)) + 2[e° + K(g)](w - v)} dµg >- 0

for all w e .*', where we again use V as an abbreviation for the gradient with respect to the metric g. Inserting w = min{K, v} we obtain

L..

{g(Vv, Vv) + 2 [ev + K(g)] (v - K)} dµg < 0.

By the choice of K we have e° + K(g) > 0 for v > K, and we conclude that v = 0

a.e. on the set {x a M: v(x) > iC}. We thus have shown that v < K a.e. and, consequently, v lies in the interior of the set _E' with respect to the Lw-norm. The

Euler equations for v can then be derived in the usual way, proving that v is a weak solution of (2). Since the nonlinear term e' is already bounded, regularity of v follows from regularity theory (see Gilbarg-Trudinger [1] ).

312

11. The General Problem of Plateau

In the symmetric case we minimize I in the class of symmetric functions

C={ueHZ(2M):u<1+K,uoS=u}, and we find a symmetric minimizer v. By the same reasoning as above, the relation {g(Vv, Vw) + 2[e° + K(g)]w} dµ9 = 0

(5) f2 M

holds for all symmetric functions w e Hz (2M). Since g is symmetric and all functions in the integrand of (5) are symmetric, we may conclude that v is a weak solution of (2). The smooth dependance of G from g stated in our theorem follows readily

from the implicit function theorem applied to the equation (2) since the linearization of (2) with respect to v is always an isomorphism between suitably chosen Sobolev or Ck."-spaces. The theorem is now completely proved.

We now wish to characterize the tangent space to lfs, at g, as we did earlier, as a direct summand, one of whose terms consists of tensors of the form Lv9 E Ta9(_qo). This follows as in Section 11.2, with the exception that since we are considering only symmetric metrics g, we are imposing a condition on the metrics g along the curves of transition. To see what condition must be required on the subspace of T9,& complementary to let po E C3, and consider a coordinate system 'P: Gp0 --+ C for 2M which locally flattens C,. We shall require that 1P(GPo)

be the open unit disk B in the plane, and that 'P(Gpo n C3) = (- 1, 1) x {0}, T(po) = 0. Moreover, we assume that W takes symmetric points on 2M to symmetric points in B with respect to complex conjugation. In such a coordinate system we may represent g as (6)

gltdu2 + g22dv2 + 2g12dudv.

However, locally, the map (u, v) -+ (u, -v) is required to be an isometry. Thus we must have equality in expression (6) when we replace (u, v) by (u, - v) and dv by - dv. Thus we find that g21(u, v) 921 (U, - v), 91 t (u, v) = 91 t (u, - v), 922(u, v) = 922(u, -v). In particular we find that g12(u, 0) = 0 holds "on the boundary". We therefore conclude that in any coordinate system for 2M about a point po e 8M which respects symmetry (as above) the local representation of any symmetric metric must have its off-diagonal elements vanish at the boundary.3 We now state the symmetric analogue Theorem 3 of Section 11.2.

Theorem 2. Let h e T,,#S, be a symmetric two-tensor. Then h will also be S-symmetric4 and can be uniquely written as a direct sum ' This fact can be shown to be equivalent to 8M being a geodesic. The two notions of symmetry should cause no confusion.

II 3 Symmetric Riemann Surfaces and Their Teichmuller Spaces

313

h=hTT +Lvg with L,,g e 7 & ('o) (= the tangent space to the orbit of go through g). The S-symmetric tensor hTT can be expressed in conformal coordinates as

hTT = adu2 - adv2 - 2bdudv where a + ib is locally a holomorphic function. If this conformal coordinate system taken about a point on the boundary respects symmetry and maps the boundary to the real axis as above, then we have b = 0 on the real axis (thus a + ib is "real on the boundary").

Corollary. Every h e Tgsr can be expressed uniquely as a direct sum h=

(7)

L,,g

where (w)dw2 is a holomorphic quadratic differential which is real on the boundary, and the vector field V is tangent to DM5. Moreover, each holomorphic quadratic differential on M arises as such a sum (7).

As before, it follows from the theorem of Riemann-Roch that the dimension

of the space of all such holomorphic quadratic differentials real on 8M has dimension - 3x(M) = 6,q - 6 + 3k, and this therefore must be the dimension of the Teichmuller space of such symmetric surfaces. One now imitates the previous construction to show

Theorem 3. Let 2M be a symmetric surface with symmetry S as above. The St/-qo have the structure of a finite dimensional Teichmuller spaces Ws/-9o and C°°-manifold of dimension -3x(M). The tangent space at a point [g] e -&S-1 /-90 the real parts of holomorphic (with respect to can be identified with the complex structure induced by g) quadratic differentials which are real on OM,

We should like to point out that J-s = ,#'-,/go may be embedded as a finite dimensional C°°-submanifold.E into ..ts t, in such a way that 7t: E S t /.90 is onto and that T91' +O Tgag(-9o) = T,-#s,

It also follows from Fischer-Tromba [4] that #s-,1!20 is diffeomorphic to the Euclidean space of dimension - 3x(M), which concludes our discussion of Teichmuller spaces for oriented surfaces. Remark. One can in a similar way consider Teichmuller theory for unoriented surfaces M by passing to a Z2-cover Al. The Teichmuller space is then obtained as above from those metrics which satisfy an additional Z2-symmetry. This Teichmi ls This last condition on V means that the family of diffeomorphisms generated by V map each boundary component to itself.

11. The General Problem of Plateau

314

ler space is needed for the existence theory of non-oriented minimal surfac't ~' shall not develop these ideas here.

We shall end this section with a discussion of the Weil-Petersson met 1-' L: `l (M).

First, there is an L2-metric on

tr ace(HK)dµg

<(h, k)>9 =

(8)

x T,#-, --+ R, givc r"

<<,)):

I'M

_, and H = g-1 h, K2 = g-1k are the (1, 1)-tensors on M obt 11 i 1104 from h and k via the metric g, i.e., by where h, k e TgA

HB = g"1h,p,

where (gdd) denotes the inverse of (gay), and similarly for K. In local coordinates equation (8) can be written as 1

(8)

<% = 2

Stga11g7'ha,,kfla}dµg M

The inner product (8) is -90-invariant (actually, s-invariant). This invari
(9)

But

f

<
2

2

By the change of variables theorem this equals I 2 SM

t r(HK)dµg = >9

which proves (9).

Thus -qo acts smoothly on '#_1 as a group of isometries with respect to this metric, and consequently we have an induced metric on

(M) in such a way t 1-11.1 t

the projection map it: #_1 -> ,.1#_,/-qo becomes a Riemannian submersion- As it turns out (see Fischer-Tromba [3]) this induced metric is precisely the metric originally introduced by Weil, now called the Weil-Petersson metric. Let <,) be the induced metric on 9-(M). We can characterize <, > as follows. every h e Tg.#_, can be written as We know that, given g e (10)

h = hTT + Lvg

where L,,g is the Lie derivative of g with respect to some (unique) V, and h-"-7- is a trace free, divergence free, symmetric tensor. Moreover the decomposition (10)

11.4 The Mumford Compactness Theorem

315

is L2-orthogonal. We call hTT a horizontal tangent vector in T9,#_1 and L,g a vertical tangent vector. Let h, k e T,,9-(M). Then for any g v n-' [g] there exist unique horizontal tangent vectors hTT and kTT such that D7ta(hTT) = h and D7t9(kTT) = k. Then
By s-invariance this is independent of the choice of g e 7z-'[g].

11.4 The Mumford Compactness Theorem We now come to the compactness theorem for the moduli space which is fundamental to any existence proof for minimal surfaces of higher genus within a given boundary configuration. In its original form the theorem is due to Mumford [1]. We present another proof given by Tomi-Tromba [4] using only basic geometric notions instead of the uniformization theorem; this method also works for symmetric surfaces as well as for unorientable ones. For completeness we include the flat case since the proof requires only a few additional comments. Theorem 1. Let M be a closed connected smooth surface, and {g"} be a sequence of smooth metrics of curvature -1 or 0 respectively on M such that all their closed geodesics are bounded below in length by a fixed positive bound. In the flat case we assume furthermore that the areas of the (M, g") are independent of n. Then there exist smooth diffeomorphisms f" of M which are orientation preserving if M is oriented, such that a subsequence of { f"*g"} converges in C° towards a smooth metric. If M admits a symmetry S which is an isometry for all g", then the maps f" can also be chosen to be S-symmetric and to map each half of M to itself.

Before proceeding with the proof we should note that this is a compactness The corresponding statement for theorem for Riemann's moduli space the Teichmuller space would be false. To see this we recall that 9/-9o is an infinite discrete group. Let f" be a sequence of diffeomorphism such that its classes e 1-9/.9p have no convergent subsequence. Let g be any metric, and consider the orbit class { f"*g}. If the Mumford theorem were true for Teichmuller's moduli space, this would imply the existence of a sequence h" a -9o such that h* (f"*g) = (f" o h")*g converges in #_1 (say, in the HS-topology, s > 2). By Section 11.2, and in particular by the Ebin-Palais lemma, this implies that f, o h, has a convergent subsequence. Thus has a convergent subsequence, and we have found a contradiction. We can now proceed to the

Proof of the Mumford theorem. Since on a negatively curved surface there are no conjugate points along any geodesic, it follows that every geodesic arc is a relative minimizer of the arc length (with fixed end points). Therefore, any two

316

11. The General Problem of Plateau

geodesic arcs with common endpoints cannot be homotopic with fixed endpoints; otherwise, by a common Morse-theoretic argument (see Milnor [1]), there would exist a non-minimizing geodesic arc joining these endpoints. Hence we may conclude that a lower bound 1 on the lengths In of the closed geodesics of g" implies a bound on the injectivity radii pn of M" = (M, g"), pn >- p >- 1/2. It follows that on each open disk BR(p), p e M" and R < p, one can introduce a geodesic polar coordinate system. By a classical result in differential geometry

which can easily be derived from the results of chapter 1, the metric tensor associated with g" in these coordinates assumes the form 0 (glP = 0 f(r) 1

(1)

>

f(r) =

(sinh r)2 1r2

if R(g") = -1 ifR(g") = 0

where r denotes the polar distance. For the area of BR(p) we obtain from (1) the simple estimate area BR(p) > irR2.

The genus of the manifolds M" being fixed, the total area of M" is determined by the Gauss-Bonnet formula if R(g") = -1. It follows that there is an upper bound, only depending on R, for the number of disjoint open disks BR(p) in M". Let us now take R = 1p, and let N(n) be the maximal number of open disjoint disks of radius in M. By passing to a subsequence we can assume that N(n) = N 4R

holds independently of n. It follows that for each n e N we can find points

pj e M", j = I, ... , N, with the property that the disks B114R(p7) are disjoint while

the disks B1,2R(pj") cover M". Let us now denote by H the Poincare upper halfplane in the hyperbolic case6 and the Euclidean plane in the flat case. We pick an arbitrary point Co e H, e.g., Co = i, the imaginary unit, and introduce geodesic polar coordinates on M" and on B4R(Co) c H, respectively. The corresponding metric tensors assume the same form (1) in each of both cases,

and we may therefore conclude that there exist isometries cj

B4R(Pj) --- B4R(S0),

(Pi (Pi= CO SLet

then In denote the set of all pairs (j, k),/ 1 < j, k < N, such that 0. B2R(Pj) n B 2R(Pk) #

By passing to a subsequence we can assume that In = I is independent of n. For (j, k) e 1, the transition mappings

Zik _ j o ((Pk)-t (Pk [B4R(Pj) n B4R(Pk)] -a (P [B4R(Pj) r B4R(Pk)] are well defined local isometries of H. Before proceeding further with the proof we first want to show that any such local isometry in fact extends to a global one. We only consider the hyperbolic case, the flat one being trivial.

'This is the half plane (u, v) a L2 with v z 0 endowed with the metric {(du)' + (dv)Z }/4v2. The scalar

curvature of this metric is =- -1.

11.4 The Mumford Compactness Theorem

317

Lemma 1. Let f : U -+ H be a Ct isometry on an open connected subset U of the hyperbolic plane. Then

f(w) _ Aw+B

A, B, C,DE18,

Cw+D'

and AD - BC= 1. A" + B AD - BC = 1 with real coefficients is Cw + D'

Proof. . The class of maps w -.

the group of isometries of the Poincare metric. Thus we must show that a local isometry is also a global isometry.

It is clear that we can take f to be orientation preserving. Then an easy calculation shows that f must be holomorphic and has to satisfy the nonlinear condition Im

(2)

If'(w)I = Im ww)

One can check that every map of the form w -.

Aw + B

Cw+D

as above satisfies

condition (2) and that the set of maps satisfying (2) from a fixed domain to itself froms a group. Therefore, by composition with an appropriate element of the three-dimensional conformal group of H we may assume that f satisfies the following additional conditions: f is defined in a neighbourhood of i E H, and

f'(i) = in f(i).

(3)

Now, writing w = u + iv and using (2), we have

(log f')' = {Re(log f')},,, = {loglf'I}w (Imf )w

vw

Imf

V

- - if' Imf

i V

By (3), (log f')'(i) = 0. Similarly

(log f)

-['-] -if"

I

+ [Vi]

i2 if, (Imf )2 (f ) + v2

Again we see that (log f')"(i) = 0. Proceeding inductively we obtain

(log f')"(i) = 0 for all n E N. Thus, since log f' is holomorphic in a neighbourhood of i, it follows

that log f' is constant, and so the mapping w - f'(w) is constant; therefore f(w) = w. Since we normalized f by the isometry group of H, this proves that

318

11. The General Problem of Plateau

our initial map f must be in this isometry group, and the proof of the lemma is complete. For (j, k) E 1 we have pk e B4R(pj ), and hence qfk := gf(pk) e B4R('o, since qpj" is

an isometry. It is obvious from the definition that qjk = tjk(gkj)

(4)

We are now going to construct a limit manifold of the sequence M" = (M, g"). For this purpose we prove Lemma 2. The family of transition mappings (Tjk)" E N is compact for each (j, k) e 1.

Proof. By Lemma 1, each Tfk is a global isometry of H and there are a fixed compact subset K of H and points qj, e K such that (4) holds. From this the assertion follows at once in the flat case, since each Tj"k decomposes into a rotation and a bounded translation. In the hyperbolic case, by composition with a conformal map of H onto the unit disk .4 c R2 we may assume that each Tfk is a conformal map of -4 onto itself and (suppressing the indices j, k) that there are points p" strictly staying away from as such that also T"(p") stays away from aa. Each t" is of the form w - an d"

1 - a"

w,

where la"I < 1, d"I = 1. It suffices to show that la"I stays strictly below 1. If not, we can assume a" --> a, I a I = 1, and d" -+ d, IdI = 1. The limit map

t(w)=d

w-a

law -1

1-aw -ad 1-aw}

-ad

then collapses the disk onto a point on aB which is a contradiction.

We can now continue with the proof of Mumford's theorem. Passing to a subsequence we can by Lemma 2 assume that (5)

tjk --> Tjk

as (n -+ oo).

We now define a limiting manifold M as the disjoint union of N disks BR(to) c H, labelled as B, , ..., BN with the identifications

pEBj equals geBkc>(j,k)eI and p=tjk(q). It is clear that l 1 is a differentiable manifold carrying a natural Riemannian metric which on each Bj coincides with the Poincare metric or Euclidean metric, respectively. We claim that M is compact. Assume to the contrary that there were a point q e aBR(t'o) such that q 0 Tjk(BR(Co)) for some j and all k with (j, k)yya 1. Then it would follow that, for sufficiently large n, we have q 0 T;kCB(3/4)R(SO)1 which means that (rpf)-'(q) 0 B(,/4)R(p1). This, however, would imply that

11.5 The Variational Problem

B(1ia)R[((pj")-t(9)] n B(114)R(Pk) = 0

319

for k = 1, ..., N,

contradicting the choice of N as the maximal number of disjoint disks in M" of radius R/4. The remainder of the proof rests upon the following Lemma 3. There are diffeomorphisms f": M -* Mn, f"(BB) c BZR(p7) such that (6)

cps o f" --, id

in C on each BB, as n -+ oo .

In the symmetric case f" can be chosen to be symmetric, i.e. to commute with the symmetries S and S" on M and M" respectively, S"of^=

fn O S.

The proof of this lemma is somewhat technical, and we refer the reader to Tomi-Tromba [4] for a proof. Let us quickly finish the proof of Mumford's theorem assuming the lemma. Denoting by g the Poincare metric or the Euclidean metric, respectively, we have from (6) that f"*Tj"*g -+ g

as

n --,. 00

on each B;. Since, however, (pj" was an isometry between g and g" on M", this means that

f"*g"-+g asn -goo on M. Choosing now any (symmetric) diffeomorphism f : M - M, we obtain

(f" o f)*g" - f *g asn -+ oo, which proves Mumford's theorem.

11.5 The Variational Problem Given a smooth compact surface M with k boundary components and k disjoint

curves Jr, ..., r, in f8', N > 2, with prescribed orientations we would like to prove the existence of a "minimal surface" X: M --+ R' such that -'IOM parametrizes T1 u ... u I'k, and in the case that M is oriented we require £fl3M to preserve orientations. The question of whether such a minimal surface is immersed or not is as in the case of a disk a separate question (see Gulliver [7]). Also, as in the disk case, the need to control parametrizations in finding surfaces of least area spanning T1, ..., Fk leads one to the definition of a minimal surface as a conformal harmonic map ': M --+ R' on a given "parameter surface" M into l". The topological type of ' is given by M. As M is no longer a planar domain, we have now to define what we understand to be a conformal harmonic mapping . ' of M into if8'. The following is in a way a repetition of what we have said in Section 3.6. .

11. The General Problem of Plateau

320

Suppose that g is a metric on M, and let d9 be the Laplace-Beltrami operator corresponding to g. In local coordinates 9: G -> 682 on M we have (P-1

('49-T) e

=

det

g

P

{fci

y aus

x}

where det g = det(gae). The mapping X: (M, g) -+ 0 N is said to be harmonic if

d9, '=0.

(1)

It turns out that Y: M --> RN is harmonic if and only if, for any system of conformal coordinates on M defined by cp: G - R2, the pull-back X := ' o cp-'

is harmonic in the classical sense, i.e., JX = 0. Note that the definition of harmonicity by means of the equation (1) is intrinsic, i.e., independent of the chosen coordinates (u', u2) on M (see Chapters 1 and 2). Also, the definition of harmonicity of . ' by means of the equation AX = 0 for X = X o co` and a conformal map cp: G -+ R2 is intrinsic since the transition map between two conformal coordinate system is holomorphic, and harmonic mappings of domains in R2 remain harmonic after composition with holomorphic mappings. We call a mapping _X: (M, g) -+ R' conformal if, for any system q : G -+ 682

of conformal coordinates w = u' + iu2 on (M, g), the pull-back X =

'o

satisfies (Xw, Xw> = 0,

(2)

that is, if X(w) = (X'(w), X2(w),..., V(w)), then

(2)

(XW)2 +

(X,2)2

+ ... + (Xw)2 = 0.

Clearly, this definition of conformality of a mapping el is intrinsic as well. Let c be a complex structure on M. Then we can equally well view . ' as a mapping from (M, c) into R", and we can call t : (M, c) -* RN harmonic and conformal if, for any chart (G, (p) e c, the function X = . ' o q is harmonic

and satisfies (2). These two definitions mean the same since, because of the one-to-one correspondence 16 H/Y, for every conformal equivalence class of metrics g on M there is exactly one complex structure c = c(g) generated by g. The precise formulation of this bijective correspondence between complex struc-

tures on a manifold M with boundary and symmetric Riemannian metrics on M, genus M > 1, of constant Gauss curvature -1 was given in Section 11.3. This leads us to the following two equivalent formulations (90) and (9) of Plateau's problem: (9o) Given M and F,,..., Ti as above, determine a complex (or conformal) structure c on M (not just on M) and a map .t: M - R', continuous on M and smooth in the interior, such that (a) . 18M : OM - r, u ... u Ti is one-to-one and preserves orientations if M is oriented. (b)

.

' is conformal and harmonic with respect to c.

11.5 The Variational Problem

321

(b') Given M and rt, ..., rk as above, determine a smooth symmetric metric g on 2M of Gauss curvature - 1 and a map 9: M --+ RN, continuous on M and smooth in the interior, such that (a) X 10M -. r1 u ... u rk is one-to-one and preserves orientations if M is oriented,

(b) the induced metric

.

'*gN (gN the Euclidean metric on RN) is conformal

to g; i.e.,

A > 0,

*9N = 29, and , ' is g-harmonic; i.e., d9 y = 0.

Let us now introduce the generalized Dirichlet integral E(9%, g) as a functional on mappings X: M --> RN and on metrics g. The functional E is defined by

E(Y, g) = 2 J g(x) [V;Y', V91] dp9

(3)

M

where g(p): TM x TpM R is a Riemannian metric on M, V9."` is the ggradient of the i-th component function of T, and gg is the classical volume measure induced by the metric g. In local coordinates (u', u2) on M, the functional E is given by E(o , g) = 2

f

M 9

p

v

,o>dft9(u1, u2)

det (g,,) du' due, and we are using the standard conventions of differential geometry. With a slight abuse of notation we have written X(w) instead of X(w) for X = w o cp-1 and a chart (G, (p). Obviously, the functional E(e", g) depends only on the conformal class of g, i.e., where dµ9(u1, u2) =

E(-T, 9) = E(.r, 29)

for any positive function 2 on M. One also readily verifies the invariance of E under diffeomorphisms f of M,

E(T,g)=E(91of,f*9). If g is a symmetric metric on the double 2M (equivalently, if OM is a geodesic

for M), the conformal invariance of Dirichlet's energy and the fact that, given any g, there is a unique 2 such that the Gauss curvature K(2g) _ -1 immediately implies that we may restrict Dirichlet's energy to symmetric metrics of Gauss curvature -1, i.e., to g e M1, Now let r be an oriented Jordan curve (or a collection of Jordan curves r1, ... , rk) in RN, and set *

rl'r = {.°.ir': M -* RN, .f1 a H2 (M, RN) n C°(M, RN'),

': aM -+ rmonotonically}.

Let E: ?l,- x 1 -. R be Dirichlet's energy. For fixed g, suppose we have a minimum X'° satisfying

11. The General Problem of Plateau

322

E(10, g) 5 E(1, g)

for all . ' e qr. In the case that M is the disk, this implies that -To is harmonic and conformal, i.e., a minimal surface. However, this is not any longer true in the higher genus case. Instead one also has to vary the metrics g on M. Minimizing over all metrics we shall be able to produce a conformal map. Suppose now that we had a minimum pair (.2°0, go) for E, that is, E(" o, go) E(-T, g)

for all (.1,g)enr x mss,. In general such a minimum will not exist as we shall presently see. For the moment we would like to understand why a minimum is a harmonic conformal map.

The relation E(T0, go) <_ E(.Y, go)

for all ' e rlf immediately implies as in the disk case that To is harmonic,

d3"o = 0, whence X. e C°° (I I, RI).

If we knew that the minimum Yo would be smooth up to the boundary aM provided that r were smooth, say, of class C°, then we would be able to deduce the additional boundary condition

ado aXo _ 0 on aM

(4)

aN ' aT

i.e., the normal derivative would be perpendicular to the tangential derivative. In a local conformal coordinate system flattening aM this would be equivalent to

a.' ax

au TV

= 0. In the case of the disk, equation (4) suffices to insure that

aXo ago

az' az

=°'

i.e., that -To is a minimal surface.

Interestingly, one can show that (4) holds up to the boundary aM even though To may not a priori be known to be smooth up to the boundary. We have the following result. Theorem 1. Fix some metric g and let Xo be a minimum for Then the expression (6)

_ jY

' - E(', g), T e rlr.

[!]2iw2 aw

is a holomorphic quadratic differential of class C° (up to the boundary) which is

11 5 The Variational Problem

real on 3M; i.e.,

a.o a o au

323

= 0 in any local conformal coordinate system flat-

au

tening 3M.

Proof. That g is a holomorphic quadratic differential of class C°° on M follows immediately from the fact that .C'Q is C° smooth on M. Moreover, c is of class C°° up to the boundary and real on OM as was proved in the disk case in Tromba

[13], and since this is only a local result, the same proof applies with only superficial alterations. We refer the reader to this source for a proof. We would now like to prove Theorem 2. If (YO, go) is a minimum for the functional

Sr,

(-1,g)-*E(31,g), then

o:(M,go)--SRN is harmonic and conformal, i.e.,

j

N C -Wo] dw2

Y

= 0.

Proof. We have already established harmonicity. Let us calculate the derivative of g - E(e"o, g) with respect to the variable g. Recall (cf. Section 11.3) that 7A' t can be decomposed as an L2-orthogonal direct sum, with h e T,,#'-, expressed as

h=Lg+hTT where hTT is the real part of a holomorphic quadratic differential vanishing on

3M;hTT=ReC. Again suppose for the moment that To is smooth up to OM and let f be a one parameter family of diffeomorphisms with fo = id, df

dt ,=o

V.

Then d

dt(f *go) = Lvgo -

(7)

is the derivative of go, and d (8)

'o) = d.To(V)

is the derivative of X. in the direction V. By the invariance of Dirichlet's energy (3) we obtain (9)

E(f*.E0, f ,*go) = E(to, go)

11. The General Problem of Plateau

324

Therefore it follows that

dtE(f *Yo, f*9o)l,=0 = 0, and we arrive at the differential relation (10)

- (Io, go) [d1o(V)] +

a9

Here we have used the notations aE and

(-To, go) [Logo] = 0.

OE

for the "first variations" of E with

9

respect to the first and the second arguments respectively, i.e., dtE(f*(o, 9o)Ir=o,

ax (To, 9o)Ld-'o(V)7:= a9

(fo, go) [Lv9o] :

dt E(110, f *9o)I,=o

Now, if 2 do is a minimum (or even merely a critical point) for the mapping -> E(2', go), it follows that 8E

8° .(To, 90) = 0

(11)

which implies that (12)

a9

(.To, go) [Logo] = 0

for all vector fields V arising as infinitesimal symmetric diffeomorphisms. This

says that, formally, at a mimimum (or more generally at any critical point) derivatives of Dirichlet's energy with respect to g in the vertical directions Lvg are zero and therefore yield no additional information. Therefore, if we are to conclude that i; = 0 we should consider derivatives of E with respect to g only in horizontal directions (cf. formula (10) of Section 11.3). Lemma 1. Let (.d", g) e t1 r x Ms 1, and let p = hTT be a trace free divergence free (0, 2)-tensor, p = Re {v(w)dw2 } where v is a holomorphic quadratic differential that

is real on 8M. Then (13)

ag

(X, 9)p = -2 <>9.

Here <<, >> denotes the L2-inner product introduced at the end of Section 11.3, and

[%J]2 is a (0, 2)-tensor of class L2 on M.

(dw)2

11.5 The Variational Problem

325

Proof of Lemma 1. In local coordinates 1

E( , g) = 2

>dg(1, u2).

L

Now the derivative of the function g calculated to be

pg in the direction h can easily be

h -* (i trgh)ug

where trgh = g"%#. Therefore, if h is trace free, the derivative of this term vanishes. Just as easily one computes that the derivative of g -> g'J in the direction his

h-> -h", where haft = g"lgflbhyh,

Then we obtain for dE(" ', g)p :=

-g,', g)p the formula

dE(-', g)p = -21

(14)

JM

p"P<Xu T'u,)dpg.

In conformal coordinates (u 1, u2) = (u, v) we have

p" = A-2p"p,

go = A8"#,

dpg = Adudv,

and we see that (14) is equal to 1

2p1Adudv,

dE(', g)p = - 2 TM

where I"e := <.'o, 2'uf> .

Since p is trace free, we have Pi 1 = - P22, and we obtain

dE(f, g)p = -2 JA_2{pii(1ii

- 122)+ 2p12112}2dudv.

As N

Re = Re Y (e

)2(dw)2 = (l 11 - 122)[(du)2 - (dv)2] + 4I12dudv.

i=1

we obtain z

(Re z)11 = -(Re 02 2 =111-122,

(Re 012 = (Re 021 = 2112 .

Therefore, <<(Re ), p>> = 2

g

i'M

g(Re

pdpg

326

11. The General Problem of Plateau

=

1

2 JM

f

A-2 (Re )aapaadp,

-2{2Pii(lti - Izz) + 4Ptzliz}.idudu

M

= - 2dE(,1, g)p which yields Lemma 1.

We are now in a position to complete the proof of Theorem 2. By Theorem 1 the C°-quadratic differential is holomorphic and real on M. By Lemma 1 we have

dE(Xo, go)p = -'((Re , p>>90 = 0 for all p = Re(v(w)dw2), where v is any quadratic differential which is holomorphic and real on OM. This immediately implies that

= 0.

As we have already observed, the second variable of Dirichlet's energy belongs to .A' 1. Fortunately it is possible to reduce the infinite dimensional space

M1, to a finite dimensional C'-manifold, without changing the critical points, in considering Dirichlet's energy as follows. The bundle ir:Xs 1 - . #', loo = 9-(M) is a trivial principal fibre bundle of class C' and admits a C°°-smooth section r :.T (M) - . #s , (see Fischer-Tromba [1]). Let E c mss, be the image of such a section. Then E is a smooth finite dimensional C°-submanifold of M1, everywhere transverse to the group of symmetric diffeomorphisms homotopic to the identity, and whose dimension is equal to that of 9-(M). The reader may now verify the following Theorem 3. Consider Dirichlet's energy restricted to gr x E, (15)

E: rig x E (among them the minimizers) are precisely the minimal

surfaces spanning T.

Now that we know that the critical points of (15) are minimal surfaces, it seems natural to try to generate minimal surfaces of a given genus spanning a curve Tby finding a minimizer of (15). In general there will not exist a minimizer of prescribed topological type. For example, a plane circle cannot bound any minimal surface apart from a disk. In order to generate a minimal surface, one would take a minimizing seE gr x E and attempt to find a convergent subsequence, quence, say, (X,,, such that (X, (TO, go) and again denoted by E(`1,, go) < liminf E(1,,, g.). n-co

11.5 The Variational Problem

327

Fig. 1. Four terms of a degenerating minimizing sequence with shrinking necks. This phenomenon is excluded by the condition of cohesion.

To see what can go wrong in attempting to produce such a minimizing sequence let us consider the simple case of two coaxial circles I1, f`2 of the same radius R, although this case does not fit into the picture we have developed so far. Here M is an annulus, and 2M is a torus whose genus is one. In this case the

Teichmiiller space consists of symmetric metrics of zero curvature and fixed volume modulo the group of symmetric diffeomorphisms homotopic to the identity. Thus there is still a E, but it is not a submanifold of 01, but of the manifold of symmetric flat metrics of prescribed volume. Yet this example is instructive as it will show what may go wrong in the general case. If the distanced is small enough then the boundary configuration d*, and choose a minimizing sequence { ,,, g,, } for E. It will develop a narrower and narrower neck, and the images of the X will be degenerating in topological type by tending to two disjoint disks spanning F, and r2. Notice that, as the T,, degenerate, the images of homotopically nontrivial loops shrink to zero in length. This observation is the basis of Courant's condition of cohesion described below. The first to cope with minimal surfaces of higher topological type (than the disk) and with the phenomenon of degeneracy was Douglas. He formulated the following result.

Theorem of Douglas. Let a denote the infimum of the Dirichlet integrals of all oriented connected surfaces of genus g spanning the given curves T,, ..., Tk and let a* be the corresponding infimum over all oriented connected surfaces of genus less than g and all oriented, disconnected surfaces of total genus f consisting of two or

more components spanning proper, non-empty, disjoint subsets of r, u... U Tk whose union equals T, u ... u Ij. If a < a* then there exists an oriented minimal surface of genus p and having I', u ... u Tk as boundary. Since in any topological class the infimum of Dirichlet's integral coincides with the infimum of area (see Chapter 4 for the case of a disk, and Tomi-Tromba [4] for the general case), the above theorem can be rephrased replacing Dirichlet's integral by area.

328

11. The General Problem of Plateau

Another approach to the general Plateau problem is due to Courant and Shiffman. They attacked the problem by staying in a class of surfaces of fixed topological type. Courant noted that degeneration of minimizing sequences can be excluded by imposing a certain additional condition, the so-called condition of cohesion. Courant's condition of cohesion. A family ,F of (differentiable) mappings X: M ---> R,

satisfies the condition of cohesion if there is a positive lower bound for the length of the images under any. "' e F of all homotopically non-trivial closed loops on M. In their work on minimal surfaces in Riemannian manifolds Schoen and Yau [2] used the same condition under the name of "incompressibility condition". While Courant applied his method only to surfaces with schlicht parameter domains Shiffman [3] later extended the method to the higher-genus case. He essentially proved the following result. Theorem (Shiffman). If there is a minimizing sequence for Dirichlet's integral in the class of all surfaces of genus spanning I'1, ..., Fk which satisfies the condition of cohesion then there is a minimizing minimal surface in this class.

Douglas' proof of his theorem was quite ingenious but not entirely satisfactory; in fact, some parts are not established at all. The proofs of Courant and Shiffman are completely stringent; however, Shiffman did not explicitly derive Douglas's theorem from his result; possibly he considered this more or less evident. A presentation of the Courant-Shiffman method can be found in Courant's treatise [15]. The case k = 2 is also presented in Nitsche's Vorlesungen [28]. Modern and complete proofs of the results of Douglas and Shiffman are given in Tomi-Tromba [5] and in Jost [6]. In the following section we want to present sufficient geometric-topological criteria for a system of rectifiable Jordan curves in R3 which guarantees the existence of a minimal surface of prescribed topological type spanning the configuration . This result will not be as general as the theorems of Douglas-Courant-Shiffman. On the other hand it is rather satisfactory as it replaces the "sufficient condition of Douglas" or "Courant's condition of cohesion" by concrete geometric conditions that can, in principle, be checked.

11.6 Existence Results for the General Problem of Plateau in l83 We want to derive a sufficient condition ensuring the existence of a minimal surface of prescribed topological type within a configuration of rectifiable Jordan curves F j.

11.6 Existence Results for the General Problem of Plateau in t

329

We first set up our variational problem from a different point of view which

allows the introduction of an artificial constraint, or "obstacle", in form of a 3-dimensional submanifold T with boundary which is of sufficient topological complexity depending on the topological type of surfaces we wish to produce. We will then restrict the class of admissible surfaces to those surfaces contained in the submanifold T. By a topological condition on the position of F in T we can ensure that homotopically nontrivial loops on an admissible surface are also homotopically nontrivial in T and therefore bounded below in length. This will guarantee (via the condition of cohesion) the existence of a minimizing surface which in general, however, need not be a minimal surface in IR3 since it can touch the boundary of T along portions of arbitrary size. If, however, T is H-convex, i.e., if the inward mean curvature of aT is nonnegative, then we may apply a maximum principle and conclude that a minimizing surface in T is actually contained in the interior

of T and hence is a minimal surface in R'. We treat the case of one boundary curve and oriented genus-one minimal surfaces in complete detail and shall later indicate the modifications necessary for tackling higher genus, several contours, and unoriented surfaces. We begin with a few preliminaries. Again let 21 be of class H' (M, lN), and let us consider Dirichlet's energy, where M is as before and

E(X, g) = 2 Jgs<, X.,)dy . Lemma 1. Let X be a map of Sobolev class H' (M, RN) and g a smooth metric on M.

(i) If the pair (d, g) is stationary for E with respect to all smooth variations of g compactly supported in M = M - iM, then the metric X*gN induced by the Euclidean metric gN is conformal to g, i.e..1*gN = Ag a.e. for some A >_ 0.

(ii) If the pair (.', g) is stationary for E with respect to all smooth variations of X compactly supported in M, then °..C' is smooth and g-harmonic.

Proof. We need only prove i) since again ii) is a standard result from the calculus of variations (see also chapter 4). With the abbreviations

y= det(gap),

lap = <X,-,

we compute for any smooth symmetric 2-tensor h = (ha,) on M that d9E(

{_ghv

g) =

+

JM 2

{_gvipgaa

= 2 fm

+ 19aelapgva} hvavdu' due.

F rom our hypothesis it therefore follows that gvalaflgea = 2gva

a.e. on M with Z = igaPlap,

330

11 The General Problem of Plateau

which can easily be rewritten as 1,,# = Ag.fi ,

11

It is now obvious that each critical point of E on a suitable space of pairs (.T, g) will furnish a solution to Plateau's problem, and vice versa. In what follows we shall solely be concerned with absolute minima of E. The next theorem, a basic result of this section, illustrates the importance of the cohesion condition formulated in Section 11.5. Theorem 1. Let M be a compact smooth surface which is not simply connected and has k > 1 boundary components C,, ..., Ck with the genus of the Schottky double 2M greater than one (this excludes the annulus), and let F,, ..., Tk be pairwise n e %, disjoint rectifiable Jordan curves in 11N. Furthermore, let a sequence (Y.,

be given where each X. is a mapping of class C° n H2 '(M, R') which maps C3 monotonically onto 1 (j = 1, ..., k), and where the g are smooth metrics on M. T We suppose that the Dirichlet integrals E(.1t , as well as sup are uniformly bounded and that the family satisfies the condition of cohesion. Then there exists a smooth metric g on M and a map . ' a HZ (M, I') such that .%"iay is continuous and maps each C; monotonically onto F j and such that

E(et, g) < lim inf

In case that M is oriented and that all X map C, onto T with a prescribed fixed orientation, then T can be chosen to map C; onto T in the same orientation. The proof of Theorem 1 will be carried out in several steps. The following theorem links the condition of cohesion with the hypothesis of Mumford's compactness theorem (see Section 11.4) on a lower bound of the length of closed geodesics on (M, g). This result is a simple consequence of the collar theorem of Halpern [1] and Keen [1]; its usefulness for minimal surfaces

was observed by Schoen and Yau [2], and in what follows we present their argument.

Theorem 2. Let (M, g) be a closed oriented surface with R(g) = -1 and let X: M -> RN be a map of class Co n H2(M, R') such that for all homotopically non-trivial closed C1-loops a on M the length of 9' o a is bounded below by 6 > 0. Then the length I of any closed geodesic y on (M, g)) can be estimated by

12min{1,282[ -2 arctg 0:] Remark. Corresponding estimates are obtained for unoriented surfaces by passing to the oriented cover.

11.6 Existence Results for the General Problem of Plateau in l83

331

Proof. By standard results in differential geometry (cf. Section 11.4), there is an isometry 0 of a neighbourhood U of y with a region Tin the Poincare half plane,

T = {re`O: 1
0<00<2,

(y)={ir: 1
and the points e`0, e" have to be identified. It is the content of the Halpern-Keen collar theorem that the area of T can be estimated in the form

21 cot 00>_.

(1)

5

Since the curves {reie: 1 < r < e`}, 00 < 0 < rr - 00, are homotopically nontrivial closed loops, we have by hypothesis

feI I ff,(r, 0)I dr > b

for almost all 0. Using Schwarz's inequality and integrating over 0, we obtain ea

62(7r - 200) <

re1

a

dr

1

Y drd0 <_ 2lE(.I, g).

r Jr fe'. The assertion of Theorem 2 then follows immediately from (1).

In the next lemma we show how the condition of cohesion extends from a surface with boundary to the corresponding closed symmetric surface. Lemma 2. Let 2M = M u SM be a symmetric surface where aM has the components C1,. .. , Ck. Let T'k be a collection of pairwise disjoint rectifiable Jordan curves in l with the minimal distance p > 0 of any two of them. For S > 0 denote

by a(b) the suprernum of the lengths of all shortest subares of T't u... u F joining any two points at spatial distance not exceeding b. (Clearly, a(b) -+ 0 as b -). 0.) Furthermore, let .1: 2M --. 1N be a symmetric continuous map which maps CC monotonically onto T, j = 1, ..., k, and let a be a homotopically nontrivial loop on 2M such that the length L(,' o a) < p. Then there exists a homotopically non-trivial loop ao on M with

L(91 o ao) <_ L(.f o a) + a(L(" o a)).

Proof. If a is totally contained either in M or in S(M), we may set ao = a or a0 = S o a, respectively. In the remaining case let a1, a2,... be the maximal (open) subarcs of a contained in S(M) with endpoints pj, q3 on M. We infer from IX(pi)

- X(q,)I <_ L(9' c a;) < L(. ' o a) < p

332

11. The General Problem of Plateau

that p; and q1 are always contained in the same component of 3M. We can therefore connect p; and q; by some arc f3; in OM with the property that 1 o /3; is the shortest subarc of r, u ... u Tk joining . (pj) and '(q;). If the arcs a; and f are homotopic in S(M) with fixed endpoints for all j, then also Sc, and S/3, = /i; are homotopic in M with fixed endpoints, and it follows that

_ a0

a Saj

on a-1(M)

on a, (S(M)) 1

(I = 1, 2, ... )

is a loop in M homotopic to a which satisfies L(' o ao) = L(Y o a). In the other case there is some j such that a; and flj are not homotopic in S(M) with fixed endpoints. Then a;$;1 is not contractible in S(M), and hence ao := S(a;/ij-t) _ (Sa1)/3 1 is not contractible in M. We obtain the trivial estimate L(."axo)
)+o(L(Ioa)),

thereby proving the lemma also in this case. We shall also need

Lemma 3. Let (M, g) be a smooth compact surface with boundary. Then there is a smooth closed surface (2M, g,) together with an involutive isometry S 0 id such that there is a natural inclusion (M, g) -+ (2M, g,) which is conformal. Moreover, we have 2M = M u S(M), and aM is precisely the fixed point set of S.

Proof. Let z be an atlas of M such that in local coordinates the metric tensor g is in isothermal form, gQp = )5 and, moreover, that the range of each boundary chart is the upper half unit disk {(ut, u2) a 112: (u1)2 + (u2 )2 < 1, u2 >_ 0}. Let M'

be a second copy of M, endowed with an atlas z' in which each chart of a is replaced by its complex conjugate. Then let 2M be the disjoint union of M and M' with the boundary points identified. The charts for points of aM = 3M' are constructed by glueing together each boundary chart of M with its complex conjugate in the obvious way. We thus produce an atlas a, on 2M, the transition maps of which are conformal maps in the plane and therefore real analytic. The map S assigns to each point of M the same point in M', leaving OM fixed. The metric g is extended from M onto 2M such that S*g = g. This extended metric, however, need not be smooth. In order to construct a smooth metric on M having S as isometry we cover OM = 3M' with coordinate neighbourhoods G...... G" from the atlas a,. In local coordinates of each Gk we define a 2-tensor gk by gap = Sap. Consider functions rlk of class Co (Gk), 1 < k < n, such that r7k = rjk 0 S,

and Elk < 1, Erik = I in a neighbourhood of 3M. We then define the metric g, on 2M by

gs=L1- k=1 IkJg+ Yrlkgk. k=1 It is obvious from the construction that on M the metric g, differs from g only by a conformal factor. This concludes the proof of the lemma.

11.6 Existence Results for the General Problem of Plateau in R

333

We can now turn to the Proof of Theorem 1. Replacing (_T, gn) by a subsequence we can assume that lim E(X,,, exists. By the preceding lemma we can conformally embed (M, gn) n-co

into a closed symmetric surface (2M, gn') of genus greater than one. Therefore we can apply Theorem 1 of Section 11.3 and replace g' by a symmetric metric G,, of Gauss curvature -1 that is conformally equivalent to Due to the conformal invariance of Dirichlet's integral we have E(ec, gn) = E(f, g') = E(.', Gn).

We now extend X,, to a symmetric mapping T'.: 2M --> RN. We know from Lemma 2 that the family again satisfies the condition of cohesion on 2M. By means of Theorems 2 and 1 of Section 11.4, we obtain the existence of symmetric diffeomorphisms f,,: 2M --> 2M satisfying f ,(M) = M, which are orientation preserving if M is oriented such that (after passing to a subsequence) ff*G converges in C`° to a smooth symmetric metric Gs. We observe that E(X.,

E(n o fn, f*G.). If we therefore replace X,, by X,, o f and g by we may assume that the sequence {gn} converges in C°° to a smooth metric g and, after passing to a further subsequence, that {Xn } converges weakly in H2 '(M, RN) to some ' e H2 '(M, RN ). Since any nonnegative continuous quadratic form on a Hilbert space is weakly

lower semicontinuous, we have E(. ', g) < lim inf E(Yn, g) = lim E(. ',,, gn). n_op

n-CO

It remains to be shown that the boundary values of X are continuous and map C; monotonically onto T, j = 1, ..., k. For this purpose it suffices to show that the boundary values of the sequence .I,, o f,, (again denoted by Gln) are equicontinuous. As in Lemma 2 we denote by c(8) the supremum of the lengths of all shortest subarcs of r1 u - u I'k joining any two points at a spatial distance not exceeding

6. Let then po e C; c OM be arbitrary, and fix some coordinate chart (G(po), 9) where cp(po) = 0 and cp: G(po) -+ ip(G(po)) _ {w = u' + iu2: Iw1 < 1, u2 >_ 0}.

For 0 < p < 1 let y(p) denote the arc in G(po) with endpoints p(p), q(p) e 8M which corresponds to a halfcircle of radius p in cp(G(po)) centered at 0. Since we

can assume that the metrics g converge uniformly to a smooth metric g, the Dirichlet integrals (with respect to the Euclidean metric) of .%"n o cp-t are uniformly bounded. We can therefore apply the Courant-Lebesgue lemma and obtain sequences pn,,, > 0 such that (2)

1 < P..,, < V

1

,

y

(n, v e N),

334

11 The General Problem of Plateau

and length it

(3)

where d(p)

0 for p -* 0. In particular, we have

I ° (p(Pv)) -

(4)

divide C1 into two subarcs C,, C;;,v where C., is were shorter than contained in G(po). If, for some pair (n, v), the arc 2,,(C',v), then we have by (4)

The points p(p j and

length 2(Cn,v) would form a closed loop homotopic and and the sum of the two arcs image under X. has a length not exceeding to C1 (and hence nontrivial) whose S(p,,,v) + o(S(p Y)). This contradicts the condition of cohesion provided that v is sufficiently large. We can therefore conclude that, for v > vo, the curve and hence is always shorter than

o(b(p,, ,)) In view of (2) this shows the equicontinuity of the theorem is proved.

at the point po e M, and

Observing the compact embedding of HZ(M) into L2(M) and Lemma 1 we immediately obtain the following generalization of Shiffman's theorem.

Corollary 1. Let M, F, ..., F. be as in Theorem 1, and let K be some compact subset of R". We consider the variational problem E(e', g) -+ min in the set of all pairs (.', g) where g is a smooth metric on M and ' a map of Sobolev class H2(M, R") with continuous boundary values mapping C1 monotonically onto T,

j = 1, ..., k, and with X (p) e K for almost all p e M. If this problem admits a where all X are continuous and satisfy the condition minimizing sequence (.,,, of cohesion, then the problem possesses a solution (s', g) such that ' is conformal with respect to g.

Remark 1. It follows from the proofs of Theorem 2, Lemma 2 and Theorem 1 that the condition of cohesion needs only to be checked for simple, piecewise smooth curves on M with finitely many interior and boundary edges meeting under acute angles. Remark 2. In order to avoid technical complications we have used the condition of cohesion only in connection with continuous maps.

We are now in a position to use our previous results to obtain existence theorems for higher genus minimal surfaces in R3.

Let us start with a topological consideration based on the fact that the fundamental group of an oriented surface of genus z >_ 1 with one boundary

11.6 Existence Results for the General Problem of Plateau in R3

335

component is a free group on 2g generators, and that the fundamental group of

a solid 2g torus is also a free group on 2g generators, see Massey [1]. The following theorem then is an easy consequence of algebraic results due to Zieschang [1, 2]. Theorem 3. Let M be an oriented surface of genus 1 with one boundary component and let T be a solid 2-torus (i.e. the connected sum of two tori). Furthermore let r

be a Jordan curve in T and ': M - T a continuous map which maps am monotonically onto F. Finally, suppose a base point po e 3M is fixed. Then in order that the induced map

*: nl(M, Po) - It 1(T, x0),

xo =

(Po),

be an isomorphism it is necessary and sufficient that there are two generators of 7c1(M, po) and two corresponding generators of 7rl(T, xo) such that the classes of 3M in 7t1(M, po) and of F in ir1(T, xo) respectively are represented by the same word with respect to the above sets of generators. Proof. Necessity: Let c1, c2 be free generators of ir1(M, po). If ,, is an isomorphism, then clearly X*(cl), T'*(c2) generate nt(T, xo) freely, and obviously the

word for [3M] with respect to c1, c2 is the same as for [T] = ; '*([3M]) with respect to .'*(c1), .*(c2). Sufficiency: Let us suppose that the hypothesis on F is fulfilled for generators

c1, c2 of irl(M, po) and generators 71, y2 of 7rI(T,xo). Then we can define an isomorphism cp: 711(T, xo)- it1(M, Po)

such that p(y,) = cj for j = 1, 2. By hypothesis we have

q ([r]) = [3M]. From the canonical models of surfaces (see Massey [1]) we see that in the oriented case we can now choose generators A, B, of 7r1(M, po) such that

[3M] = ABA-1B-1. That is, OM is the commutator with respect to the basis A, B. If follows that (X*(ABA`B-1) = QP([X*(3M)]) = rp([T]) = [3M] =

ABA-1B-1

i.e., (pX, fixes the commutator ABA-1B-1. It now follows from Zieschang's results [1], [2] that cpX,, is an isomorphism of ir1(M). O We now have the following existence result. Theorem 4. Let M be an oriented surface of genus 1 with one boundary component

and let F be a rectifiable Jordan curve in 683. We assume that (i) there exists a

336

11. The General Problem of Plateau

H-convex solid 2-torus T of class C3 in l3 such that F c T, and that (ii) with respect to suitably chosen base points and generators, the class of I in n,(T) is represented by the same word as the class of 3M in 71,(M), respectively. Then there exists a minimal surface M --+ l3 mapping OM topologically onto I' and such that . " (M) is contained in T.

Proof. We consider the class of all pairs (', g) where g is a smooth metric on M and : "' E H'2 (M, R3), X (M) c T, and ' is mapping 3M onto F continuously and monotonically. We must show that this class is not empty. By approximation it clearly suffices to produce a continuous map : M --+ T satisfying the boundary condition. By the classification of surfaces (see Massey [1]) we obtain a topological model of M in form of an annulus c whose inner boundary circle (of radius z ) corresponds to 3M, whereas the outer circle (of radius 1) is broken up into 4 consecutive segments C...... C4 such that all vertices are identified with one point a, and the segments are identified according to the rule C3 = C1 ', C4 = CZ'. It follows that A := C,, B := C2 generate the fundamental group of M with base point a and that 3M is freely homotopic to ABA-'B-'. Connecting an arbitrary base point b on 3M with a by means of some simple arc y, and replacing A, B by

their conjugate A := yAy-', B := yBy-', we see that A, B generate rt,(M, b) and that ABA-'B` is the class of 3M in n,(M, b). It follows therefore from hypothesis (ii) of the theorem that there are closed loops a, /3 based at some point

c c Tgenerating n,(T, c) and such that [T] = a/t a-'/3-'. Let h: [0, 1] x S' -+ T be a homotopy between l and y = a/3 a-' f3'. We can clearly choose the parametrization of y in such a way that Y1c, = a,

YIc2 = 13,

YIc, = a',

YIc,

Then we define a map ':.sad -+ T by e'(re`B) = h(r - 1, e'°). It is clear from the construction that .1 respects the identifications of .4 which make sad a topological equivalent of M, and therefore 3' can be considered as a map .': M --+ T with '(8M) =.r. As the class of admissible maps is not empty, we can now pick an energy minimizing sequence

g }. By approximation we may

assume all X. to be continuous. It then follows from Theorem 3 that any homotopically nontrivial loop on M is mapped by .I into a homotopically nontrivial loop in T whose length is therefore bounded below by a positive constant depending only on T Thus .C' satisfies Courant's condition of cohesion. Corollary 1 now provides the existence of minimizing conformal pair (X, g). From the regularity theory of variational inequalities, cf. Tomi [4], Hildebrandt [12, 13], and also Section 7.12, No. 15 we obtain that X e C'(M - 3M) and X e C" (M - (8M u ?"-' (0T))). If both sets

(M - 0M)n,('(8T) and M - (8M u X-'(aT)) ' That is, the mean curvature of OT with respect to the inward normal to OT is nonnegative.

11.6 Existence Results for the General Problem of Plateau in 1

337

were non-empty, then there would exist a point where the surface . ' touches OT tangentially. This, however, contradicts E. Hopf's strong maximum principle if T is H-convex (see Hildebrandt [11], Theorem 8 and Section 6.5, no. 4). Let us next consider the case when _T(M) c aT. Then ' is a harmonic map from (M, g) into the submanifold OT of D3 which, by assumption, is of class C3. It follows

that X E C3(M - OM) and that, apart from isolated branch points,

.

'

is

immersed, and its mean curvature H with respect to the interior normal v equals that of OT whence H > 0. On the other hand, since ' is area minimizing among all surfaces in T with boundary r, we obtain from the first variation of area that .

dA(.T)<,N> = -2 fm <,Y, vH> dvol > 0 for all compactly supported smooth variations V with <&, v> >- 0. It follows that

H < 0, and hence H = 0, i.e., " is a minimal surface. In the last case when "(M - 8M) c T - OT it is clear anyway that .?' is minimal. Thus Theorem 4 is .

proved.

Theorem 4 gives the existence of a surface whose boundary is the same as a classically known physical example (see Fig. I in the introduction to Chapter 4) since the boundary of this curve can be considered as the commutator with respect to a basis of a solid 2-torus which is the same as [aM], M an oriented surface of genus one.

Fig. 1. (a)-(c) Three types of H-convex surfaces as building blocks lead to minimal q-tori of arbitrary genus g, e.g. to the classical minimal surface in the configuration (d).

11. The General Problem of Plateau

338

The example pictured in Fig. I of Chapter 4 is embedded. For our existence

theorem it can be deduced from results of R. Gulliver [7] that the minimal surfaces resulting from existence Theorem 4 are immersed on the interior. How-

ever, J. Jost [9, 17] has shown in the situation of Theorem 4 and under the additional hypothesis that the curve T lies on the boundary of the torus T that r spans an embedded minimal surface.

These remarks apply to a generalization of Theorem 4 concerning the question of existence for oriented and unoriented minimal surfaces spanning one

or several contours. For the case of several contours we refer the reader to Tomi-Tromba [4]. These cases may require the Teichmiiller theory for orientable

and nonorientable symmetric surfaces of genus one (which we have not developed here) and of genus greater than one.

In the case of one boundary curve I' one can also prove the following existence result (see Tomi-Tromba [4], p. 70). Theorem 5. Let M be a surface of genus y > I with one boundary component and let T be a solid mean convex q-torus (that is, the connected sum of. q-solid tori) of class C3 with q = 2y in case M is orientable and q = q if not. Then both fundamental groups will have the structure of a free group on q generators. Assume either of the following hypotheses on I' and M. (i) M oriented of genus y and I' homotopic in T to the commutator product v

I [aj, fij] for some set at, ..., j=1

fl

of free generators of rz1(T);

(ii) M non-orientable of genus y and

in T to II xj where a1, .... j-1

av are free generators of 7t1(T); (iii) M non-orientable of genus Y = 2k + I and I' homotopic in T to k

H laj, l1),z

j=1

where a1, .... Nk, y are free generators of n1(T); k--I

1l

j=1

(iv) M non-orientable of genus y = 2k, k >_ 1, and I' homotopic in T to 10(i,

/

11

Nj]Jak//

F'k ak

1 Rfk, for generators a1, ..., fk of n1(T)

Then there is a minimal surface :f of the topological type of M spanning r, i.e., a minimal surface Y: M -+ R3 mapping (IM topologically onto T such that .4'(M) is contained in T and .x?" restricted to the interior of M is an immersion.

Analogously to Theorems 4 and 5, Tomi and Tromba [4] have proved the following result for boundaries r consisting of several components. Theorem 6. Let M be a surface of genus y and of k = m + I boundary components, and let r = be a boundary configuration consisting of m + 1 disjoint Jordan curves contained in some solid, mean-convex, (q + m)-torus T of E3 where

11.7 Scholia

339

q = 2g if M is orientable and q = y if not. Choose some base point x e T and paths b; connecting x with points x; a T, and define the loops Qj = bjFjjb; 1, j = 0, 1, ..., m. Assume also that there are elements a,, /3k, ..., a,, /3y E ar1(T x) in the oriented case and a,, ..., ay a n,(T, x) in the unoriented case such that Q,, ..., am, a,, ..., fl, (or al, ..., am, al, ..., ay respectively) generate n,(T, x) and that the relations a0 ... am F1 [aje J3j] = 1 j=1

or

ao ... am H ajf = 1 J=1

hold in the respective cases. Then there exists a minimal surface X: M -+ T such that _X(OM)=

11.7 Scholia 1. If one wants to study the classes of all conformally equivalent structures on a general topological 2-manifold M, one obviously has to study the moduli space A(M). Unfortunately, the topology of the moduli space is fairly complicated; in particular, it does not have the structure of a manifold. This led Teichmuller to introduce another set which has become known as "Teichmuller space" (M). This space has much better properties than the moduli space and is accessible to the methods of real and complex analysis. Classical tools of Teichmuller theory are quasiconformal mappings whose study was initiated by Grotzsch, Ahifors and, in a completely different way, by Morrey, and the theory of Fuchsian and Kleinian groups. For a presentation of classical Teichmuller theory we refer to the monographs of Lehto [1] and Kra [1]. More recently, harmonic diffeomorphisms have proved to be an at least equally valuable tool which admits a more geometric

approach to Teichmuller theory. We refer the reader to presentations of this approach in the forthcoming monographs of Tromba [20] and of Jost [17]. These ideas will certainly play an important role in the future development of global analysis as they show a way to handle bifurcation processes when geometric objects change their topological type. 2. Chapter 11, essentially prepared by A. Tromba, is a minor revision of his joint work with F. Tomi [5]. For a more penetrating study of the subject, the reader should consult Tromba's original papers as well as his joint work with A. Fischer and F. Tomi. 3. The first to study general Plateau problems for minimal surfaces of higher topological type was Jesse Douglas; his work is truly pioneering, and his ideas and insights are as exciting and important nowadays as at the time when they were published, more than half a century ago. It seems that Douglas was the first to grasp the idea that a minimizing sequence could be degenerating in topological type, and he interpreted such a conceivable degeneration as a change in the conformal structure. He based his notion of degeneration (which he termed "reduction") on the representation of Riemann surfaces as branched coverings of the sphere. Then degeneration meant "disappearance of branch cuts". The

340

11. The General Problem of Plateau

intuitive meaning of degeneration is the shrinking of handles and the tendency to separate the Riemann surface into several components. Since degeneration is unavoidable in general, Douglas had the idea of minimizing not over surfaces of a fixed topological type but also over all possible reductions of the given type. In this set of Riemann surfaces of varying topological type Douglas introduced a notion of convergence as convergence of branch points in the representation of the surfaces as branched coverings of the sphere. The compactness of this set of Riemann surfaces seemed to be a trivial matter to him since his whole argument

reads: "This is because the set can be referred to a finite number of parameters, e.g., the position of the branch points ...... This reasoning is, however, rather

inaccurate since the position of branch points alone does not determine the structure of the surface. Douglas also argued on a rather intuitive level when it

came to the lower semicontinuity of Dirichlet's integral with respect to the convergence of surfaces. Taking the compactness of the above set of Riemann surfaces and the lower semicontinuity of Dirichlet's integral for granted, it is then obvious that an absolute minimum of Dirichlet's integral in the class of surfaces considered by Douglas must be achieved, either in a surface of desired (highest) topological type or in one of reduced type. In this way Douglas was led to his celebrated theorem that we stated in Section 11.6. Courant and Shiffman gave the general results of Douglas a solid basis by solving the variational problem within a class of surfaces of fixed topological type. In his treatise [15], Courant gave a very clear exposition of his method for minimal surfaces defined on schlicht domains. A careful presentation of the Plateau problem for two boundary curves can be found in Nitsche's Vorlesungen [28]. In particular, he derives the following result of Douglas [13] : If two closed recifiable Jordan curves I', and F2 are linked, then the sufficient condition of Douglas is satisfied. Consequently there is an annulus-type minimal surface bounded by F, and F2 whose area is less than the sum of the areas of two area minimizing surfaces of the type of the disk which are bounded by T1 and F2, respectively (see Fig. 1).

As we have remarked before, modern and complete proofs of the results of Douglas and Shiffman were given by Tomi-Tromba [5] and Jost [6].

Fig. 1. An annulus-type minimal surface bounded by two interlocking closed curves.

Bibliography

The following references are not complete with respect to the early literature but cover only some of

the essential papers. A very detailed and essentially complete bibliography of the literature on two-dimensional minimal surfaces until 1970 is given in Nitsche's treatise [28] (cf. also Nitsche [37]). Nitsche's bibliography is particularly helpful for the historically interested reader as each of its more than 1200 items is discussed or at least briefly mentioned in the right context, and the page numbers

attached to each bibliographic reference make it very easy to locate the corresponding discussion. We have tried to collect as much as possible of the more recent literature and to include some cross-references to adjacent areas; completeness in the latter direction has neither been aspired nor attained. We particularly refer the reader to the following Lecture notes: MSG:

Minimal submanifolds and geodesics. Proceedings of the Japan-United States Seminar on Minimal Submanifolds, including Geodesics, Tokyo 1977, Kagai Publications, Tokyo 1978

SDG:

Seminar on Differential Geometry, edited by S.T. Yau, Ann. Math. Studies 102, Princeton 1982

SMS:

Seminar on Minimal Submanifolds, edited by Enrico Bombieri. Ann. Math. Studies 103, Princeton 1983

TVMA: Theorie des variete minimales et applications. Seminaire Palaiseau, oct. 1983-juin 1984. Asterisque 154-155 (1987) CGGA: Computer Graphics and Geometric Analysis. Proceedings of the Conference on Differential Geometry, Calculus of Variations and Computer Graphics, edited by P. Concus, R. Finn, D.A. Hoffman. Math. Sci. Res. Inst. Conference Proc. Series Nr. 17. Springer, New York 1990

We also refer to the following report by H. Rosenberg which appeared in May 1992; therefore its bibliographical references are not included in our bibliography: Rosenberg, H.

Some recent developments in the theory of properly embedded minimal surfaces in IV. Sbminaire BOURBAKI, no. 759 (1992), 64 pp. Abikoff, W.

1. The real analytic theory of Teichmuller space. Lect. Notes Math. 820. Springer, Berlin Heidelberg New York 1980 Abresch, U.

1. Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374, 169-192 (1987)

Adam, N.K. 1. The physics and chemistry of surfaces. 3rd. edn. Oxford Univ. Press, London 1941 2. The molecular mechanism of capillary phenomena. Nature 115,512-513 (1925) Adams, R.A.

1. Sobolev spaces. Academic Press, New York 1975 Adamson, A.W. 1. Physical chemistry of surfaces. 2nd edn. Interscience, New York 1967

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1. Riemann's mapping theorem for variable metrics. Ann. Math. (2)72,385-404 (1960) Ahlfors, L., Sario L. Riemann surfaces. Ann. Math. Stud., Princeton Univ. Press, Princeton 1960 1 Alexander, H., Hoffman, D., Osserman, R. 1. Area estimates for submanifolds of euclidean space. INDAM Sympos. Math. 14, 445-455 (1974) Alexander, H., Osserman, R. I Area bounds for various classes of surfaces. Am. J. Math. 97, 753-769 (1975) Alexandroff, P., Hopf, H. 1. Topologie. Erster Band. Springer, Berlin 1935 Allard, W.K. 1. On the first vanation of a varifold. Ann. Math. (2) 95, 417-491 (1972) 2. On the first variation of a manifold: boundary behavior. Ann. Math. 101, 418-446 (1975) Allard, W.K., Almgren, F.J.

1. Geometric measure theory and the calculus of variations. Proceedings of Symposia in Pure Mathematics 44. Amer. Math. Soc., Providence 1986 Almgren, FJ. 1. Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem Ann. Math. (2) 84, 277-292 (1966) 2. Plateau's problem. An invitation to varifold geometry. Benjamin, New York 1966 3. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4, no. 165 (1976) 4. The theory of varifolds; a variational calculus in the large for the k-dimensional area integrand. Mimeographed Notes, Princeton 1965 5. Minimal surface forms. Math. Intelligencer 4, 164-172 (1982) 6. Q-valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Am. Math. Soc. 8, 327-328 (1983) and typoscript, 3 vols, Princeton University 7. Optimal isopenmetric inequalities. Indiana Univ. J. 35, 451-547 (1986) Almgren, F.J., Simon, L. 1. Existence of embedded solutions of Plateau's problem. Ann. Sc. Norm. Super. Pisa Cl. Sci. 6, 447-495(1979) Almgren, F.J., Super, B. 1. Multiple valued functions in the geometric calculus of variations. Asterisque 118, 13-32 (1984) Almgren, F.J., Taylor, J.E. 1. The geometry of soap films and soap bubbles. Scientific American 82-93 (1976) Almgren, F.J. Thurston, W.P. 1. Examples of unknotted curves which bound only surfaces of high genus within their convex hulls. Ann. Math. (2) 105, 527-538 (1977)

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Index of Names Page numbers in roman type refer to this volume, those in italics to volume I

Agmon, S. 9 Ahlfors, L. V. 185, 272 Aleksandrov, A. D. 320 Alexander, H. 422 Almgren, F. J. 281, 284, 285, 424 Alt, H. W. 250, 284, 291-292; 279, 358, 362, 366 Andersson, S. 195 Athanassenas, M. 423

Barbosa, J. L. 88, 135. 195 Barthel, W. 149 Beckenbach, E. F. 421 Beeson, M. 197; 286, 292 Beltrami, E. 51, 193 Bernoulli, J. 51 Bernstein, S. 85, 86 Bers, L. 50, 85 Berwald, L. 52 Bianchi, L. 175, 194 Bieberbach, L. 272 Bjorling, E. G. 193 Blaschke, W. 48, 133, 149, 421

Bliss, G. A. 349 Blum, Z. 195 Bohme, R. 270-271, 292-296, 424 Bolza, 0. 52, 349 Bombieri, E. 86, 284, 424 Bonnet, 0. 50, 193

Bovin,J. 0.

195

Brezis, H. 138-139; 278 Biich, J. 286 Burago, Y. D. 422

Caffarelli, L. A. 139 Calabi, E. 100 Callahan, M. J. 192, 198 Caratheodory, C. 59; 349 Carleman, T. 421

Catalan, E. 193 Chern, S. S. 48 Cheung, L. F. 43 Choe, J. 269 Christoffel, E. B. 51 Ciarlet, P. G. 229 Cohn-Vossen 177 Colares, A. G. 135, 195 Concus, P. 229 Coron, M. 278 Costa, C. J. 195 Courant, R. 43, 128, 133, 328, 340; 76, 221, 271, 278-279, 289-290, 293, 365, 423

Darboux, G. 52, 133, 135, 149, 194, 277

Davids, N. 365 De Giorgi, E. 86, 284 Dierkes, U. 292; 87, 278, 424-426 Do Carmo, M. 48, 88 Dombrowski, P. 52, 80 Douglas, J. 9, 340; 221, 277 Dubrovin, B. A. 48 Dziuk, G. 131-132, 141, 197,229;424 Earle, G. 307 Eberson, L. 195 Ebin, D. 306 Ecker, K. 292 Eells, J. 307 Eisenhart, L. P. 26, 52 Enneper, A. 133, 193 Ericsson, B. 195 Euler, L. 48, 49, 51

Federer, H. 292; 284 Feinberg, J. M. 395, 422 Finn, R. 85

Fischer, A. E. 298, 300-304,303-304,306, 313-314,326,339 Fischer-Colbrie, D. 87, 88 Fischer, W. 195 Fleming, W. H. 284 Fomenko, A. T. 48 Frehse, J. 136, 138-139 Fujimoto, H. 185

Gackstatter, F. 197 Gage, M. 424 Galilei, G. 420 Galle, A. 52 Garnier, R. 277 Gauss, C. F. 49-52 Gergonne, J. D. 276, 345 Gerhardt, C. 138, 139 Gericke, H. 420 Germain, S. 53 Giaquinta, M. 139; 245, 349

Gilbarg, D. 4, 85, 86 Giusti, E. 140; 85, 86, 284 Glaeser, L. 251 Goldhorn, K. 131 Gornik, K. 138 Goursat, E. 116 Greenberg, M. J. 272, 311 Gromoll, D. 30, 48, 178 Gromov, M. 424 Grater, M.

131; 344, 366,

417, 424-425

Gulliver, R. 284, 291-292, 338; 278-279, 282, 288, 290, 358, 366, 424

Haar, A. 85, 277 Haefliger, A. 88 Halpern, N. 330

Hardt, R. 282, 285 Harth, F. P. 97; 365

398

Index of Names

Hartman, P. 141, 196 Harvey, R. 86, 88 Hattendorf 192 Haubitz, I. 149 Heinz, E. 106, 129-130, 139, 196-197; 70, 85, 86, 278, 293, 343, 422 Heppes, A. 300

Hicks, N. J. 48 Hilbert, D. 177 Hildebrandt, S. 41, 55, 106, 129-131, 136-139, 229, 235, 292, 336-337; 86-87, 215, 245, 278, 288, 297, 301, 349, 365-366, 417, 422-425 Hoffman, D. A. 135, 192,

195, 198-199, 422 Hopf, E. 85 Hopf, H. 320 Hyde, S. T. 195

Jager, W. 106, 130, 131, 197 Jarausch, H. 229 Jenkins, H. 208 Jorge, L. P. M. 197, 198, 199

Jorgens, K. 85 Jost, J.

132, 197, 328, 338-340; 50, 87, 271,

344-345,365

Karcher, H.

132, 135, 195,

199, 208-209, 217, 366 Kaul, H. 136, 137; 278, 297, 422, 424 Keen, L. 330 Kellogg, O. D. 130 Kinderlehrer, D. 33, 129,

131,136,138-140;302 Klein, F. 52 Klingenberg, W. 30, 48, 178 Kobayashi, S. 48 Koch, E. 195 Koebe, P. 335 Koiso, M. 294 Korn, A. 85 Kra, I. 339 Kruskal, M. 291 Kiister, A. 328, 365, 422,

423,424 Lagrange, J. L. 49, 53, 192, 276

Lambert, J. H.

49

Laplace, P. S. 50, 53 Larsson, K. 195 Lawson, H. B. 85, 86, 366 Levy, P. 271, 290 Lehto, O. 339 Leichtweiss, K. 48, 195 Lemaire, L. 278 Lesley, F. D. 129, 284; 279 Levi-Civita, T. 51 Lewerenz, F. 293 Lewy, H. 41, 128, 130, 136, 139-140,292;424 Lichtenstein, L. 130; 50 Lidin, S. 195 Lilienthal, R. 133, 193, 194 Lin, F. H. 283 Li, P. 422 Lipkin, L. J. 365 Lipschitz, R. 51

Marx, I. 197 Massari, U. 85 Massey, W. 335, 336 Meeks, W. H. 135, 192, 195-196, 198-199, 282-283, 291, 344, 365 Mensbrugghe, van der, G. 292

Meusnier, J. B. M. C.

48,

50, 53 Meyer, W. 30, 48, 178 Minding, F. 51

Miranda, M. 85 Mittelmann, M. D. 229 Monge, G. 48, 49, 52 Morgan, F. 288, 422 Morrey, C. B. 41, 47, 54, 112; 4, 50, 278, 298 Mo, X. 89, 185 Mumford, D. 315 MUntz, G. H. 85

292, 302, 343, 365-366, 372, 417, 421-424 Nomizu, K. 48 Novikov, S. P. 48

Osserman, R. 284, 291-292; 85-86, 89, 185, 196-197, 279, 358, 421-422, 424 Otto, F.

251, 292

Painleve, P. 130 Palais, R. 344 Peng, C. K. 88 Pepe, L. 139 Peterson, I. 193 Pinkall, U. 177, 366 Pitts, J. T. 344-345, 366

Plateau, J. A. F. 299 Poenaru, V. 286 Polthier, K. 195 Quien, N. Radd, T.

286, 294 85, 221, 270, 278,

279, 421

Reid, C. 277 Riemann, B. 50-52, 133, 192-194 Riesz, F. 128; 265 Riesz, M. 128; 265 Ritter, F. 365 Robbins, H. 423 Rosenberg, H. 196 Royden, H. L. 284,

291-292;279,424 Ruchert, H. 286, 294 Rummler, H. 88 Sacks, J. 344, 366 Sasaki, S. 138 Sauvigny, F. 197, 235; 293, 294

Neovius, E. R. 192, 277 Nesper, R. 195 Nielsen, J. 300 Ninham, B. W. 195 Nirenberg, L. 9, 47, 83; 185, 302

Nitsche, J. C. C.

33, 128-129, 131, 138, 140, 229, 292, 328; 83, 85-86, 88-89, 100, 133, 149, 175, 194, 215, 270, 276-279,

Scherk, H. F. 193 Schiffer, M. 422 Schneider, R. 138; 270 Schnering, H. G. v. 195 Schoen, A. H. 195, 215 Schoenflies, A. 212 Schoen, R. 328, 330; 87-88, 288, 365, 422 Schubert, H. 311 Schiiffler, K. H. 293, 295 Schumacher, H. C. 49

Index of Names

Schwarz, H. A. 128, 197; 83, 100, 133, 140, 149, 175, 193-194, 212, 339 Serret, J. A. 193 Serrin, J. 208 Shiffman, M. 328 Simon, L. 87, 282, 285, 344, 366, 424 Simons, J. 284 Smale, N. 291 Smith, F. 344, 366 Smyth, B. 339, 366 Solomon, A. 302 Spivak, M. 48 Springer, G. 76 Spruck, J. 278, 282 Stackel, P. 52 Stampacchia, G 136, 139 Steffen, K. 278 Steinmetz, G. 229 Sterling, I. 366 Stessmann, B. 212 Strohmer, G. 293 Struwe, M. 271, 278, 279, 344, 366 Sullivan, D. 88

Tallquist, H. 277 Tausch, E. 424 Taylor, J. 300, 302 Thiel, U. 293, 295 Thurston, W. P. 281 Tolksdorf, P. 298, 366, 368 Tomi, F. 106, 129-130, 136, 298, 315, 327-328, 336, 338-340; 271, 282, 286, 293, 295, 297, 355, 362, 366 Tonelli, L. 221

Tromba, A. 298, 300, 303-304,306,313-315, 326-328,338-340;215, 270-271, 282, 286, 293-296, 301 Trudinger, N. S. 4 Tsuji, M. 128

Uhlenbeck, K. 344, 366 Vekua, I. N. 147; 4, 50 Vogel, T. I. 423 Volkmer, R. 149 Voss, K. 195

399

Wagner, H. J. 229 Warner, F. 48, 81 Warschawski, S. E. 33, 129, 197

Weierstrass, K.

133, 193,

194, 277

Weingarten, J 51 Wente, H. C. 278 Weyl, H. 51, 52 White, B. 296, 422, 424 Widman, K.-O. 87 Wintner, A. 141, 196 Wohlgemuth, M. 195 Wohlrab, 0. 229; 379 Xavier, F.

185, 199

Yau, S. T. 328,330;87, 282-283, 291, 344, 365, 422 Ye, R. 131, 132; 366 Young, L. C. 50, 53

Zalgaller, V. A. 422 Zieschang, H. 335

Subject Index Page numbers in roman type refer to this volume, those in irulie to volume I

absolutely continuous (AC) ACM-representative 305-306 boundary values 260, 310 functions in the sense of Morrey

(ACM) 305-306 representative 305-306 ACM-representative 305-306, 312, 316 adjoint surface 91-92 admissible boundary coordinates (1e, g} 62 centered at x() 62 normalization 63 admissible functions (see also class of admissibble functions) 232, 133, 252, :56, 257-258. 297, 306, 313, 318-321 generalized admissible sequence 323

admissible variation type I (inner variation) 330 type II (outer variation) 330 Almgren-Simon theorem 282 Almgren-Thurston example 281 almost complex structure 300 canonical 300 Alt, Gulliver, Osserman-theorem 279 Alt-Tomi theorem 356-357 analyticity of a minimal surface 63 analyticity of the movable boundary main result 273 annulus-type solutions of the general Plateau problem 294, 340 stationary surfaces in (T, S) 204-205 a priori bounds differential inequalities 21-32 harmonic functions 7-21 minimal surfaces see m.s. Poisson equation 7-21 a priori estimates area

384, 388, 390, 395-396, 405-406, 407, 414-418

derivatives 402-405

length of the free dace

t96-?V'. 4(1<_4t)n,

407, 414-418. 424 area

9, 227- 234

absolute nununa of 80-85, :53 Hilbert's independent Integral of the 83 of a minimal surface 104 relative minima of 80-85 area element Q. 20 associate surface 96-1(10, 114 Assumption (A) ofChapter5 on ti 1;,(S) 311-312 (GA) of Section 4.7 259 of Chapter 5 on supporting sets 306 of Section 62, Enclosure Theorem 11 379 of Section 6.2 on 11

372

(Al) of Section 8.1 143 (A2) of Section 8.1 143 (A3) of Section 8.1 156 (A) of Section 8.2 164 A of Section 8.3 174

A of Section 8.4 187 A of Section 9.3 206 (B) of Section 8.2 165-166 (B) on support surfaces 62

asteroid on Henneberg's adjoint surface 204-205 asymptotic expansions of minimal surfaces see nt.s. of solutions of differential inequalities 142-162, 196-197 main theorems 143, 144, 157 Section 8.1, assumption (A 1) 143 Section 8.1, assumption (A2) 143 Section 8.1, assumption (A3) 156 asymptotic expansions at boundary branch points minimal surfaces

117-121

order of branch point tangent

119, 121

120-121

tangent plane 120 asymptotic line (curve)

21, 22, 23

Subject Index of a minimal surface

96-99, 110, 114, 125,

132, 134-135, 193, 200 of Goursat transform 116 of the adjoint surface 99 of the associate surfaces

99

axis of curvature 14 axis of symmetry (see also line of symmetry) 123 balanced curves 337 Beltrami differentiator first 42, 51 second 43, 51 Beltrami operator 43-44 of the Gauss map 73 bending of minimal surfaces 96-100 of Catalan's surface 98, 128-129 of Enneper's surface 97, 147-149 of Henneberg's surface 164-166 of the catenoid 141, 142, 143 of the helicoid 141, 142, 143 bending of the frame 199-206 Bernoulli's theorem 51 Bernstein's theorem 65-70, 86-87, 185 in IIt2 in IR°

67, 185

Biich

286

86-87 bifurcation Beeson-Tromba 286 Nitsche 285 Ruchert 285-286 bijective equivalence ,ni and le 302

.4 and 11/?

302 302 4/9= 303

'B and .,11/91

X_1 and 4_ 1/2 and holomorphic quadratic differentials 305

.,!!_ 1/9,''/20, and 9'

306

binormal vector 14 Bjorling's problem 202; 120-121, 193 Schwarz's solution 121 uniqueness

121

Bohme's 4n+e-theorem 270 Bohme-Tromba index theorem 271, 294-295 Bombieri-De Giorgi-Giusti theorem 85-86 Bonnet's transformation see bending boundary behaviour of minimal surfaces with rectifiable boundaries 259-266 of solutions of 91(1', S)

322

of solutions of 9(17, S), 9+(S) 322 of the adjoint surface 259-260 boundary branch points 279, 357

401

boundary class [XI c] for X e '(S) 312 boundary configuration 224. 255

(r,S)

255

weakly connected

389-390

symmetric (T, S) 206 boundary continuity differential inequalities 27, 30-31 of harmonic functions 14-15, 20 of minimal surfaces see minimal surfaces boundary curve 231 rectifiable 233-234 boundary estimates differential inequalities 27, 28-29 of harmonic functions 16, 20 of minimal surfaces see minimal surfaces boundary frame see boundary configuration boundary regularity of minimal surfaces see minimal surfaces boundary values [XI c]

312

absolutely continuous

equicontinuous

free

260

238

57, 255-256, 318-322

homotopy classes 305-318, especially 312 integration by parts 266 length of the free trace 396-420 limits 237-239, 257-259, 322-326 natural boundary classes 312 normal derivatives 260 of an H}-function 305-318 rectifiable 259-260 tangential derivatives 260 three-point condition 236, 238-239 total variation 259-260 trace theorems 307, 310 uniform continuity 236-237 Bout surfaces 149 Boy surface 177 branch point(s) of a minimal surface 92-93 boundary 279 boundary b.p. see minimal surfaces false 273, 290-291; 279, 358 interior 101-107, 110, 114, 279-280 nonexistence of boundary branch points 357

normal form at a 102-104 number of 126-127, 138 of even order 120-121, 273, 289 of odd order 120, 273, 275 order of a 93 true

273, 290-291; 279, 358

branch point of an H-surface see Sections 7, 8 bridge principle 43, 136 see also bridge theorem bridge theorem 43, 136; 290-292

402

Subject Index

capillarity 85 capillary phenomena 85, 221 Catalan's surface 98, 120, 121, 169-174, 193

bending 98, 128-129 Bi6rling's problem

169

branch point 103 cycloid 120, 169, 173 lines of symmetry 170-172, 268 Planes of symmetry catenary 135, 349-351 125, 170-172, 267 catenoid(s) 135-140, 192, 195, 203 adjoint 138 as a surface of revolution 135-137 associate surfaces 140, 141 bending 141 fence of 209-210 With handles 210 Cauchy-Riemann

equations 301; 260 center of curvature 14 center of the osculating sphere 14 Chen-Gackstatter surface 210, 212 Cheung's example 43-45

chord-arc condition 45 Christoffel symbols 26, 27, 28, 29 of the first kind 26 of the second kind 26 circle of curvature class

-*(1)

14

253,255 254

--'V Vj, 12)

254

S) in Section9.10, no. 3 237 of Pseudoregular surfaces 691(52) 121-122 class of admissible sces

`1(f)

W* (r)

`B (T

232

second fundamental form 17 third fundamental form 17 cohesion condition see condition of cohesion coincidence set 139-140 collar theorem 329, 331 commutator 41-42 compactness of a minimizing sequence 238-239 of minimizers 364-365 theorem of Federer-Fleming 284 complete figure 350 complete Riemannian manifold 178-179 complex structures 299; 175 space of, = f(M) 299 space of symmetric, Ws 308 symmetric 308 condition of cohesion 328, 330-332 conditions (Cl), (C2), (Cl*) of Section 9.10 236, 237

cone theorem 371 configuration see boundary configuration conformal 35 conformal functions 34 conformal mapping 49-50 conformal parameters 28, 74-77 conformal parametrization 76, 230 conformal projection 49 conformal representation 64, 68 conformal surfaces 35, 36 conformal type 36, 335 conformality of minimizers 242-253 conformality relations c-conformal mappings 252-253 generalized (Riemannian) 65 in a Riemannian manifold 76,251 in IR" 65; 76, 90, 246 of the adjoint surface 91 conformally equivalent 36, 181 contact set see also set of coincidence of dS and the free trace E classification 208-218 cusp 199-206, 221 loop 199-206, 221 tongue 199-206,221 types I, II, III 209 inua of solutions 288-290, 347-356 inuity iudjoint surface 262 boundary values 234-239, 256, 259-266 2.(r)

L(XJc) 260

tour see boundary curve, boundary configuration, boundary frame

Subject Index

contractible curve (in 310 convex hull theorem 369 coordinate charts 299 Costa surface 195-196, 212-213 Courant function 293-294 Courant-Cheung example 43-45 Courant-Lebesgue lemma 235, 239-242, 253, 259

Courant-Levy examples 271, 290-292 Courant's condition 365 Courant's condition of cohesion see condition of cohesion Courant's examples 43, 133-136

Courant's formula for aD(X,,)

246

covariant derivative 46-48 covariant differentiation 40-41, 46-48, 51 critical point of Dirichlet's integral 330, 334-335 currents

281-282, 283-284

curvature invariance properties 24 of a curve 13 curvature line 23 of a minimal surface 96-99, 110, 114, 125, 127, 129, 132, 134-135, 193, 200

of the adjoint surface 99 of the associate surfaces 90 of the Goursat transform 116 curve of separation 140 curve(s) balanced 337 contractible in Tn(S) 310 holomorphic 91 isotropic 91-92, 114-115 knotted 280-281 cusp 199-206,221 cycloid 120, 172-173 degeneracy

339-340

-9-equivariant 301 De Giorgi's lemmata 78-79 difference-quotient technique 83-84 differential inequalities 21-32 boundary estimates 27 Dirichlet integral 22 gradient estimates 23, 26, 28-29 Holder continuity of the gradient 30-31 Dirichlet boundary condition 47 Dirichlet integral 227-234 bounds

22

critical point 330, 334-335 Dirichlet growth theorem 47, 49-50, 54 D(X, h), thread problem 254 generalized 321-322; 45, 251

403

lower semicontinuity 257 Morrey estimates 49-50, 137 polarization 230 reproducible Morrey norm 90 generalized 321-322 stationary point in '(T, S) 48-116, 199-247

Disquisitiones generates 49-50 distance function d(x) = dist (x, S)

105

d(p, q) on a Riemannian manifold 178 ds(P) = dist (P, S) 307 g(A,B) 322 divergence 43, 79 divergent path 178-180 Douglas condition 289-290, 366 Douglas functional 277 Douglas problem see general Plateau problem Douglas sufficient condition 298, 327 Douglas theorem 327; 289 DuBois-Reymond lemma 280-281 elliptic point 21, 22 elliptic type of a point 21, 22 enclosure of M 377-378

enclosure theorems 368-372, 374, 375, 380, 381 en d

198

catenoid 198, 203 flat (planar) 198, 203, 204 Enneper surface 97, 112, 144-149, 201, 210 associate surfaces

bending

147-149

97

Gauss map 147 higher order 202-203 Enneper-catenoid 207 Enneper-Weierstrass representation formula 108-111 equivalence class of conformally equivalent surfaces 36 equivalent 9, 10 CS-

9

strictly 9, 10, 24 estimate(s) of E. Heinz 70 of the area from above 382-396 of the area from below 104 of the length of the free trace 396-420, 424 Euler characteristic 309 Euler equation of the Dirichlet integral 45, 247, 258 energy functional 46-48

404

Subject Index

Euler equation of the functional EB(X) + VB(X)

251

numerical treatment free trace

generalized Dirichlet integral 65; 4S

cusp

1(v)

discontinuous

310

length functional 47-48 volume functional 251 generalized Dirichlet integral 65 Euler-Poincare characteristic 39, 40 examples with cusps of the free trace 199-206

132-133

experiments see soap films

fence of catenoids 209-210 fence of Scherk towers 211 field construction 82-83 field theory 80-86, 87-88, 280 finite connected minimal surfaces 369 finite connectivity 40 finite topological type 195 finiteness problem 290 first fundamental form 13, 25, 27, 93 of a minimal surface 94, 109, 114-115, 182, 200 first variation of the area 55-57 Dirichlet integral 47-48 length 47-48 foliation 77-83 by minimal surfaces 80-83 leaves of a 77 normal field of a 77 free boundary condition 57, 257-258, 318-322 existence

199-206, 221-222

327-328

free boundary problems (see also minimal surfaces)58, 258, 313, 321, 328-365 asymptotic expansions 117-121, 173-186, 186-196 boundary branch points 117-121 boundary regularity 48-116, 163-173

44, 133-136

199-206,221-222

regularity 48-116; see also minimal surfaces tongue

199-206, 221-222

types I, 11, III unbounded

examples of Almgren-Thurston 281 Cheung 43-45 Courant 43-45, 133-136 Courant, P. Levy 271, 290-292 enclosures 378-379 Gulliver-Hildebrandt 288-289, 351-356 Kiister 314, 414-415, 424 Lewerenz 293 Morgan 288, 422 Osserman 185 Quien-Tomi 286-288 White 422 Ye

loop

229-234

209

44

free trace E 322, 337 estimates of the length 396-420, 424 Fujimoto's theorem 184-185 fundamental form 13 coefficients of a 17 first 13, 25, 27, 93 second 13, 20, 25 third 13, 25

Gackstatter surface 203, 210, 212 Gauss curvature 303; 17, 19, 25, 30, 50 of a minimal surface 70, 95-96, 110, 114, 200

sign of 25 Gauss equations 29, 31 Gauss map 244; 11, 20, 22, 23, 50 also: normal map, spherical map, spherical image of a minimal surface 60-61, 73-74, 93-94, 111-115, 176, 181-191, 200 Gauss prize-essay 52 Gauss representation formulas 26 Gauss-Bonnet formula 37, 38 bound on the number of branch points 126-127, 138

branched minimal surfaces 121-128 genus zero, Plateau problem 126 H-surfaces 122, 126-127 of a Riemann surface of genus > 0 309 pseucloregular surfaces 122 solutions of a thread problem 127-128 stationary surfaces with free boundaries 127 Gauss-Bonnet theorem 37, 50 general assumption of Section 4.7 259 general Plateau problem (Douglas problem) 223,226 existence of solutions 289-290, 366 nonexistence of solutions 372 generalized admissible sequence for P(17, S) minimizing sequence for Y(17, S) generalized conformality relations 76, 251 generalized Dirichlet integral 320-321

Subject Index

generalized Plateau problem (Marx, Shiffman, Courant, Heinz, Sauvigny) 293-340, 293-294 condition of cohesion 328, 330-332 (go) and (9^) 320-321 definition 298 Douglas sufficient condition 298 Douglas theorem 327 examples 293-299, 340 existence 328-339 minimal surface 319-320, 326 Mumford compactness theorem 315-319 Shiffman theorem 328 surfaces of lower type 298 Te ic h muller theory for Riemann surfaces

298

with boundaries 307-315 without boundaries 299-307 theorem of Douglas 327 theorem of Shiffman 328 theorems of Tomi-Tromba 330, 335-338 genus (of a surface) 39 geodesic 21, 45-48, 51 on a minimal surface 125, 127, 129, 134-135, 193 geodesic curvature 15, 25, 32-34, 46-47, 50-51 geodesic curvature Gauss-Bonnet 121-128 thread problem

273, 275-278, 292

geodesic line see geodesic geometric variational problems 138 Gergonne's problem 345-346 Gergonne's surface 346 global minimal surface 388-389 Goursat transformation 115-116 gradient 42 gradient estimates differential inequalities 23, 26, 28-29 harmonic functions 16, 20 minimal surfaces see m.s. Poisson equation 8, 9, 11 Green function 7 Grater-Jost theorem 344 Gulliver-Hildebrandt examples 288-289, 351-356 Gulliver-Lesley theorem 279 Gulliver-Spruck theorem 282

hairy disk 228 halfspace theorem 199 strong 199 Hardy class 265 Hardy function 264

405

harmonic diffeomorphism 241, 339 harmonic function continuity at the boundary 14-16 gradient bounds 16, 20 Korn-Privalov theorem 17 harmonic mapping 238-239, 240-241, 319-320; 175 Hartman-Wintner method 141-162, 196-197 H-convexity 283, 355, 374, 380, 381, 426 Heinz's estimate 70 Heinz estimates 21-32 helicoid 135-140, 192, 193, 195, 347-348 adjoint

138

as a ruled surface 138-140 associate surfaces 140, 141 ben di ng 141 helicoidal type 143-144 hemisphere theorem 338 Henneberg's surface 133, 159-169 adjoint surface 205; 161, 163 associate surfaces

cusps

164-166

202-204

minimizer for (1", S) 206 MSbius strip 167-168 Neil parabolas 160-161 one-sided surface 167-168 planes of symmetry 125 Hessian form 45, 79

Hessian tensor 45 higher order Enneper surface 202-203 higher order saddle tower 205-206 Hilbert's independent integral 83 Holder continuity of boundary values gradient bounds 20 gradient (differential inequalities) 30-31 Korn-Privalov theorem 17 minimal surfaces see m.s. hole-filling 81, 116 holomorphic quadratic differentials see quadratic differentials

homotopy 310 homotopy class of boundary values 312 [XI c] 312 E. Hopis lemma 211,241;377 H. Hopis observation 32, 95 H-surfaces 32, 71-74, 74-76, 79-80, 372-373 boundary regularity for the Plateau problem 33-38 enclosure theorems 368-372,372-382 length of the free trace 416-420 partition problem 417 hyperbolic point 21, 22 hyperbolic type of a minimal surface 181

Subject Index

406

hyperbolic type of a point 21, 22 hyperboloid theorem 370 index theorem infimum a(T)

CZ

252 255

d'=d-(I,L)

255

d-=d-(T,L)

255

252

323 323 e(T) 232 e*(T) 233 e

e*

inner variation by a vector field 242 of Dirichlet's integral 246, 330 of a functional 245 of a surface 54, 330 integration by parts 266 interior estimates differential inequalities 23, 26 Poisson equation 8-9, 11-12 isometric deformation of minimal surfaces see bending of minimal surfaces isoperimetric inequality 238, 382-396, 420-424 examples of Morgan, White 422 experimental proof 422-424 for disk-type minimal surfaces 384 general version 390 linear isoperimetric inequality 387-388, 422 sharp

421-422

two boundary components 395-396 isoperimetric problem 85 isotropic curve (map) 91, 107-108, 114-115 isotropy relation 91

Lagrange multipliers 278-279 A-graph condition 396-402, 405-406 Laplace-Beltrami equation 45 Laplace-Beltrami operator 310; 43-44 Laplace operator 31, 79 Laplacian see Laplace operator least area problem 253 leaves

4-noids

206

n-noids

205-206

associate surface

77

lemma on c-conformal mappings (Morrey's lemma) 252-253 length functional 47-48 length of the free trace 396-420, 424 Lewerenz example 293 Lewy's reflection principles 39-40, 107-109 Lewy's theorems 38, 106, 130 Lichtenstein's theorem 37, 50, 63, 68, 75, 230 line element of a Riemannian manifold 76, 178 on a minimal surface 94, 109, 114-115, 182, 200

on a surface see first fundamental form line of curvature see curvature line line of symmetry 120-135, 201, 267-269 linking condition 318, 365

linking number 319-321 Lipschitz continuity of ds(P) liquid edges 299, 302 loop

Jenkins-Serrin theorem 208 Jordan curve (closed) 231 rectifiable 233-234 Jorge-Meeks catenoid (3-noid)

9-11,11-13

Green 7 Poisson 7 Kinderlehrer's theorem 140 Kneser's transversality theorem 82, 350 knotted curves 296; 280-281 Korn-Privalov theorem 17 Krust's theorem 118, 201, 341 Ktister's examples 414-415, 424 Kuster's torus example 314-316

271, 294-295

d=d(TL) e(T)

kernel

307

199-206,221

lower semicontinuity see semicontinuity maximum principle 99

99

Jost's theorems 344, 345 Karcher's surfaces 195 analogue of T- Wp 216 helicoidal saddle towers 208 higher order saddle towers 205

, -d-maximum principle 424-425 minimal surface equation 275-276 Nitsche's generalized m.p. 276 mean curvature 17, 19, 25, 50, 71-74, 192 of the leaves of a foliation 77, 79 sign of 25 zero 56 Meeks-Yau theorems on embedded minimal surfaces 282-283 Mercator projection 49 Milnor curves 286-287

Subject Index Minding's formula 32-34, 50 minimal surface adjoint (conjugate) 91-92 analyticity of a 63 annulus-type 214 associate 96-100, 114 asymptotic expansions at vertices of a curve 173-186 8.3, Assumption A

174

main theorems 175-176 asymptotic lines on a 96-99, 110, 114, 125, 132, 134-135, 193, 200 bending of a 96 boundary branch points 117-121 asymptotic expansions 119-120, 196-197 exclusion of 275, 290-291, 292 number of 126-127, 138 order of branch point 119, 121 tangent

120-121

tangent plane

120

boundary regularity at free boundaries 48-116

Dirichlet growth theorem 49-50, 54 Holder continuity of minimizers 55-56 Holder continuity of stationary surfaces

65-68,77-78

Holder estimates of minimizers

49-50,

Holder estimates of stationary surfaces 82, 132-133 Lewy's theorem 106 LP estimates for PX 94 real analyticity 106, 107 L2-estimates for P2X 83-89, 94 Xe C',0 if dS O 0 83, 100-102, 115

XeC2' ifaS=O X5C"'flifdS=O PX e C°

102-106 103-105, 106

95,98-100

boundary regularity Plateau problem 33-43

Lewy's theorem 38 real analyticity 33, 38 Riemannian manifolds 40 X e C" , C2,a 33

XeC'"°

disk-type 231 embedded 195, 198, 280-285 embedded complete 195, 198 enclosure theorems 368-372,372-38Z 424-426 expansions of stationary surfaces in (T, S) 186-196 8.4, Assumption A 187 main theorems 187-189 finite connected 369

finite total curvature 196-198 first fundamental form 94, 109, 114-115, 182, 200

Gauss curvature 70, 95-96, 110, 114, 200 Gauss map of a 60-61, 73-74, 93-94, 111-115, 176, 181-191, 200 Gauss-Bonnet formula 126-127 general definition 76, 90 general Plateau problem see g.P.p. geodesics on a 125, 127, 129, 134-135, 193 global 175-181, 388-389 gradient estimates at boundary corners 163-173 8.2, Assumption (A) 8.2, Assumption (B)

38

branch points 101-107, 114 characterization of a 58-64, 71-74 complete 195 complete global 178-181 conformal equivalence 181 conformal parametrization 64, 68, 76 construction of a 199-212 curvature lines on a 96-99, 110, 114, 125, 127, 129, 132, 134-135, 193, 200

164

165-166

main theorems 163 in a Riemannian manifold K1-surface

55

407

77-78

196-198

length of free the trace 396-420, 424 line element 94, 109, 114-115, 182, 200 nonconstant 104 nonexistence 335-339, 372, 381-382, 424-426 nonexistence of continua 271, 355, 356-357, 364 nonorientable 294, 296; 167-168, 177, 222 nonparametric 58-61 nonparametric representation 275, 376 obstacle problems 6, 137-140 thin obstacles 198-247 of finite topological type 196-198 of helicoidal type 143-144 of higher topological type 222-224, 280 of hyperbolic type 181, 183 of infinite genus 227 of parabolic type 181, 183 of revolution 135-140 on a punctured sphere 201-208 on a punctured torus 209-212 one-sided see nonorientable parameter domain 175-181 parametric 56 periodic 132, 194-195, 212-217 plane 338, 339, 341-343

408

Subject Index

minimal surface planes of symmetry 120-135, 267-269 regular points 103, 114 regularity theorem 237 representation of a 93-94, 100-101, 107-120 ruled 138-140, 193 second fundamental form 95, 110-115, 200 stable 87-88 stationary 58, 328-335, 335-339 stationary solutions in (T, S) 48-116, 199-247 absence of cusps 201, 222 asymptotic expansion for type II 218-219 asymptotic expansions for types I, III 216-217 boundary regularity 48-116 Gauss map 244 nonparametric representation 213, 243, 245-247 numerical treatment 229-234 surfaces of types I, III 209 symmetry 214 uniqueness

213, 234-247

straight lines on a 123-125, 127, 129, 267-268 strictly stable 87-88 surface normal 60-61, 111-115, 134 systems 299-302 thread problem see t.p. total curvature 126-127 touching 369-370 triply periodic 195, 212-217 umbilical points 95-98, 112, 114 universal covering of a 180 with a free boundary 224-225, 253-259, 304-364 minimal surface equation

H=0 56, 192 in codimension > 1 85-86 nonparametric 58-61, 85-86, 192 minimal surfaces in Riemannian manifolds boundary regularity for Plateau problem 40-43 minimizing cone 284, 369, 371 minimizing sequence 235, 237, 254, 298, 323 generalized for 9(17,S) 323 minimum problem

9(T)

232, 234, 248, 253, 254 233 9(T, S) 256

9a*(T)

9'(a, S) 313 9(17,S) 321 +(S) 321

Y(T,L) 254 Yo, 3 320-321 9 (. C) 298

Mdbius strip as a minimal surface

167-168,

222

modular group T= .9/.9o 300 monodromy principle 270-271 monotonicity of functionals 45, 70, 75, 131 monotonicity of the boundary values 231-232 total variation 2.(r) 260 monster surface 227 Morgan's examples 288, 422 Morrey seminorm reproducible 47, 90 Morrey's lemma 252-253 Morse index 293-294, 294-296 Struwe's Morse inequalities 296 Tromba's Morse equality 296 Morse theory 294-296 Mumford compactness theorem 315-319

natural boundary classes 312 natural boundary condition 329 necessary conditions for the existence of multiply connected minimal surfaces 368-382

stationary minimal surfaces 335-339 Neil's parabola 203; 160-161 Neumann boundary condition 47, 95, 96, 102,207,213-214 Nitsche's 4rz-theorem 270, 280 conjecture

290

Meeks-Yau 283 Ruchert

Sauvigny

285-286 293

Nitsche's 67r-theorem 92 nonexistence of continua of solutions 271, 355, 356-357, 364 multiply connected minimal surfaces 372, 381-382, 424-426 solutions of free boundary problems 335-339

stationary minimal surfaces 335-339 nonparametric minimal surface 58-61 nonparametric representation 213, 223-228, 243, 245-247

of a minimal surface 58-61, 275, 376 of an H-surface 376 nonparametric surface 58 nonsolvability of 9 (a, S)

313

409

Subject Index nonuniqueness (partially) free problems 345-365 Plateau problem 270-276, 285-294, 294-296

normal exterior, v 7 principal 13 side 14 surface

9, 14

normal coordinates 62-63, 103-104, 105-106;407-408 normal curvature 15, 25 normalization (admissible boundary coordinates) 63 normal map cf. also Gauss map, spherical map, spherical image 244; 11, 20, 22, 23, 50

normal plane 14 normal section 15 normal variation 78, 83 normal vector 9, 10, 14 number-of-solutions problem 271 numerical solutions 229-234 numerical treatment of q(T, S) 229-234

observation of H. Hopf 32, 95 observation of Riemann-Beltrami 64-65 obstacle problem 6, 199-234, 251; 224-226, 256-257,296-299,365 artificial obstacles 329 coincidence set, regularity 139 Kinderlehrer's theorem 140 regularity 137-138, 139-140, 336 obstacles thick 136-138; 297 thin 6, 139-140, 199-234; 297 order of connectivity 40 orientable surface 40 oriented 10 equally 10, 20 oppositely 10, 20 orthogonal parameter curves 28 orthonormal frame (1, a, 91) 32 osculating plane 14, 21 Osserman-Schiffer cone 378 Osserman's example 185 Osserman's theorem 86, 185 outer variation 64 of a surface 331 parabolic point 21, 22 parabolic type of a minimal surface 181 of a point 21, 22

parameter domains thread problem 252-254 parametrization 11 local

11

partially free boundary problem 224-225, 253-259,328-335,335-341,345-365 partition problem 417 periodic minimal surfaces 132 periods of Weierstrass data 200 phenomenon of degeneracy 327, 339-340 Pitts' theorems 344, 345 plane of symmetry 120-135, 267-269 Plateau's problem 221 bridge theorem (principle) 290-292 conformality of solutions 242-253 Courant-Tonelli method 234-253 disk-type solutions 222, 223, 231 Douglas method 277 existence of embedded solutions 280-285 existence of immersed minimizers 279-280 existence of solutions 221, 248 finiteness problem 290 for H-surfaces 278-279 fundamental existence theorem 279 Garnier's method 277 generalized 293-294 in Riemannian manifolds 278-279 index theorem 293 Levy-Courant construction 291-292 main theorem 248 Morrey's method 278-279 Nitsche's conjecture 290 non-orientable solution 222 nonparametric 277 nonuniqueness 222-224, 270-276, 285-294,294-296 polygonal boundaries

293-294

Radio's method 277-278 rigorous formulation 231 solutions of higher topological type 223, 227 space of solutions 292-293 uniqueness 270-276, 285-294 unstable solutions 293 with obstacles 224-226 Plateau's rules for systems of minimal surfaces 227, 299-302 Poincare inequality 81, 111 Poisson integral 8, 14-15 Poisson kernel 7 Poisson's formula 260-261 polygonal boundaries 197 asymptotic expansions 175-176 gradient estimates 164-166

222,

410

Subject Index

polygonal boundaries Holder continuity 49-56 number of branch points 138 potential-theoretic results 7-21 principal curvatures 16, 25 principal 9o-bundle (n,.,0 -1, -i/moo) 306-307 principal directions of curvature principal normal 13 principal radii of curvature 16 prize-essay of Gauss 52 of Schwarz 276-277

representation formula 199-200 integral-free

16

problem 9(I; S) 237 Problem of least area

253

of Plateau see Plateau problem, general

Plateau problem proper action 306 quadratic differentials 304-305, 312-313, 322-323 quadrilateral 276, 277 quasi-minimal surfaces 197 quasiregular set 137 Quien-Tomi examples 286-288 quotient space

T=299 %/2o .,K/Y 302

303

-#-,/go ,9', slice of .11-i

°I = '/moo

303 299 305

299

Rad6's lemma 211-212, 223-228, 242-243; 272

Rado's theorem 270, 271-276 radius of curvature 14 rectifiable boundary curve 233-234, 259-266 rectifiable Jordan curve 233-234 rectifying plane

14

reflection principles 123-125, 200, 267-269 Choe 269 regularity of liquid edges 302 of solutions of Y(F) 237 regularity of the coincidence set

139-140

free trace 49-50, 55-56, 65-68, 77-78, 83, 100-102, 102-106, 107, 115, 117-121 regular points 103, 114 set of 108-109, 114

93-94, 107-120, 193,

113

of Enneper-Weierstrass 108-111, 193 of Monge 100-101, 193 of Weierstrass 112-115, 116-118, 193 representation formula (normal coordinates) 407-408 representation formulas 8, 13-14 representation theorem 213, 223-228, 287 Riemann curvature tensor 29, 30, 41-42 Riemannian line element 76, 178 Riemannian manifold 76, 178 complete 178-179 Riemann mapping theorem 248-249 Riemann-Roch theorem 306, 313 Riemann's periodic minimal surface 193, 211-212 Riemann surfaces 299-307 symmetric 307-315 F. Riesz theorem 265 F. and M. Riesz theorem 265 R-sphere condition (two-sided) 406, 407, 424 ruled minimal surface 138-140, 193 saddle towers 205-208 Sauvigny's theorems 294 scalar curvature 300 Schauder estimates 9 Scherk's doubly periodic surface see Scherk's first surface Scherk's first surface 151-159, 193 lines of symmetry 124 Scherk's saddle tower (Scherk's fifth surface) 204-205 Scherk's second surface (Scherk's doubly periodic minimal surface) 140-144, 193 as a general screw surface 143-144 associate surfaces 142 Scherk surfaces

193

schlicht domain 40 E. Schmidt's inequality 90, 147 Schneider's estimate (on the number of branch points) 138 A. Schoen's surfaces 195 H'-T-surface 215 I-Wp-surface 216 S'-S"-surface 216 gyroid 217 R. Schoen's uniqueness theorem 288 Schottky double 307-308 Schwarz boundary continuity of Poisson's integral 14

Subject Index reflection principle 128 solution of the Bjorling problem 202 Schwarzian chain 130, 345-346 generalized 133, 347, 365 Schwarzian chain problem 127, 130-133, 158-159, 277 Schwarz's CLP-surface 215 H-surface 214 P-surface 214 (periodic) surface 174-175, 213 prize-essay 276-277 reflection principles 123-125, 200, 267-269 solution of Bjorlings problem 121, 133-135

stationary minimal surface in a tetrahedron 339

screw surface see helicoid second fundamental form 13, 20, 25 of a minimal surface 95, 110, 114-115, 200 second variation of area 83-85 operator 85 selfintersections 281 semicontinuity of 1(r) 261 .PB(X) 298 mass 282 the Dirichlet integral

255

the generalized Dirichlet integral 326, 340 Shiffman function 293-294 Shiffman's theorem 320 side curvature 51 side normal 14, 57 signorini problem 6; 297 singular points of a minimal surface see branch points Smyth's theorem 340 soap films attaching to 2S 97, 199-234 general plateau Problem 293-297 in stable equilibrium 221 selfintersecting 280 systems 299-302 thin obstacles (d S O0) 199-201 thread problem 250-253 touching an obstacle 298-299 solution of a free boundary problem 58, 321 partially free boundary problem (semifree boundary problem) 224, 258 partition problem 417 solution of the minimal surface equation 63

411

space of

almost complex structures,.W(M)

300

complex structures, ' _ W(M) 299 s-maps homotopic to the identity, Q0(M) 299

moduli, yf (M), 39 _ /g =

/I'

299

of positive C°°-functions, 91 301 of Riemann C`°-metrics, ,d( 301 orientation preserving C°`-diffeomorphisms,

2=_9(M) 299 Riemann metrics with K=-I, M-1 space 9/9 of equivalence classes of

303

conformally equivalent metrics 302 symmetric complex structures, les 308 symmetric metrics with K= -1, ..#s , 312 "symmetric" Teichmuller space, 9(2M) _ `Ws/Do 308 Teichmuller space, 9-(M), .°l =W/9o 299

sphere condition R-sphere condition 406, 407, 424 oo-sphere condition 328 spherical image (see also Gauss map, normal map, spherical map) 11, 20, 36, SO of a minimal surface 36 spherical map (cf. also Gauss map, normal map, spherical image) 11, 20, 22, 23, 50 star-shaped domains 379, 381 stationary H-surface 416 stationary minimal surfaces existence 343-345 in a convex surface 344-345 in a polyhedron 339-341, 345 in a simplex 339-341 in a sphere 341-343 in a surface 344 in le(S), +(S), x(17, S) 334-335 in le(T, S) 329 in r 33-43 regularity

33-43

in (T, S) 48-116 83, 100-102, 115, 198-242 nonparametric representation 213 regularity 48-116 symmetry 214 uniqueness 213, 234-247 necessary conditions 335-339 nonuniqueness 345-366 stationary point of Dirichlet's integral d S =A 0

in' (T, S) 330 in 6-(S), +(S), "1(17, S), 334-335 stationary surface 58, 328-335, 335-339, 416 Steffen-Wente theorem 280

412

Subject Index

stereographic projection 110-111 of the Gauss map of a minimal surface 111-112 strictly equivalent 9 strip 120, 134 strong halfspace theorem 199 Struwe's Morse inequalities 296 subharmonic functions 21 supporting manifold 57-58 irregular 255, 257, 304 support surface admissible 61 assumption (B) 61-62 6, M-chord arc condition 49 chord arc condition 45, 48, 49 surface 7, 10

compact 40 complete 178-179 conformally parametrized 28, 35-36, 64, 68, 76, 90, 230 embedded 8, 20 equivalent 9, 10 general 10 immersed 8 in lR3

10

nonorientable 177 nonparametric 58 not embedded 8 of class C' 11 of constant mean curvature 32, 71-74 of least area 221 of prescribed mean curvature 71-74, 74-76, 79-80, 372-373 orientable 40 parametric 8

pseudoregular, 8(52)

121-122

regular 7, 281 strictly equivalent 9, 10, 24, 33 surface normal 244; 14 symmetric boundary configuration Assumption A of 9.3 206

(r, S) 206 symmetric two-tensors 304 splitting 304, 313 symmetry axis 123 Scherk's (first) surface 124 symmetry plane 123 Catalan's surface 125 systems of minimal surfaces 299-302 Jean Taylor's theorem 301-302 liquid edges 299 Plateau rules 299-300 singular part 299-300, 302

tangential vector fields 40-48 along a curve 46 tangent space 7, 24, 93 Tg s i 312 TgJP_i 304 tangent vector of a curve 13

to the orbit Og(.9),ge304 J. Taylor's theorem 301-302 Teichmuller space 299, 307, 339 for nonoriented surfaces 313-314 of symmetric Riemann surfaces 307-315 oriented Riemann surfaces with boundaries 307 Weil-Petersson metric 314, 315 test cone for nonexistence 372 theorema egregium 30, 50 theorem(s) of Almgren-Simon 282 Alt, Gulliver, Osserman 279 Alt-Tomi 356-357 Bernstein 67 Bohme (4zc+e-theorem) 270 Bohme-Tromba 271, 294-295 Bombieri-De Giorgi-Giusti 86-87 Douglas 327; 289 Fujimoto 184-185 Gauss-Bonnet 37, 50-51 Gulliver-Lesley 279 Gulliver-Spruck 282 Grater-Jost 344 Hardt-Simon 282, 285 Jean Taylor 301-302 Jenkins-Serrin 208 Johann Bernoulli 51 Jost 344, 345 Kinderlehrer 140 Krust 118, 201, 341 38, 106 Lichtenstein 37, 50, 63, 68, 75, 230 Meeks-Yau 282, 283 Nitsche (4n-theorem) 270 Nitsche (6n-theorem) 292 Osserman (minimal surfaces in codimension > 1) 85-86 Osserman (on the Gauss map) 185 Pitts 344, 345 Rado 270, 275-276 F. and M. Riesz 265 F. Riesz 265 Sauvigny 294 Shiffman 328 Lewy

Smyth 340 Steffen-Wente

280

Subject Index

Tomi 271, 355

types I, II, III

Tomi-Tromba 335-339, 282, 295 Tsuji 128 thin obstacles 6, 139-140, 199-205 third fundamental form 13, 25 Thomsen surfaces 149-151 thread problem 250-292

D(X,B) 254 analyticity of the movable boundary 271-291 (see also anal. mov. boundary) boundary maps pB (u), p8 __(u)

253

branch points on the thread

273, 275,

284-291

class of parameter domains -4 253 class of parameter domains .4*(T, L) 255 constant geodesic curvature 278

(T,L)in 10.3

273

9(T, L) in 10.1 254 existence of solutions 255, 264, 269, 274, 291-292 experiments 250-253

infima d, d', d

255

Lagrange multipliers 278-279 lengths of traces 1(c,1'),1(y) 254 necessary condition 254, 255, 269, 272 parameter domains 253 regularity of the movable boundary 271-291, 292 stationary thread problem 274, 275-278, 281

total curvature of r 281 total (Gauss) curvature 37 total geodesic curvature 37 trace 7, cusp 199-205, 221-222 discontinuous 44, 133-136 free 322, 337 length

396-420

loop 199-205,221-222 of an HZ-surface 305-318 of a surface 7, tongue

199-205,221-222

trace theorem 307, 310

209

unbounded 44 trace theorem 307, 310 triply periodic minimal surfaces 195, 212-217 Tromba's degree method 296 Tromba's Morse equality 296 Tsuji's theorem 128 tubular p-neighbourhood T (S) 306 types I, II, III of free trace 209 umbilical point 16, 36, 95-96, 98, 112, 114 uniformization theorem 50, 76 uniform three-point condition 238-239 uniqueness partially free problems 213, 234-247 uniqueness Plateau problem 270-276, 285-294 universal covering 179-180

variation first 47-48, 55-57 inner 54, 242, 330 normal 78, 83 of a functional 245, 246, 330 of a surface S4, 330, 331

outer 331 second 83-85 variational inequality in l6 (T, S) 64-65 thread problem

sufficient condition 255, 263, 269 three-point condition 233, 236, 276 uniform 238-239 Tomi's finiteness theorems 271, 355-357 Tomi-Tromba general index theorem 295 Tomi-Tromba theorems 330,335-338;282 tongue 199-205, 221 topological mapping by boundary values 248 topological type finite 195

413

278-279

variational problem 3/(T) 232, 234, 248, 253, 254 3/*(T) 233 9(T, S) 256 9(17,S) 321 3/(Q, S)

313

91(97, C) 298 91' (S)

321

3/(T, L)

254

Yo, 9 320-321 vector field 10 along a surface 10 normal 10 parallel 46-48 tangential 10, 40-48 vector tangent

13

volume constraint

280

functional

251

weak lower semicontinuity (with respect to sequences) Dirichlet integral 257

414

Subject Index

weakly connected 389-390 weakly (sequentially) closed subset (* (T) 255 re* (T, S) 256 16*) 298

weak transversality relation 106 Weierstrass data 200 Weierstrass field theory 80-86, 87-88, 280 Weierstrass function 8 112, 114 Weierstrass functions G, H 116-118 Weierstrass functions g, h 199 Weierstrass functions u, v 108 Weierstrass representation formula 112-115, 116-118, 199-200

Weierstrass theorem (on algebraic minimal surfaces) 113 Weil-Petersson metric 314 Weingarten equations 18, 23, 31, 51 Weingarten map 11, 16, 24, 93 Wente surface 33 White's examples 422 Willmore surface 177 Wirtinger's inequality 383 Ye's example

132-133

zero mean curvature

56

Index of Illustrations Minimal Surfaces II

The illustrations in this index are sorted by topic. Figures explaining only notations are omitted from this index. Costa's surface:

An analogue of higher topological order ..............................

Frontispiece

Notions:

A Riemann surface and its Schottky double ...........................

11.3 Fig. I

Plateau's problem:

A contour spanning a disk-type surface and a MSbius strip .............. The cloverleaf, spanning a one-sided minimal surface ................... A non-orientable solution of higher topological order ...................

11.1 Fig. 6 11.1 Fig. 7 11.1 Fig. 8

Plateau's problem generalized:

Various solutions of higher topological order .......................... Soap film catenoids spanned between pairs of circles .................... Doubly connected surfaces bounded by two curves ..................... Two interlocked curves spanning an annulus-type surface ................ A minimal surface of genus 0 bounded by three closed curves

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

The catenoid and relatives bounded by several closed curves .............

A one-sided surface of genus 2 in a closed curve ........................ Two-sided surfaces of higher genus bounded by a single curve ............ A Jordan curve bounding an area-minimizer of infinite genus ............

A degenerating minimizing sequence ................................. H-convex blocks used to build solutions of higher genus ................. Two interlocked curves spanning an annulus-type surface ................

Plate 3 11.1 Fig. 1 11.1 Fig. 2 11.1 Fig. 3 11.1 Fig. 4 11.1 Fig. 5 11.1 Fig. 9 11.1 Fig. 10 11.1 Fig. 11 11.6 Fig. 1 11.6 Fig. 2 11.7 Fig. 1

Semi free boundary problems:

Soap films with partially free boundaries on a halfplane .................

The shapes of the trace: Loop - cusp - tongue ......................... A solution for four almost semi-circular arcs and a plane ................ Unbounded solution on a surface without chord-arc condition ...........

No a priori estimates for stationary solutions .......................... Set-up for the Hildebrandt-Nitsche uniqueness theorem ................. The shape of the arc leading to a tongue/cusp-shaped trace .............. The shapes of the traces while bending the boundary curve ..............

Position of the curve r giving a tongue/cusp/loop ...................... A bending process without cusps on the trace .......................... Views of the cusp in Henneberg's surface .............................. Two cusps in a Schwarzian chain problem (Henneberg surface) ...........

A configuration with a disk and a closed Jordan curve .................. A solution with four cusps (adjoint of Henneberg's surface) ..............

Plate I Plate 2 Plate 4 7.4 Fig. 1 7.6 Fig. 1

9.1 Fig. I 9.1 Fig. 2 9.1 Fig. 3 9.1 Fig. 4 9.1 Fig. 5 9.2 Fig. 1, 2 9.2 Fig. 3 9.2 Fig. 4 9.2 Fig. 5

416

Index of Illustrations Minimal Surfaces II

A discretization of the semi-disk ..................................... Numerical solutions for several curves on rectangles ....................

9.10 Fig. 1 9.10 Fig. 2

The geometric setting for the Hildebrandt-Sauvigny

uniqueness theorem ................................................ Conditions (Cl), (C2) ..............................................

9.10 Fig. 3 9.10 Fig.4, 5

Thread problems: Thread experiments performed by the Institut fur

Leichte Flachentragwerke in Stuttgart ................................ A parameter domain and a solution having two components ............. Figures explaining only notations were omitted from this index.

10.1 Fig. 1 10.1 Fig. 2

Index of Illustrations Minimal Surfaces I

The illustrations in this index are sorted by topic. Figures explaining only notations are omitted from this index.

Analysis - general theory:

Maps with zero boundary values in the Sobolev spaces H,' (B) ............

5.1 Fig. 1

Catalan's surface:

... built from the cycloid using Bjorling's principle ...................... The planes and lines involved in the bending process .................... Bending the fundamental part - view from y> 0

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

Bending the fundamental part - view from y <0 .................... . ..

A global view from x>0 ........................................... A global view from x <0 ........................................... Parallel projections of the fundamental piece 0 < u < 27t, v > 0 ............

Reflection at the z, x-plane .......................................... The fundamental piece reflected in the x-axis

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

A global view of the surface - (Iul <6z, Ivi
... constructed via Bjorling's problem for the cycloid ....................

3.4 Fig. 1 3.4 Fig. 5 3.4 Fig. 6 3.4 Fig. 7 3.5 Fig. 30 3.5 Fig. 31 3.5 Fig. 32 3.5 Fig. 33 3.5 Fig. 34 3.5 Fig. 35 3.5 Fig. 36

Catenoid:

A stable and an unstable catenoid with the same boundary .............. ... generated by rotating a catenary (= hanging chain) .................. Three picturesque views of different parts of the catenoid ................

... bent into the helicoid ............................................

Plate 1 a

3.5 Fig. I 3.5 Fig. 2 3.5 Fig. 4

Catenoid-type surfaces:

Fences of catenoids ................................................ Higher genus minimal surface based on the catenoid ....................

Plate 8 3.8 Fig. 16

Costa's surface:

A view of the central part of the surface ...............................

Four views of the central part .......................................

Frontispiece 3.8 Fig. 20

Differential geometry:

The Gauss curvature defines the orientation of the Gauss map ........... The tangent plane & the Gauss map on a bell-shaped surface .............

The Gauss map on a torus .......................................... The Gauss map on a monkey saddle near a parabolic point .............. An immersed closed curve bounding no immersed disk ..................

The 10 geodesic nets on the 2-sphere ..................................

1.2 Fig. 3 1.2 Fig. 5 1.2 Fig. 6 1.2 Fig. 7 4.10 Fig. 4 4.10 Fig. 12

Enclosure theorems:

Two cones enclosed by a family of catenoids ........................... A star-shaped set enclosed by its homothetic relatives ................... A enclosure theorem for an H-surface with boundary ...................

6.2 Fig. 4 6.2 Fig. 5 6.2 Fig. 6

Index of Illustrations Minimal Surfaces I

418

Enneper's and related surfaces:

Its Gauss map is the stereographic projection ..........................

3.3 Fig. 2

Three views of the part Iul, lvI <2 ...................................

3.5 Fig.7

Increasing parts conveying the large scale behaviour ....................

3.5 Fig. 8

The Gauss map of four increasingly large parts

3.5 Fig.9

........................ The bending process on the disk I wI <2, a mere rotation ................ ... of order 1: g(w) = w . ............................................. ... of higher order, g(w) = w 2,

w 3 .....................................

Large scale behaviour of a higher order surface (g(w) = w2) ..............

An Enneper-catenoid ............................................... Doubled Enneper surfaces ..........................................

... and Chen-Gackstatter surfaces with one/two handles .................

3.5 Fig. 10

3.8 Fig.2 3.8 Fig. 3 3.8 Fig. 4 3.8 Fig. 11 3.8 Fig. 12 3.8 Fig. 17

Estimates and properties of the boundary:

The estimate for the length of the trace on a graph is sharp .............. There is no a priori estimate for the length of the trace .................. There is no a priori estimate for the trace in terms of H or K ............. Free boundary problems: Rules for systems of soap films

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

6.4 Fig. 1 6.4 Fig. 2 6.4 Fig. 4 4.10 Fig. 11

A bizarre support set ...............................................

5. Fig. I

A minimizing sequence with a prescribed homotopy class ................

5.1 Fig. 4 5.3 Fig. I 5.3 Fig. 2 5.6 Fig. I 5.7 Fig. I 5.9 Fig. 7 5.9 Fig. 8 5.9 Fig. 9

A support surface linked with a polygon .............................. A support surface bounding no stable minimal surface ..................

A stationary minimal surface in a tetrahedron ......................... A stationary catenoid and a stationary disk in a sphere .................. A support surface with a continuum of annulus-type minima ............. A support surface with a continuum of disk-type minima ................ Some of infinitely many area-minimizing surfaces in S .................. Helicoid:

The double-helix and a helicoid ......................................

A ruled minimal surface ............................................

Plate 1 b

3.5 Fig. 3

Henneberg's surface:

The part corresponding to 0.27r < u <0.4n ............................ The whole v-axis is mapped onto a straight line segment .................

The two Neil parabolas contained in the surface ........................ Projections of the part I u I <37z/10, 0 < v < 7r/2 onto y = 0 &x=0 ........

The large scale behaviour of the surface ...............................

Views of the adjoint surface ......................................... Bending the adjoint surface back into the surface .......................

... is non-orientable ................................................

A Mobius strip contained in the surface ...............................

Projections of the Mobius strip onto the coordinate planes ...............

3.5 Fig. 20 3.5 Fig. 21 3.5 Fig. 22 3.5 Fig. 23 3.5 Fig. 24 3.5 Fig. 25 3.5 Fig. 26 3.5 Fig.27 3.5 Fig. 28 3.5 Fig. 29

Isoperimetric inequality:

Rugen, an island for which A << L2/47r .............................. Experimental proof: A thread spanned by a plane soap film

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

Experimental proof: A soap bubble between two planes .................

6.5 Fig. I 6.5 Fig. 2 6.5 Fig. 3

k-noids

The Jorge-Meeks 3-noid ............................................

Plate lc

The Jorge-Meeks 3-noid, limit of saddle towers ........................ A 4-noid with two orthogonal symmetry planes ........................

3.8 Fig. 8 3.8 Fig. 9 3.8 Fig. 10

Other views of a 4-noid .............................................

Index of Illustrations Minimal Surfaces I Karcher's surfaces: The T-Wp-surface ..................................................

Minimal surfaces - general theory: Associate surfaces of a minimal surface (Enneper's surface) Associate surfaces of Catalan's surface The Jorge-Meeks 3-noid and an associate surface The convergence of the tangent plane near a branch point A large part of Catalan's surface generated by Bjorling's principle A Willmore surface, a critical point of JH2dA The deformation of a catenoidal end into an Enneper end

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

Plate 6e

3.1

Fig. I Fig. 2

3.1

Fig. 3

3.1

3.2 Fig. 1 3.4 Fig. 2 3.6 Fig. 1 3.8 Fig. 5

Minimal surfaces with planar ends:

A Hoffman-Meeks surface ................... . Three examples with one planar end

. .

. .. . .

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

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

Neovius surface:

A fundamental cell of the surface ..........

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

Plate 2a 3.8 Fig. 6 Plate 7b

Nonexistence phenomena:

A catenoid torn apart Two suitable cones give a nonexistence result

.

.

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

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

6. Fig. I 6.1 Fig. I

Nonuniqueness:

A closed Milnor curve bounding two minimal surfaces .................. A closed Milnor curve bounding three minimal surfaces .................

................ A curve bounding non-denumerably many disk-type surfaces ............ Three circles bounding a continuum of minimal surfaces

Application of the bridge principle ...................................

4.10 Fig. 5 4.10 Fig. 6 4.10 Fig. 7 4.10 Fig. 8 4.10 Fig. 9

Notions:

A parametric surface ............................................... An embedded surface - immersed surface - branched covering ...........

The Gauss map of a surface (Enneper's surface) ........................

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

Normal plane, osculating plane, rectifying plane of a curve An elliptic - hyperbolic - parabolic point

The genus of domains and surfaces ...................................

The stereographic projection ........................................ The linking number of two closed polygons ............................ A balanced and an unbalanced curve on a support surface ............... A family of domains depending continuously on a parameter a ........... An enclosure of a simply connected set

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

Weakly connected curves and curves not weakly connected ..............

The R-sphere condition ............................................. The inradius and the smallest curvature radius ......................... Obstacle problems:

Two different ways to treat obstacle problems

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

1.1 Fig. 1 1.1 Fig. 2 1.2 Fig. 1 1.2 Fig. 2 1.2 Fig. 4 1.4 Fig. I 3.3 Fig. 1 5.2 Fig. 1 5.5 Fig. 1 6.2 Fig. 2 6.2 Fig. 3 6.3 Fig. 1 6.4 Fig. 3 6.4 Fig. 5

4.10 Fig. 10

Periodic minimal surfaces:

Riemann's surface - an example with translational symmetry ............. Gergonne's problem leading to a periodic minimal surface ............... Plateau's problem: A contour spanning a disk-type surface and one of genus 1

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

A contour spanning a disk-type surface and a Mobius strip ..............

Two non-congruent solutions for the same contour .....................

3.8 Fig. 1 5.9 Fig. 1

4. Fig. 1 4. Fig. 2, 3 4. Fig. 4

419

420

Index of Illustrations Minimal Surfaces I

A non-orientable solution of higher topological order ................... An area-minimizer of infinite genus in closed Jordan curve ..............

A hairy disk minimizing the area functional ........................... A disk with one hair minimizing the area functional .................... A knotted curve bounding a surface of higher topology .................

An example of Almgren and Thurston ................................ Three minimal surfaces bounded by a curve on Enneper's surface .........

4. Fig. 5 4.1 Fig. I 4.1 Fig. 2 4.1 Fig. 3 4.10 Fig. 1 4.10 Fig. 2 4.10 Fig. 3

Plateau's problem generalized:

Two interlocked curves spanning an annulus-type surface ................ Minimal surfaces spanning two parallel concentric circles ................

4. Fig. 6 4. Fig. 10

Reflection principles:

Straight lines in Scherk's surface as lines of symmetry ................... Planes of symmetry in Catalans's and Henneberg's surface ...............

Catalan's surface reflected at the z, x-plane ............................ Catalan's surface reflected at the x-axis

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

3.4 Fig. 3 3.4 Fig. 4 4.8 Fig. 1 4.8 Fig. 2

Scherk's first surface:

A view from z = -- ............................................... Defined on the black squares of an infinite checker board ...............

The level and gradient lines on IxI, IYI < rr/2 ..........................

A view of the surface near z = 0 ..................................... Construction of the complete surface from a single saddle ............... Corresponding domains in a parametric representation ..................

3.5 Fig. 12 3.5 Fig. 13 3.5 Fig. 14 3.5 Fig. 15 3.5 Fig. 16 3.5 Fig. 17

A part solving a Schwarzian chain problem for two perpendicular planes and a

straight line .......................................................

The corresponding part of the adjoint surface ..........................

3.5 Fig.18 3.5 Fig. 19

Scherk's second surface:

Four members of the family .........................................

Three members of the family seen from y = +. ........................

3.5 Fig. 5 3.5 Fig. 6

Scherk-type surfaces:

Saddle towers of higher topological type ..............................

Less symmetric saddle-towers .......................................

Helicoidal saddle towers ............................................ The Jenkins-Serrin theorem for the hexagon (n = 3) .................... A fence of Scherk towers, a doubly periodic toroidal surface ............. A conjugate pair of doubly periodic embedded surfaces .................

3.8 Fig. 7 3.8 Fig. 13 3.8 Fig. 14 3.8 Fig. 15 3.8 Fig. 18 3.8 Fig. 19

Schoen's surfaces:

The H'-T surface - dual lattice - hexagonal & trigonal cell ............... From Schwarz's P-surface to Schoen's S'-S"-surface ....................

Plate 3 Plate 4

A fundamental cell of the I-Wp-surface ............................... The H'-T-surface in a trigonal cell ....................................

Plate 7a 3.8 Fig. 25 3.8 Fig. 26 3.8 Fig. 27 3.8 Fig. 29

S'-S"-surface solves a free boundary problem for a cube ................. The I-Wp-surface solves a free boundary problem for a cube ............. The gyroid, an associate surface to Schwarz's surface ................... Schwarz's surfaces: A part of the P-surface .............................................

Plate 2b

The CLP-surface ..................................................

Plate 5

The H-surface .....................................................

Plate 6a-d

... spanning a quadrilateral

3.5 Fig. 37 3.5 Fig. 38

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

Extension of the surface in the quadrilateral by reflection ................

Index of Illustrations Minimal Surfaces I

421

A part of the periodic surface ........................................

3.5 Fig. 39

...bounded by a quadrilateral .......................................

3.8 Fig. 21

The P-surface - its part contained in four cubes ........................ The H-surface - its part contained in a hexagonal cell ...................

3.8 Fig. 22 3.8 Fig. 23

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

3.8 Fig. 24

An analogue to the 1-Wp-surface with hexagonal symmetry ..............

3.8 Fig.28

The CLP-surface

Schwarzian chains:

Gergonne's problem yielding a periodic minimal surface ................. Schoen's S'-S' surface bounded by the faces of a cube ..................

Schwarz's H-surface in a hexagonal prism ............................. Two perpendicular planes connected by two smooth curves .............. Henneberg's surface bounded by two curves and two planes

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

3.4 Fig. 8 3.4 Fig. 9 3.4 Fig. 10 3.4 Fig. 11 3.4 Fig. 12

Semi free boundary problems:

A solution for a partly circular arc ending on a plane ................... A curve spinning around a plane disk and its solution ................... The solution for a curve spinning around a plane disk ...................

Solutions for support surfaces with boundary .......................... More solutions for support surfaces with boundaries .................... An irregular support surface admitting a solution

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

Multiple solutions for two problems .................................. Two planes and two curves bounding Henneberg's surface ............... A cylinder and two curves bounding infinitely many helicoids ............ A cylinder and one curve bounding infinitely many helicoids ............. Catenaries and wavefronts - and the complete figure .................... Two of a continuum of surfaces of minimum area in i'(T, S) .............

4.

Fig. 7

4. Fig. 8

4. Fig. 9 4.6 Fig. 1 4.6 Fig. 2 4.6 Fig. 3 4.6 Fig. 4 5.9 Fig. 2 5.9 Fig. 3 5.9 Fig. 4 5.9 Fig. 5 5.9 Fig. 6

Thomsen's surface:

Four views of a part of the surface ...................................

3.5 Fig. I l

Thread problems:

The thread represents a curve of constant curvature .....................

6.5 Fig. 4

Uniqueness:

A Plateau problem with a unique solution - Radb's theorem .............

4.9 Fig. 2

Wente's surface:

A view of the constant mean curvature surface ......................... Figures explaining only notations were omitted from this index.

1.3 Fig. 1

Sources of Illustrations Minimal Surfaces II

Boix E., Hoffman J., Wohlgemuth M.: Plates 3, 4; Frontispiece; Section 11.1 Fig. 5

Hahn J., Polthier K.: Section 11.1 Fig. 2b, 10a Haubitz I.: Section 9.2 Fig. 2 Hildebrandt S., Nitsche J. C. C. [3]: Section 9.1 Fig. 1-3, 5, Section 9.3 Fig. 1, Section 9.8 Fig. 1

Institut fur Leichte Flachentragwerke Stuttgart - Archive: Plates 1, 2; Section 10.1 Fig. 1, Section 11.1 Fig. I

Polthier K.: Section 11.1 Fig. 2, 4

Polthier K., Wohlgemuth M.: Section 11.1 Fig. 10b, IOc Tomi F., Tromba A. J.: Section 11.6 Fig. 2

All the other illustrations were produced by the authors.