Grundlehren dermathematischen Wissenschaften 339A Series of Comprehensive Studies in Mathematics

Series editors

M. Berger P. de la Harpe F. HirzebruchN.J. Hitchin L. Hörmander M. KashiwaraA. Kupiainen G. Lebeau F.-H. Lin B.C. NgôM. Ratner D. Serre Ya.G. SinaiN.J.A. Sloane A.M. Vershik M. Waldschmidt

Editor-in-Chief

A. Chenciner J. Coates S.R.S. Varadhan

The Costa surface. Courtesy of D. Hoffman

For further volumes:http://www.springer.com/series/138

Ulrich Dierkes � Stefan Hildebrandt �

Friedrich Sauvigny

Minimal Surfaces

With assistance and contributions by A. Küster and R. Jakob

Revised and enlarged 2nd edition

With 139 Figures and 9 Color Plates

Ulrich DierkesFaculty of MathematicsUniversity of Duisburg-EssenCampus DuisburgForsthausweg 247057 DuisburgGermanyulrich.dierkes@uni-due.de

Friedrich SauvignyMathematical InstituteBrandenburgian Technical UniversityKonrad-Wachsmann-Allee 103044 CottbusGermanysauvigny@math.tu-cottbus.de

Stefan HildebrandtMathematical InstituteUniversity of BonnEndenicher Allee 6053115 BonnGermany

With contributions by

Ruben JakobFaculty of MathematicsUniversity of Duisburg-EssenCampus DuisburgForsthausweg 247048 DuisburgGermanyruben.jakob@uni-due.de

Albrecht KüsterWalter-Flex-Str. 1947809 KrefeldGermany

This volume is the first part of a treatise on Minimal Surfaces in the series Grundlehren der mathemati-schen Wissenschaften. Part Two is Vol. 340 ISBN 978-3-642-11699-5, Part Three is Vol. 341 ISBN 978-3-642-11705-3. A 1st edition of the treatise appeared as Vols. 295 and 296 of the same series.

ISSN 0072-7830ISBN 978-3-642-11697-1 e-ISBN 978-3-642-11698-8DOI 10.1007/978-3-642-11698-8Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010930922

Mathematics Subject Classification (2010): 49Q05, 53A05, 53A07, 53B20, 35J20, 35J4

c© Springer-Verlag Berlin Heidelberg 1992, 2010This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations areliable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.

Cover design: VTEX, Vilnius

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is the first volume of a treatise on minimal surfaces consisting of al-together three volumes, which can be read and studied independently of eachother. The central theme is boundary value problems for minimal surfaces,such as Plateau’s problem. The present treatise forms a greatly extended ver-sion of the monograph Minimal Surfaces I, II by U. Dierkes, S. Hildebrandt,A. Kuster, and O. Wohlrab, published in 1992, which is often cited in the liter-ature as [DHKW]. New coauthors are Friedrich Sauvigny for the first volumeand Anthony J. Tromba for the second and third volume.

The topic of the treatise, belonging to differential geometry and the cal-culus of variations as well as to the theory of partial differential equationsand functions of a complex variable, may at a first glance seem rather special.However, we believe that it is both attractive and advantageous to considermathematical ideas in the light of special problems, even though mathemati-cians nowadays often tend to prefer the opposite approach, namely to em-phasize general theories while relegating specific problems to play the modestrole of examples. Both ways to present mathematics are equally valuable andnecessary, but the theory of minimal surfaces is a good case for the first ap-proach, to study in some detail examples which are as fascinating as they areimportant.

Our intention in writing this book is best characterized by a quote fromCourant’s treatise Dirichlet’s principle which in several respects has been amodel for our work: “Enlightenment, however, must come from an under-standing of motives; live mathematical development springs from specific nat-ural problems which can easily be understood, but whose solutions are difficultand demand new methods of more general significance.”

One might think that three books are more than enough in order to give amore or less complete presentation of the theory of minimal surfaces, but wefailed in many respects. Thus the reader should not expect an encyclopedictreatment of the theory of minimal surfaces, but merely an introduction tothe field, followed by a more thorough presentation of certain aspects which

v

vi Preface

relate to boundary value problems. For further study we refer to our extensivebibliography as well as to comments and references in the Scholia attachedto each chapter. In particular, we mention the various lecture notes, citedat the beginning of our bibliography, as well as the treatises by Rado [21],Courant [15], Osserman [10], Federer [1], Nitsche [28,37], Giusti [4], Massariand Miranda [1], Struwe [11], Simon [8], Jost [17], and Giaquinta, Modica,and Soucek [1].

As Courant remarked, “in a field which has attracted so many mathemati-cians it is difficult to achieve a fair accounting of the literature and to appraisethe merits of others.” By adding Scholia to each chapter we have tried to givea sufficiently detailed account of how the theory of minimal surfaces has de-veloped and what are the basic sources of information and inspiration, andwe hope that not too many were omitted.

We thank M. Beeson, F. Duzaar, K. Große-Brauckmann, R. Jakob, J.Jost, E. Kuwert, F. Muller, M. Pingen, F. Tomi, H. von der Mosel, and D.Wienholtz for pointing out errors and misprints in [DHKW]. Special thankswe owe to Ruben Jakob who studied and corrected most of the new materialadded to [DHKW], thereby eliminating numerous mistakes. His assistancewas invaluable. Moreover, Chapter 6 of this volume is substantially inspiredby his diploma thesis [1]. We also thank Robert Osserman for providing uswith Example 5 in Section 3.7, and Albrecht Kuster for his cooperation inwriting [DHKW], and for numerous illustrations supplied by him.

We should also like to thank David Hoffman, Hermann Karcher, KonradPolthier and Meinhard Wohlgemuth for permitting us to use some of theirdrawings of complete and of periodic minimal surfaces, and Imme Haubitzfor allowing us to reproduce some of her drawings of Thomsen surfaces. Weare grateful to Klaus Bach, Frei Otto and Eric Pitts for providing us withphotographs of various soap film experiments.

The continued support of our work by the Sonderforschungsbereich 611 atBonn University as well as by the Hausdorff Institute for Mathematics in Bonnand its director Matthias Kreck was invaluable. We also thank the Centrodi Ricerca Matematica Ennio De Giorgi in Pisa and its director MarianoGiaquinta for generous support of our work.

We are especially grateful to Anke Thiedemann and Birgit Dunkel whoprofessionally and with untiring patience typed many versions of the newtext.

Last but not least we should like to thank our publisher and in partic-ular our very patient editors, Catriona Byrne, Marina Reizakis, and AngelaSchulze-Thomin, for their encouragement and support.

DuisburgBonnCottbus

Ulrich DierkesStefan HildebrandtFriedrich Sauvigny

Contents

Part I. Introduction to the Geometry of Surfaces and to MinimalSurfaces

Chapter 1. Differential Geometry of Surfaces in Three-DimensionalEuclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Surfaces in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Gauss Map, Weingarten Map. First, Second and Third

Fundamental Form. Mean Curvature and Gauss Curvature . . 91.3 Gauss’s Representation Formula, Christoffel Symbols,

Gauss–Codazzi Equations, Theorema Egregium, Minding’sFormula for the Geodesic Curvature . . . . . . . . . . . . . . . . . . . . . . 24

1.4 Conformal Parameters, Gauss–Bonnet Theorem . . . . . . . . . . . 331.5 Covariant Differentiation. The Beltrami Operator . . . . . . . . . . 391.6 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 2. Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1 First Variation of Area. Minimal Surfaces . . . . . . . . . . . . . . . . . 542.2 Nonparametric Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 582.3 Conformal Representation and Analyticity

of Nonparametric Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . 622.4 Bernstein’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.5 Two Characterizations of Minimal Surfaces . . . . . . . . . . . . . . . 722.6 Parametric Surfaces in Conformal Parameters. Conformal

Representation of Minimal Surfaces. General Definitionof Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.7 A Formula for the Mean Curvature . . . . . . . . . . . . . . . . . . . . . . . 782.8 Absolute and Relative Minima of Area . . . . . . . . . . . . . . . . . . . 822.9 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

vii

viii Contents

Chapter 3. Representation Formulas and Examples of Minimal Surfaces . 91

3.1 The Adjoint Surface. Minimal Surfaces as Isotropic Curvesin C

3. Associate Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . 933.2 Behavior of Minimal Surfaces Near Branch Points . . . . . . . . . . 1043.3 Representation Formulas for Minimal Surfaces . . . . . . . . . . . . . 1113.4 Bjorling’s Problem. Straight Lines and Planar Lines of

Curvature on Minimal Surfaces. Schwarzian Chains . . . . . . . . 1243.5 Examples of Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.5.1 Catenoid and Helicoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.5.2 Scherk’s Second Surface: The General Minimal

Surface of Helicoidal Type . . . . . . . . . . . . . . . . . . . . . . . . 1463.5.3 The Enneper Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.5.4 Bour Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.5.5 Thomsen Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.5.6 Scherk’s First Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.5.7 The Henneberg Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.5.8 Catalan’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.5.9 Schwarz’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

3.6 Complete Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833.7 Omissions of the Gauss Map of Complete Minimal Surfaces . 1903.8 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Color Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Part II. Plateau’s Problem

Chapter 4. The Plateau Problem and the Partially Free BoundaryProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

4.1 Area Functional Versus Dirichlet Integral . . . . . . . . . . . . . . . . . 2464.2 Rigorous Formulation of Plateau’s Problem and of the

Minimization Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2514.3 Existence Proof, Part I: Solution of the Variational Problem . 2554.4 The Courant–Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 2604.5 Existence Proof, Part II: Conformality of Minimizers of the

Dirichlet Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.6 Variant of the Existence Proof. The Partially Free

Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2754.7 Boundary Behavior of Minimal Surfaces with Rectifiable

Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2824.8 Reflection Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2894.9 Uniqueness and Nonuniqueness Questions . . . . . . . . . . . . . . . . . 2924.10 Another Solution of Plateau’s Problem by Minimizing Area . 2994.11 The Mapping Theorems of Riemann and Lichtenstein . . . . . . 3054.12 Solution of Plateau’s Problem for Nonrectifiable Boundaries . 314

Contents ix

4.13 Plateau’s Problem for Cartan Functionals . . . . . . . . . . . . . . . . . 3204.14 Isoperimetric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3274.15 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Chapter 5. Stable Minimal- and H-Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 365

5.1 H-Surfaces and Their Normals . . . . . . . . . . . . . . . . . . . . . . . . . . 3675.2 Bonnet’s Mapping and Bonnet’s Surface . . . . . . . . . . . . . . . . . . 3715.3 The Second Variation of F for H-Surfaces and Their

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3765.4 On μ-Stable Immersions of Constant Mean Curvature . . . . . . 3825.5 Curvature Estimates for Stable and Immersed cmc-Surfaces . 3895.6 Nitsche’s Uniqueness Theorem and Field-Immersions . . . . . . . 3955.7 Some Finiteness Results for Plateau’s Problem . . . . . . . . . . . . 4075.8 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Chapter 6. Unstable Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

6.1 Courant’s Function Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4266.2 Courant’s Mountain Pass Lemma . . . . . . . . . . . . . . . . . . . . . . . . 4386.3 Unstable Minimal Surfaces in a Polygon . . . . . . . . . . . . . . . . . . 4426.4 The Douglas Functional. Convergence Theorems for

Harmonic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4506.5 When Is the Limes Superior of a Sequence of Paths Again

a Path? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.6 Unstable Minimal Surfaces in Rectifiable Boundaries . . . . . . . 4636.7 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

6.7.1 Historical Remarks and References to the Literature . . 4726.7.2 The Theorem of the Wall for Minimal Surfaces

in Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4736.7.3 Sources for This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 4746.7.4 Multiply Connected Unstable Minimal Surfaces . . . . . . 4746.7.5 Quasi-Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Chapter 7. Graphs with Prescribed Mean Curvature . . . . . . . . . . . . . . . . . . 493

7.1 H-Surfaces with a One-to-One Projection onto a Plane,and the Nonparametric Dirichlet Problem . . . . . . . . . . . . . . . . . 494

7.2 Unique Solvability of Plateau’s Problem for Contours witha Nonconvex Projection onto a Plane . . . . . . . . . . . . . . . . . . . . . 508

7.3 Miscellaneous Estimates for Nonparametric H-Surfaces . . . . . 5167.4 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

Chapter 8. Introduction to the Douglas Problem . . . . . . . . . . . . . . . . . . . . . 531

8.1 The Douglas Problem. Examples and Main Result . . . . . . . . . 5328.2 Conformality of Minimizers of D in C(Γ ) . . . . . . . . . . . . . . . . . 538

x Contents

8.3 Cohesive Sequences of Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 5528.4 Solution of the Douglas Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5618.5 Useful Modifications of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 5638.6 Douglas Condition and Douglas Problem. . . . . . . . . . . . . . . . . . 5688.7 Further Discussion of the Douglas Condition . . . . . . . . . . . . . . 5788.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.9 Scholia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Appendix 1. On Relative Minimizers of Area and Energy . . . . . . . . . . . . . 589

Appendix 2. Minimal Surfaces in Heisenberg Groups . . . . . . . . . . . . . . . . . 597

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

Introduction

This text on minimal surfaces is arranged in three volumes, which in the sequelwill be quoted as Vol. 1 (Minimal Surfaces), Vol. 2 (Regularity of MinimalSurfaces), and Vol. 3 (Global Analysis of Minimal Surfaces). Each volume isdivided into two parts.

The present volume is in many ways an introduction to differential geome-try and to the classical theory of minimal surfaces, and the first four chaptersshould more or less be readable for any graduate student. For these the onlyprerequisites are the elements of vector analysis and some basic knowledgeof complex analysis. After an exposition of the basic ideas of the theory ofsurfaces in three-dimensional Euclidean space given in Chapter 1, we beginChapter 2 by introducing minimal surfaces as regular surfaces which are sta-tionary points of the area functional. This is equivalent to the fact that themean curvature H of such a surface vanishes identically. Then we show thatany minimal surface can be represented both in an elementary and a geomet-rically significant way by conformal parameters. In general this representationwill only be local. However, invoking the uniformization theorem, we are led toglobal conformal representations. This reasoning will suggest a new definitionof minimal surfaces that includes the old one but is much more convenient:a minimal surface X(w) is defined as a nonconstant harmonic mapping froma parameter domain Ω in the complex plane into R

3 which satisfies the con-formality relation 〈Xw, Xw 〉 = 0. Note that such a mapping X may haveisolated zeros of its derivative Xw = 1

2 (Xu − iXv), called branch points. Hencea minimal surface in this general sense need not be a regular surface, i.e. animmersion, and therefore one occasionally speaks of a branched minimal sur-face. Mostly we do not use this notation; for us a minimal surface X : Ω → R

3

is a harmonic mapping with conformal parameters. If the parameters of X arenot conformal, but X is an immersion with H ≡ 0, we often speak of a zeromean curvature surface (e.g. in Chapters 5 and 7), except if X is also thegraph of a real valued function z(x, y), (x, y) ∈ Ω; then X is said to be anonparametric minimal surface.

xi

xii Introduction

Other parts of Chapter 2 are concerned with basic features of nonparamet-ric minimal surfaces such as Bernstein’s theorem, stating that entire solutionsof the nonparametric minimal surface equation in R

2 have to be planes, andwith foliations by one-parameter families of minimal surfaces and their signif-icance in establishing the minimum property. Finally we derive the formulafor the second variation of area.

The third chapter deals with the classical theory of minimal surfaces whichis in particular connected with the names of Monge, Scherk, Bonnet, Weier-strass, Riemann, Enneper, and Schwarz. First we show that minimal surfacescan be viewed as real parts of holomorphic isotropic curves in C

3. This inturn leads us to representation formulas of minimal surfaces by means of ar-bitrary meromorphic functions. We shall see how the Gauss map, the secondfundamental form and the Gauss curvature of a minimal surface can be com-puted from such a representation formula. The reader might particularly enjoySection 3.5 where we present some of the celebrated minimal surfaces, mostof which have been known for more than a century, and illustrate them bynumerous drawings.

In Section 3.6 we introduce the notion of a global minimal surface and inparticular that of a complete minimal surface. The spherical image of completeminimal surfaces is then studied in Section 3.7. We present some results ofthe work of Osserman–Xavier–Fujimoto which can be viewed as a profoundgeneralization of both Bernstein’s theorem and of Picard’s theorem in complexanalysis which in turn led to Nevanlinna’s value distribution theory. In theScholia we give a brief survey of some of the more recent results on completeand on periodic minimal surfaces. Here the development of the last twentyyears has brought many new results which are not at all covered by thischapter. We refer the interested reader to the reports in GTMS (2005), theencyclopaedia article by Hoffman and Karcher, and the survey by Rosenberg(1992).

The second part of the present volume deals with the existence of minimalsurfaces which are bounded by prescribed boundary configurations. In Chap-ter 4 we treat the simplest problem of this kind, the Plateau problem. This isthe question of whether one can find a minimal surface spanning a given closedJordan curve Γ . We present the celebrated existence theorem of Douglas andRado in the form described by Courant and Tonelli. A slight variation of theirmethod then leads to solutions of partially free boundary problems. Furthersections as well as the Scholia are concerned with Schwarz’s reflection prin-ciples, obstacle problems, the existence of regular and of embedded minimalsurfaces, the isoperimetric inequality, and in particular with the question ofwhether there can be more than one solution of Plateau’s problem.

In Chapter 4 we only use the simplest method to prove existence resultswhich is based on Dirichlet’s principle. This is to say, we obtain solutions of agiven boundary problem by minimizing Dirichlet’s integral within a suitableclass C(Γ ) of mappings. This method does not give all solutions as it only leadsto minimizers and misses the unstable minimal surfaces and even the relative

Introduction xiii

minima. Furthermore we prove that minimizers of Dirichlet’s integral in C(Γ )also minimize area in C(Γ ). The same method is used to derive Lichtenstein’stheorem on conformal representation of regular surfaces and to solve Plateau’sproblem for regular Cartan functionals. In addition we show that every Jordancurve bounds a minimal surface, even if this surface cannot be obtained byminimizing area.

In Chapter 5 we study stable minimal surfaces and stable surfaces of pre-scribed mean curvature (“H-surfaces”). Here the essential tool is the stabilityinequality, which for minimal surfaces expresses the fact that the second vari-ation of the area functional is nonnegative. The basic results of this chapterare curvature estimates, field embeddings, Nitsche’s uniqueness theorem, andvarious “finiteness results”, in particular Tomi’s theorem. Some of these re-sults are used in Chapter 7 to treat the Dirichlet problem for nonparametricH-surfaces. Here we also apply results on the solvability of the Plateau prob-lem for (parametric) H-surfaces, to be proved in Section 4.7 of Vol. 2.

Chapter 6 deals with the existence of unstable minimal surfaces when themountain-pass lemma can be applied. We present Courant’s approach to thisproblem.

Finally, in Chapter 8 we present an introduction to the general problem ofPlateau that, justifiedly, is often called the Douglas problem. This is the ques-tion whether a configuration of several nonintersecting closed Jordan curvesin R

3 may bound multiply connected minimal surfaces of prescribed Eulercharacteristic and prescribed character of orientability. In a general form, theDouglas problem will be tackled in Vol. 3. Here we treat only the simplest formof the problem, namely to find a minimal surface bounded by a prescribedconfiguration which is parametrized on a “schlicht” domain in C = R

2, pre-cisely, on a k-circle domain in C. As it will be seen in Vol. 2, there is notalways a solution; however, we prove the existence of a solution if Douglas’ssufficient condition is satisfied. This solution is a minimizer both of area andof Dirichlet’s integral. For example, Douglas’s condition holds if the boundaryconfiguration consists of two linked closed Jordan curves.

In many ways the material of this volume is self-contained; but there aresome exceptions. We use a few ideas from Sobolev space theory, and in Chap-ters 4–8 we also apply basic results from the regularity theory of minimalsurfaces which will be established in Vol. 2. In fact, Volume 2 can be regardedas an exercise in regularity theory for nonlinear boundary value problems ofelliptic systems. Nevertheless, regularity results are not only an interestingexercise in generalizing classical results on conformal mappings to minimalsurfaces and to H-surfaces, but they may also have interesting applicationsin geometry, for instance in establishing compactness results, index theorems,or geometric inequalities such as estimates on the length of the free trace, orgeneralized Gauss–Bonnet formulas.

Actually, the notions of regular curve, regular surface, regularity are usedin an ambiguous way. On the one hand, regularity of a map X : Ω → R

3

can mean that X is smooth and belongs to a class C1, C2, . . . , Cs, C∞, Cω,

xiv Introduction

or to a Holder class Ck,α, or to a Sobolev class Hkp . The regularity results

obtained in Chapter 2 of Vol. 2 are to be understood in this sense. On the otherhand, a map X : Ω → R

3, viewed as a parameter representation of a surfacein R

3, is called regular or a regular surface or an immersion if the Jacobimatrix (Xu, Xv) has rank 2, i.e., if at all of its points the surface has a well-defined tangent space. If X(w), w = u + iv, is given in conformal parameters,then the singular (i.e., nonregular) points of X are exactly its branch pointsw0, which are characterized by the relation Xw(w0) = 0. In Chapter 3 ofVol. 2 we shall derive asymptotic expansions of minimal surfaces at boundarybranch points, which can be seen as a generalization of Taylor’s formula to thenonanalytic case. Chapter 1 of Vol. 2 deals with minimal surfaces having freeboundary values. This is a generalization of the partially free boundary valueproblem studied in Section 4.6 of Vol. 1. Chapter 2 presents the basic resultson the boundary behavior of minimal surfaces under Plateau or free boundaryconditions, and asymptotic expansions at branch points as well as the generalGauss–Bonnet formula for branched surfaces are derived. In Chapter 3, theHartmann–Wintner–Heinz technique for obtaining asymptotic expansions isdescribed, together with Dziuk’s expansions at singular boundary points.

The second part of Vol. 2 deals with geometric properties of minimal sur-faces and H-surfaces, furthermore with obstacle problems and the Plateauproblem for H-surfaces. As a generalization of the isoperimetric inequality,the thread problem for minimal surfaces is studied. The volume ends with anew approach by A. Tromba towards the celebrated result that a minimizerof area in a given contour has no interior branch points.

The first part of Vol. 3 investigates solutions of partially free boundaryvalue problems. Then we study various generalizations of Bernstein’s theoremfor minimal surfaces. These results, and even more so those of Part II of Vol. 3,are of a global nature. In this second part, a version of the general Plateauproblem (the “Douglas problem”) is solved by an approach via Teichmullertheory, and then the fundamental index theorems by Bohme and Tromba andby Tomi and Tromba are proved. In the final chapter of Vol. 3 methods fromglobal analysis are applied to Plateau’s problem.

The prospective reader will probably find many sections of the presentvolume elementary, in that they require only basic knowledge of analysis andthat the exposition of the principal facts is fairly broad. The presentationof Volumes 2 and 3 is somewhat more advanced although we have tried todevelop the necessary facts from potential theory ab ovo. Only a few resultsof regularity theory will be borrowed from other sources; usually this will beinformation needed for more refined statements such as higher regularity atthe boundary. For asymptotic expansions in corners we rely on some resultstaken from Vekua’s treatise [1,2] and from the work of Dziuk. Part II ofVol. 3 probably requires additional reading since we use results about Riemannsurfaces and from Teichmuller theory as well as from Global Analysis.

All the Scholia provide sources of additional information. In particular, wetry to give credit to the authorship of the results presented in the main text,

Introduction xv

and we sketch some of the main lines of the historical development. Referencesto the literature and brief surveys of relevant topics, not treated in our notes,complete the picture.

Our notation is essentially the same as in the treatises of Morrey [8] and ofGilbarg and Trudinger [1]. Sobolev spaces are denoted by Hk

p instead of W k,p;the definition of the classes C0, Ck, C∞, and Ck,α is the same as in Gilbarg andTrudinger [1]; Cω denotes the class of real analytic functions; C∞

c (Ω) standsfor the set of C∞-functions with compact support in Ω. For greater precisionwe write Ck(Ω, R3) for the class of Ck-mappings X : Ω → R

3, whereas thecorresponding class of scalar functions is denoted by Ck(Ω), and likewise weproceed for the other classes of differentiability. Another standard symbol isBr(w0) for the disk {w = u + iv ∈ C : |w − w0| < r} in the complex plane. Ifformulas become too cumbersome to read, we shall occasionally write B(w0, r)instead of Br(w0). In general we shall deal with minimal surfaces defined onsimply connected bounded parameter domains Ω in C which, by Riemann’smapping theorem, all are conformally equivalent to each other. Hence wecan pick a standard representation B for Ω: we take it to be either the unitdisk {w : |w| < 1} or the semidisk {w : |w| < 1, Im w > 0}. In the first casewe write C for ∂B, in the second C will denote the semicircle {w : |w| = 1,Imw > 0} while I stands for the interval {u ∈ R : |u| < 1}. On some occasionsit 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 transformedmutatis mutandis to the other one. This may sometimes require slight changesbut we have refrained from pedantic adjustments which the reader can easilysupply himself.