Grundlehren dermathematischen Wissenschaften 339A Series of Comprehensive Studies in Mathematics

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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

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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

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

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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 ofW 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, Imw > 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.

Part I

Introduction to the Geometry ofSurfaces and to Minimal Surfaces

Chapter 1

Differential Geometry of Surfacesin Three-Dimensional Euclidean Space

In this chapter we give a brief introduction to the differential geometry of sur-faces in three-dimensional Euclidean space. The main purpose of this intro-duction is to provide the reader with the basic notions of differential geometryand with the essential formulas that will be needed later on.

Section 1.1 discusses the notion of surfaces that is mainly used in thesenotes. Moreover, the notions of tangent space, surface normal, surface area,equivalent surfaces, as well as tangent and normal vector fields are defined.

In Section 1.2 we consider the spherical image of a surface X and itsnegative differential, the Weingarten map. This leads to the three fundamentalforms on a surface which, in turn, give rise to the definition of the principalcurvatures, and of the Gauss curvature and the mean curvature. By meansof the orthonormal frame {t, s,N} along a curve c on X consisting of thetangent t = c to the curve, the side normal s and the surface normal N,the geodesic curvature and the normal curvature of c are defined. This leadsto the standard interpretation of the principal curvatures and of the Gaussand mean curvatures. The principal curvatures turn out to be the eigenvaluesof the Weingarten map, and the Gauss curvature is interpreted as the ratiodAN

dAXof the area element of a surface X and of its spherical image N if we

take orientation into account. After defining geodesics, asymptotic curves, andlines of curvature, we note the invariance properties of the various curvaturemeasures.

Section 1.3 begins by stating the Gauss equations for the second deriva-tives X,αβ of a surface representation X which lead to the definition of theChristoffel symbols of the first and second kind. It will then be shown thatthese symbols can be expressed in terms of the coefficients of the first funda-mental form whence it follows that the same holds for the Gauss curvature.This is essentially the content of Gauss’s celebrated theorema egregium. Itis proved by connecting the Gauss curvature with the Riemann curvaturetensor. Finally, a general expression for the geodesic curvature of a curve onthe surface is computed. In addition, we provide a collection of formulas for

U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces,

Grundlehren der mathematischen Wissenschaften 339,

DOI 10.1007/978-3-642-11698-8 1, c© Springer-Verlag Berlin Heidelberg 2010 3

4 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

the Christoffel symbols and for the Gauss, mean, and geodesic curvatures inorthogonal and conformal coordinates which will be useful later.

In Section 1.4 we define conformally equivalent surfaces and conformalparameters, discuss the theorem of Gauss–Lichtenstein that surfaces X : Ω →R

3 with Ω ⊂ R2 can be mapped conformally into the plane, and finally we

state and prove different versions of the Gauss–Bonnet theorem by employ-ing conformal representations. This approach is particularly well suited forgeneralizing the Gauss–Bonnet theorem to surfaces with singularities (branchpoints) as we shall see later.

In Section 1.5 we deal with basic vector analysis on surfaces X. Afterintroducing the covariant differentiation of tangential vector fields, we in par-ticular define the X-gradient ∇Xf of a scalar function f as a tangential vectorfield, and the X-divergence of a tangential vector field. Of basic importance isthe Laplace–Beltrami operator ΔX on X, a linear elliptic differential operatorwhich is defined as the divergence of the gradient. After providing an invari-ant form of the Gauss integration theorem involving the Laplace–Beltramioperator, we interpret the Laplace–Beltrami equation ΔXf = 0 as the Eu-ler equation of the generalized Dirichlet integral. We close our discussion bydefining the covariant derivative DV

dt of a tangential vector field V (t) along acurve c(t). Autoparallel vector fields V (t) are defined by the equation DV

dt = 0,and geodesics c(t) on the surface are curves with D

dt c = 0. This turns out tobe equivalent to the fact that such c have zero geodesic curvature so thatthe new definition of geodesics is equivalent to the one of Section 1.2. Finallygeodesics are proved to be the stationary curves of the energy functional andthe length functional.

1.1 Surfaces in Euclidean Space

Most of the surfaces studied in this book are mappings

(1) X : Ω → R3

from a domain Ω in R2 into R

3. The points of R2 are written as w = (u, v) =

(u1, u2) or, if we identify R2 with the complex plane C, as w = u+iv = u1+iu2.Then X maps w ∈ Ω onto some image point X(w) ∈ R

3.In the following, we shall usually assume that X is at least of class C3,

and that X is regular on Ω, except for isolated points. By definition, at allregular points w of Ω, the Jacobian matrix

(2) X∗(w) := ∇X(w) = (Xu(w), Xv(w)) =(∂X

∂u(w),

∂X

∂v(w))

has the maximal rank two. At such points the tangent space TwX of X cor-responding to the parameter value w, defined by

1.1 Surfaces in Euclidean Space 5

Fig. 1. A parametric surface

Fig. 2. (a) An embedded surface X : M → R3. (b) An immersed but not embedded surface

X : M → R3. (c) A branched covering; the branch point in the center has been removed to

provide a better view of the surface. (a) and (b) are parts of Enneper’s surface, (c) a part

of Henneberg’s surface

6 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

(3) TwX := X∗(w)(R2),

is the two-dimensional subspace of R3 spanned by the linearly independentvectors Xu(w) and Xv(w). Note that we have attached the tangent space tothe parameter point w and not to the point P = X(w) on the trace S :=X(Ω) of the surface since selfintersections of S are not excluded. Thus, ifP = X(w) = X(w), w, w ∈ Ω and w = w,S could have the two differenttangent planes P+TwX and P+TwX; in other words, the tangent space TP S

of S at P would in general not be well defined, except for embeddings X.At a regular point w the exterior product Xu ∧ Xv does not vanish, i.e.

W = 0, where

(4) W = |Xu ∧ Xv | =√

|Xu|2|Xv |2 − 〈Xu, Xv 〉2.

Hence, in a neighborhood of a regular point w, the normal vector to thesurface,

(5) N =1W

· Xu ∧ Xv,

is well defined.The area AΩ(X) (or simply A(X)) of the surface X : Ω → R

3 is definedas

(6) AΩ(X) =∫

Ω

|Xu ∧ Xv | du dv.

Introducing the area element dA as

dA = W du dv = |Xu ∧ Xv | du dv,

we may write

(6′) A(X) =∫

X

dA =∫

Ω

W du dv.

Two mappings X : Ω → R3 and X : Ω → R

3 of class Cs, s ≥ 1, are said to beequivalent (strictly speaking: Cs-equivalent) if there is a Cs-diffeomorphismϕ : Ω → Ω, mapping Ω onto Ω, such that

(7) X = X ◦ ϕ or X(w) = X(ϕ(w)) for w ∈ Ω.

Let ϕ(w) be given by ϕ(w) = (α(w), β(w)) and denote the Jacobianof ϕ by

Jϕ = detDϕ =∣∣∣∣αu βu

αv βv

∣∣∣∣ .If, in addition to (7), also the condition

(8) Jϕ > 0

1.1 Surfaces in Euclidean Space 7

is satisfied, the mappings X and X are called strictly equivalent. Since

(Xu ∧ Xv) = Jϕ · (Xα ∧ Xβ) ◦ ϕ,

the transformation theorem for multiple integrals implies that AΩ(X) =AΩ(X), that is, equivalent surfaces have the same area. If the surfaces arestrictly equivalent, we get

|Xu ∧ Xv | −1 · (Xv ∧ Xv) = |Xα ∧ Xβ | −1 · (Xα ∧ Xβ) ◦ ϕ.

Denoting the normal vectors of X and X by N and N , respectively, thisimplies that

(9) N = N ◦ ϕ.

In other words, the normal vectors of two strictly equivalent surfaces are equiv-alent.

Equivalent surfaces can be considered as identical geometric objects. Theyare said to be equally or oppositely oriented if Jϕ > 0 or Jϕ < 0, respectively.

A mapping V : Ω → R3 can be interpreted as a vector field along a

surface X : Ω → R3. The proper geometric picture is to imagine that, for

each w ∈ Ω, the vector V (w) is attached to the point X(w) of the surface. Ofparticular importance are tangential and normal vector fields. We say that Vis tangential, if

(10) V (w) ∈ TwX holds for all w ∈ Ω,

and we call it normal, if

(11) V (w) ⊥ TwX for all w ∈ Ω.

Clearly, a vector field V (w) is tangential along X if and only if it can bewritten in the form

(10′) V (w) = V 1(w)Xu(w) + V 2(w)Xv(w) = V α(w)Xuα(w)

for all w ∈ Ω (summation with respect to repeated Greek indices from 1 to 2),and it is normal along X if and only if it is of the form

(11′) V (w) = λ(w)N(w) for all w ∈ Ω,

with appropriate functions V 1(w), V 2(w) and λ(w) respectively.To make these definitions precise we have to assume that all points of Ω are

regular points of X. If not, we either have to restrict ourselves to sufficientlysmall neighborhoods Ω′ of regular points (instead of Ω), or we must replaceΩ by the (open) set Ω0 of its regular points.

Often a surface X : Ω → R3 can be extended to the closure Ω or at least

to Ω ∪ C, where C is a subset of ∂Ω. Then the previous definitions may becarried over in an appropriate way.

8 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

In most cases we will restrict ourselves to surfaces which are mappingsX : Ω → R

3. Sometimes, however, we shall have to adopt a more global pointof view as, for instance, in Chapter 3. Then by a (general) surface in thethree-dimensional Euclidean space R

3 we mean a mapping

(12) X :M → R3

from a two-dimensional manifold M into R3. The image of any parameter

point p ∈ M will be denoted by X(p). Occasionally we shall also write

X = X(p), p ∈ M,

as symbol for the mapping (12). This is quite convenient although somewhatimprecise. If we have fixed a Cartesian coordinate system in R

3 with coordi-nates x, y, z, the map X(p) will be given by a triple of real-valued functions:

X(p) = (x(p), y(p), z(p)).

Let ∂M be the (possibly empty) boundary of M and M its interior.The differentiability properties of a surface X : M → R

3 are defined bymeans of the charts of an atlas A of M . If, for example, M is at least ofclass Cs, s ≥ 1, then we may say that the surface X is of class Cs up to theboundary (notation: X ∈ Cs(M,R3)), if for every chart ϕ : G → R

2 of theatlas A the local map or local parametrization

X := X ◦ ϕ−1 : ϕ(G) → R3

of the mapping X is of class Cs(ϕ(G),R3). Such a local map X yields aparameter representation of a patch of the global surface on a planar domainΩ = ϕ(G) which is of the type discussed at the beginning.

Similarly, the space Cs(M,R3) will be defined as the set of surfaces X

that are of class Cs on the interior M of M . Usually we shall only admitsurfaces X : M → R

3 which are regular (i.e., each local representation X ofX is regular) or have at most isolated singular points p ∈ M . This still leavesus with a fairly general class of geometric objects. For instance, all embeddedand even all two-dimensional manifolds immersed in R

3 belong to our generalsurfaces, be they orientable or not. Moreover, we have also included branchedcoverings.

The area A(X) of a general surface X : M → R3 can be defined by em-

ploying a partition of unity. Vector fields V : M → R3 along X will be treated

by investigation of their parameterizations V = V ◦ ϕ−1 along the local rep-resentations X = X ◦ ϕ−1 of X.

The reader who is not very familiar with manifolds need not worry. Onlyin Chapter 3 we shall assume more than the most elementary facts aboutthem. Until then we shall only study local surfaces as in (1).

1.2 Mean Curvature and Gauss Curvature 9

1.2 Gauss Map, Weingarten Map. First, Second andThird Fundamental Form. Mean Curvature and GaussCurvature

In the following we shall assume that X : Ω → R3 is a regular surface of

class C3.Let N be its normal field defined by formula (5) of the previous section.

Since |N | = 1, we can view N as a mapping of Ω into the unit sphere S2

of R3,

(1) N : Ω → S2 ⊂ R3.

This mapping will be called the normal map, the spherical map, or the Gaussmap of the surfaceX. The set N(Ω) is called the spherical image of the surfaceX : Ω → R3. However, sometimes also the map N : Ω → S2 will be calledthe spherical image of X, following an old custom of geometers.

Fix now some point w = (u, v) ∈ Ω, and consider the tangential mapN∗(w) of N at w, that is,

N∗(w) = ∇N(w) = (Nu(w), Nv(w)).(2)

Then the Weingarten map S(w) at w is a linear mapping of the tangent spaceTwX into itself,

S(w) : TwX → TwX,(3)

which maps a tangent vector

Fig. 1. A surface X : Ω → R3 and its Gauss map N : Ω → S2. X parametrizes the part of

Enneper’s surface corresponding to Ω = I × I, I = [−1/2, 1/2]

10 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

V = V 1Xu(w) + V 2Xv(w) = V αXuα(w)

onto the vector

(4) S(w)V := −V 1Nu(w) − V 2Nv(w) = −V αNuα(w).

Since 〈N,N〉 = 1, we obtain by differentiation

(5) 〈N,Nu〉 = 0 and 〈N,Nv 〉 = 0,

i.e., the derivatives Nu(w) and Nv(w) are orthogonal to N(w) and must,therefore, be elements of TwX. Hence S(w)V is indeed an element of TwX,as we have stated.

If no misunderstanding is possible, we drop the argument w and write Sinstead of S(w) for the Weingarten map.

We now claim that S is a selfadjoint linear mapping on the tangent spaceTwX equipped with the scalar product 〈V,W 〉 of the surrounding Euclideanspace R

3. In other words, we have

〈SV,W 〉 = 〈V, SW 〉(6)

for arbitrary tangent vectors V = V αXuα(w), W = W βXuβ (w). In fact, theequation 〈N,Xuβ 〉 = 0 implies

(7) 〈Nuα , Xuβ 〉 + 〈N,Xuαuβ 〉 = 0

whence

〈Nuα , Xuβ 〉 = 〈Nuβ , Xuα 〉,(8)

and therefore

〈SV,W 〉 = −〈NuαV α, XuβW β 〉 = −〈Nuα , Xuβ 〉V αW β

= −〈Xuα , Nuβ 〉V αW β = −〈XuαV α, NuβW β 〉 = 〈V, SW 〉.

Thus we can define on TwX three symmetric bilinear forms

(9) I(V,W ) := 〈V,W 〉, II(V,W ) := 〈SV,W 〉, III(V,W ) := 〈SV, SW 〉

for all V,W ∈ TwX, with their corresponding quadratic forms

(10) I(V ) := |V |2, II(V ) := 〈SV , V 〉, III(V ) := |SV |2

which are called first, second , and third fundamental form of the surface Xat w.

The first fundamental form is also called the metric form of X. If itshould be necessary to indicate that these forms depend on w, we writeIw(V ), IIw(V ), IIIw(V ).

1.2 Mean Curvature and Gauss Curvature 11

To understand the geometric meaning of the second fundamental form, weconsider an arbitrary C3-curve ω in Ω which starts at w, for example:

ω : [0, ε] → Ω, ω(0) = w, ω(t) = (ω1(t), ω2(t)).

Then c := X ◦ω is a C3-curve on the surface X with initial point c(0) = X(w)and initial velocity

c(0) = Xuα(w)ωα(0) ∈ TwX.

We note that, by definition,

|c(0)|2 = I(c(0)).

Let us temporarily assume that t is the parameter of arc length s of the curvec; therefore we write c = c(s), 0 ≤ s ≤ l. Then we have |c(s)| = 1. Moreover,

t(s) = c(s)

is the unit tangent vector of the curve c,

κ(s) = |t(s)|

its curvature, and, for κ(s) = 0, its principal normal n(s) is uniquely definedby the equation

t(s) = κ(s)n(s).

Fromt(s) = c(s) = Xuα(ω(s))ωα(s)

we infer

t(s) = c(s) = Xuαuβ (ω(s))ωα(s)ωβ(s) +Xuα(ω(s))ωα(s).

By taking the scalar product of t(0) with N = N(w) we arrive at

〈N, t(0)〉 = 〈N,Xuαuβ (w)〉ωα(0)ωβ(0).

On account of (7), we obtain

〈N, t(0)〉 = −〈Nuα(w), Xuβ (w)〉ωα(0)ωβ(0)

whence by (4)

〈N, t(0)〉 = 〈Sc(0), c(0)〉,(11)

that is,

κ(0)〈N(w),n(0)〉 = 〈Sc(0), c(0)〉.(12)

By defining the surface normal N(s) := N(ω(s)) and the side normals(s) := N(s) ∧ t(s) along c, we obtain the moving orthonormal frame

12 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

Fig. 2. Normal plane, osculating plane, rectifying plane. To describe a curve c : [0, l] → R3

in space satisfying |c| = 1 we introduce: t = c = tangent vector, κ = |t| = curvature,

ρ = 1/κ = radius of curvature, n = ρt = normal vector, b = t ∧ n = binormal vector, M =

c+ρn = center of curvature, τ = − 〈b′, n〉 = 〈n′, b〉 = torsion. The normal plane is spanned

by n and b, the tangent plane S by t and n, and the rectifying plane R by t and b. The

circle of curvature C lies in S and has ρ as its radius. Its center is M ; C is the limit of a circle

through three points P, P ′, P ′′ on the curve as P ′, P ′′ → P : The sphere of curvature has the

center M ∗ = c + ρn+ (ρ/τ)b and intersects the tangent plane S in C. It is defined as limit

of the sphere determined by four points P, P ′, P ′′, P ′′′ on the curve as P ′, P ′′, P ′′′ → P .

Taken from K.H. Naumann and H. Bodeker (in Hilbert and Cohn-Vossen [1])

(13) {t(s), s(s),N(s)}

along the curve c(s), 0 ≤ s ≤ l, where t(s) and s(s) span the tangent spaceTω(s)X, and N(s) is orthogonal to Tω(s)X.

From 〈t, t〉 = 1 we infer that 〈t, t〉 = 0. Thus t(s) is a linear combinationof s(s) and N(s), and we have functions κg(s) and κn(s) such that

(14)dt

ds= κgs + κnN.

One calls κg(s) the geodesic curvature and κn(s) the normal curvature ofthe curve c for the parameter value s. If θ(s) denotes the angle between n(s)and N(s), we have

cos θ = 〈n,N〉,(15)

and it follows that

κg = 〈t, s〉 = κ〈n, s〉 = ±κ sin θ,

κn = 〈t,N〉 = κ〈n,N〉 = κ cos θ, κ =√κ2

g + κ2n.

(16)

1.2 Mean Curvature and Gauss Curvature 13

Equation (12) is therefore equivalent to

(17) κn(0) = IIω(0)(c(0)), c =dc

ds.

If we drop the condition |c(t)| = 1 and return to an arbitrary parametrizationof the curve c, we have ds

dt = |c| =√

I(c), and the chain rule yields

dc

dt=dc

ds

ds

dt=dc

dsI1/2(c), c =

dc

dt.

We therefore infer from (17), that

(18) κn =II(V )I(V )

where I(V ) and II(V ) are the values of the first and second fundamental formsof X at w, and κn is the normal curvature of a curve c = X ◦ω : [0, ε] → R

3 onX with the initial values c(0) = X(w) and c(0) = V at t = 0. If, in particular,V is a velocity vector of a normal section c of an embedded surface X, thevectors N(w) and n(0) are collinear, i.e., κn = ±κ, or

(19) κ = ± II(V )I(V )

.

In other words, the Rayleigh quotient II/I measures the curvatures κ of allpossible normal sections of the surface X at the parameter point w, and thesign of the quotient indicates whether n(0) points in the same direction asN(w) or in the opposite direction, that is, whether the normal section curvestowards N(w) or away from it.

Since the Rayleigh quotient II/I has the meaning of a curvature, we call

κ1 := min{

II(V )I(V )

: V ∈ TwX,V = 0}

(20)

= min{II(V ) : V ∈ TwX, I(V ) = 1},

κ2 := max{

II(V )I(V )

: V ∈ TwX,V = 0}

(21)

= max{II(V ) : V ∈ TwX, I(V ) = 1}

the principal curvatures of the surface X at w. Note that κ1 = κ1(w), κ2 =κ2(w), and II(V ) = IIw(V ) = 〈S(w)V, V 〉. The numbers ρi = 1/κi are said tobe the principal radii of curvature at w.

We can find unit vectors V1, V2 ∈ TwX such that

(22) κ1 = II(V1), κ2 = II(V2).

14 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

An elementary reasoning yields that

(23) SV1 = κ1V1, SV2 = κ2V2.

In fact, we infer from the minimum property (20) that, for all ε ∈ R and allV ∈ TwX,

II(V1 + εV ) ≥ κ1I(V1 + εV )

or

II(V1) + 2εII(V1, V ) + ε2II(V ) ≥ κ1I(V1) + 2κ1εI(V1, V ) + κ1ε2I(V ).

Since II(V1) = κ1 and I(V1) = 1, we arrive at

2ε{II(V1, V ) − κ1I(V1, V )} + ε2[II(V ) − κ1I(V )] ≥ 0

whenceII(V1, V ) − κ1I(V1, V ) = 0

and therefore〈SV 1, V 〉 = κ1〈V1, V 〉 for all V ∈ TwX.

But this is equivalent to the first equation of (23), and similarly the secondequation of (23) can be proved.

In other words, the principal curvatures are the eigenvalues of the Wein-garten map S : TwX → TwX. If κ1 = κ2, we infer from (23) that 〈V1, V2〉 = 0.If κ1 = κ2 =: κ, the point w is said to be an umbilical point of X. In this case,we have

SV = κV for all V ∈ TwX.

Therefore we may choose V1 and V2 ∈ TwX such that

SVi = κiVi, |V1| = |V2| = 1, 〈V1, V2〉 = 0.

Thus the eigenvectors V1, V2 of S can always be assumed to be orthogonal.They can be considered as tangent vectors of normal sections of X at w whichhave the smallest or largest signed curvature; thus they are called principaldirections of curvature of X at w.

Of particular geometric importance are the elementary symmetric func-tions

(24) H = 12 (κ1 + κ2), K = κ1κ2

of the principal curvatures κ1, κ2. One calls H(w) the mean curvature andK(w) the Gauss curvature of X at w.

Let E be the identity map on TwX. By (23), κ1 and κ2 are the roots ofthe characteristic polynomial

p(κ) = det(S − κE)

1.2 Mean Curvature and Gauss Curvature 15

whencep(κ) = (κ − κ1)(κ − κ2) = κ2 − 2Hκ+K.

On account of the Hamilton–Cayley theorem, we arrive at

S2 − 2HS +KE = 0 (= zero map),(25)

and therefore, the identity

KI − 2HII + III = 0(26)

holds.For computations, it is often advantageous to use coordinates, namely the

coefficients of the first, second, and third fundamental forms:

gαβ(w) := 〈Xuα(w), Xuβ (w)〉,bαβ(w) := −〈Nuα(w), Xuβ (w)〉,(27)cαβ(w) := 〈Nuα(w), Nuβ (w)〉.

Obviously,

gαβ = gβα, cαβ = cβα.(28)

Because of (7) and (8), we also have

(29) bαβ = bβα = 〈N,Xuαuβ 〉.

Sometimes, the Gaussian notation

(30) G :=(g11 g12g21 g22

)=(

E F

F G

), B :=

(b11 b12b21 b22

)=(

L M

M N

)

will be used, i.e.,

E = 〈Xu, Xu〉, F = 〈Xu, Xv 〉, G = 〈Xv, Xv 〉,L = −〈Nu, Xu〉 = 〈N,Xuu〉, N = −〈Nv, Xv 〉 = 〈N,Xvv 〉,(31)M = −〈Nu, Xv 〉 = −〈Nv, Xu〉 = 〈N,Xuv 〉 = 〈N,Xvu〉.

We, moreover, write

g := detG = g11g22 − g212,

b := detB = b11b22 − b212.(32)

Then

W =√

EG − F2 =√g =

√det(gαβ),(33)

and, for V = V 1Xu + V 2Xv = V αXuα ,

16 1 Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

I(V ) = gαβ(w)V αV β = E(w)(V 1)2 + 2F(w)V 1V 2 + G(w)(V 2)2,II(V ) = bαβ(w)V αV β = L(w)(V 1)2 + 2M(w)V 1V 2 + N(w)(V 2)2,(34)

III(V ) = cαβVαV β .

We also introduce gαβ(w), setting

(35) G−1 = (gαβ).

Hence

gαβ = gβα and gαβgβγ = δγα,(36)

where δγα is the Kronecker symbol, and

(37)[g11 g12

g21 g22

]=

1W2

[G −F

−F E

].

As we know (cf. (5)), Nu and Nv are contained in TwX and, therefore, linearcombinations of the two linearly independent vectors Xu and Xv. Hence thereare uniquely determined coefficients aβ

α = aβα(w) such that

Nuα = aβαXuβ .

Then〈Nuα , Xuγ 〉 = aβ

α〈Xuβ , Xuγ 〉, i.e., − bαγ = aβαgβγ

whence−bαγg

γν = aβαgβγg

γν = aβαδ

νβ = aν

α.

Thus we have found the Weingarten equations

(38) Nuα = −bβαXuβ with bβα := bαγgγβ .

They can also be written in the form

(39) Nu = aXu + bXv, Nv = cXu + dXv

with

−L = aE + bF, −M = aF + bG,−M = cE + dF, −N = cF + dG,

(40)

or

a = W−2(FM − GL), b = W−2(FL − EM),

c = W−2(FN − GM), d = W−2(FM − EN).(41)

(Note that, in the last three formulas, the coefficient b is not to be confusedwith the determinant b = det(bαβ) = LN − M2!)