Minimal Surfaces (eBook)

Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (Autoren)

eBook Download: PDF

2010 | 2nd, rev. and enlarged ed. 2010
XVI, 692 Seiten
Springer Berlin (Verlag)
978-3-642-11698-8 (ISBN)

Infos

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X:Omega to R^3 which is conformally parametrized on Omega subset R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem and Tomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates.

Inhalt

Preface 6
Contents 8
Introduction 12
Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces 17
Differential Geometry of Surfaces in Three-Dimensional Euclidean Space 19
Surfaces in Euclidean Space 20
Gauss Map, Weingarten Map. First, Second and Third Fundamental Form. Mean Curvature and Gauss Curvature 25
Gauss's Representation Formula, Christoffel Symbols, Gauss-Codazzi Equations, Theorema Egregium, Minding's Formula for the Geodesic Curvature 40
Conformal Parameters, Gauss-Bonnet Theorem 49
Covariant Differentiation. The Beltrami Operator 55
Scholia 63
Textbooks 63
Annotations to the History of the Theory of Surfaces 64
References to the Sources of Differential Geometry and to the Literature on its History 67
Minimal Surfaces 69
First Variation of Area. Minimal Surfaces 70
Nonparametric Minimal Surfaces 74
Conformal Representation and Analyticity of Nonparametric Minimal Surfaces 78
Bernstein's Theorem 82
Two Characterizations of Minimal Surfaces 88
Parametric Surfaces in Conformal Parameters. Conformal Representation of Minimal Surfaces. General Definition of Minimal Surfaces 91
A Formula for the Mean Curvature 94
Absolute and Relative Minima of Area 98
Scholia 102
References to the Literature on Nonparametric Minimal Surfaces 102
Bernstein's Theorem 104
Stable Minimal Surfaces 105
Foliations by Minimal Surfaces 106
Representation Formulas and Examples of Minimal Surfaces 107
The Adjoint Surface. Minimal Surfaces as Isotropic Curves in C3. Associate Minimal Surfaces 109
Behavior of Minimal Surfaces Near Branch Points 120
Representation Formulas for Minimal Surfaces 127
Björling's Problem. Straight Lines and Planar Lines of Curvature on Minimal Surfaces. Schwarzian Chains 140
Examples of Minimal Surfaces 157
Catenoid and Helicoid 157
Scherk's Second Surface: The General Minimal Surface of Helicoidal Type 162
The Enneper Surface 167
Bour Surfaces 171
Thomsen Surfaces 172
Scherk's First Surface 172
The Henneberg Surface 182
Catalan's Surface 187
Schwarz's Surface 198
Complete Minimal Surfaces 199
Omissions of the Gauss Map of Complete Minimal Surfaces 206
Scholia 216
Historical Remarks and References to the Literature 216
Complete Minimal Surfaces of Finite Total Curvature and of Finite Topology 220
Complete Properly Immersed Minimal Surfaces 223
Construction of Minimal Surfaces 224
Triply Periodic Minimal Surfaces 238
Structure of Embedded Minimal Disks 244
Complete Minimal Surfaces and the Plateau Problem 244
Color Plates 245
Part II. Plateau's Problem 254
The Plateau Problem and the Partially Free Boundary Problem 255
Area Functional Versus Dirichlet Integral 262
Rigorous Formulation of Plateau's Problem and of the Minimization Process 267
Existence Proof, Part I: Solution of the Variational Problem 271
The Courant-Lebesgue Lemma 276
Existence Proof, Part II: Conformality of Minimizers of the Dirichlet Integral 279
Variant of the Existence Proof. The Partially Free Boundary Problem 291
Boundary Behavior of Minimal Surfaces with Rectifiable Boundaries 298
Reflection Principles 305
Uniqueness and Nonuniqueness Questions 308
Another Solution of Plateau's Problem by Minimizing Area 315
The Mapping Theorems of Riemann and Lichtenstein 321
Solution of Plateau's Problem for Nonrectifiable Boundaries 330
Plateau's Problem for Cartan Functionals 336
Isoperimetric Inequalities 343
Scholia 351
Historical Remarks and References to the Literature 351
Branch Points 355
Embedded Solutions of Plateau's Problem 356
More on Uniqueness and Nonuniqueness 361
Index Theorems, Generic Finiteness, and Morse-Theory Results 371
Obstacle Problems 373
Systems of Minimal Surfaces 376
Isoperimetric Inequalities 379
Plateau's Problem for Infinite Contours 379
Plateau's Problem for Polygonal Contours 380
Stable Minimal- and H-Surfaces 381
H-Surfaces and Their Normals 383
Bonnet's Mapping and Bonnet's Surface 387
The Second Variation of F for H-Surfaces and Their Stability 392
On µ-Stable Immersions of Constant Mean Curvature 398
Curvature Estimates for Stable and Immersed cmc-Surfaces 405
Nitsche's Uniqueness Theorem and Field-Immersions 411
Some Finiteness Results for Plateau's Problem 423
Scholia 436
Unstable Minimal Surfaces 441
Courant's Function Theta 442
Courant's Mountain Pass Lemma 454
Unstable Minimal Surfaces in a Polygon 458
The Douglas Functional. Convergence Theorems for Harmonic Mappings 466
When Is the Limes Superior of a Sequence of Paths Again a Path? 477
Unstable Minimal Surfaces in Rectifiable Boundaries 479
Scholia 488
Historical Remarks and References to the Literature 488
The Theorem of the Wall for Minimal Surfaces in Textbooks 489
Sources for This Chapter 490
Multiply Connected Unstable Minimal Surfaces 490
Quasi-Minimal Surfaces 490
Graphs with Prescribed Mean Curvature 509
H-Surfaces with a One-to-One Projection onto a Plane, and the Nonparametric Dirichlet Problem 510
Unique Solvability of Plateau's Problem for Contours with a Nonconvex Projection onto a Plane 524
Miscellaneous Estimates for Nonparametric H-Surfaces 532
Scholia 545
Introduction to the Douglas Problem 547
The Douglas Problem. Examples and Main Result 548
Conformality of Minimizers of D in C(Gamma) 554
Cohesive Sequences of Mappings 568
Solution of the Douglas Problem 577
Useful Modifications of Surfaces 579
Douglas Condition and Douglas Problem 584
Further Discussion of the Douglas Condition 594
Examples 597
Scholia 600
Problems 603
On Relative Minimizers of Area and Energy 605
Minimal Surfaces in Heisenberg Groups 613
Bibliography 615
Index 697

Produkt-Details

• Erscheint lt. Verlag: 16.8.2010
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Co-Autor: Ruben Jakob, Albrecht Küster
• Zusatzinfo: XVI, 692 p. 149 illus., 9 illus. in color.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: 49Q05,53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20 • Boundary value problem • Calculus of Variations • Conformal Mappings • Curvature • Differential Geometry • differential geometry of surfaces • mean curvature • minimal surface • minimal surfaces • Minimum • Partial differential equations • Plateau's problem • regularity theory
• ISBN-10: 3-642-11698-1 / 3642116981
• ISBN-13: 978-3-642-11698-8 / 9783642116988