Minimal Surfaces

by Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich

• ISBN13: 9783642116971
• ISBN10: 3642116973
• Edition: 2nd
• Format: Hardcover
• Copyright: 2010-10-30
• Publisher: Springer Nature

Summary

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume which can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces.The treatise is a greatly 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 surfacesof zero mean curvature. The final definition of a minimal surface is that of a non-constant harmónic mapping X: O-> R3 which is conformally parametrized on O-> R2 and may have branch points. Thereafter the classical theroy 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 theorem 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 suurfaces 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 dor H-surfaces and the pertinent estimates.

Table of Contents

Introduction to the Geometry of Surfaces and to Minimal Surfaces
Differential Geometry of Surfaces in Three-Dimensional Euclidean Space	p. 3
Surfaces in Euclidean Space	p. 4
Gauss Map, Weingarten Map First, Second and Third Fundamental Form. Mean Curvature and Gauss Curvature	p. 9
Gauss's Representation Formula, Christoffel Symbols, Gauss-Codazzi Equations, Theorema Egregium, Minding's Formula for the Geodesic Curvature	p. 24
Conformal Parameters, Gauss-Bonnet Theorem	p. 33
Covariant Differentiation The Beltrami Operator	p. 39
Scholia	p. 47
Minimal Surfaces	p. 53
First Variation of Area Minimal Surfaces	p. 54
Nonparametric Minimal Surfaces	p. 58
Conformal Representation and Analyticity of Nonparametric Minimal Surfaces	p. 62
Bernstein's Theorem	p. 66
Two Characterizations of Minimal Surfaces	p. 72
Parametric Surfaces in Conformal Parameters. Conformal Representation of Minimal Surfaces. General Definition of Minimal Surfaces	p. 75
A Formula for the Mean Curvature	p. 78
Absolute and Relative Minima of Area	p. 82
Scholia	p. 86
Representation Formulas and Examples of Minimal Surfaces	p. 91
The Adjoint Surface. Minimal Surfaces as Isotropic Curves in C3 Associate Minimal Surfaces	p. 93
Behavior of Minimal Surfaces Near Branch Points	p. 104
Representation Formulas for Minimal Surfaces	p. 111
Bjorling's Problem Straight Lines and Planar Lines of Curvature on Minimal Surfaces. Schwarzian Chains	p. 124
Examples of Minimal Surfaces	p. 141
Catenoid and Helicoid	p. 141
Scherk's Second Surface: The General Minimal Surface of Helicoidal Type	p. 146
The Enneper Surface	p. 151
Bour Surfaces	p. 155
Thomsen Surfaces	p. 156
Scherk's First Surface	p. 156
The Henneberg Surface	p. 166
Catalan's Surface	p. 171
Schwarz's Surface	p. 182
Complete Minimal Surfaces	p. 183
Omissions of the Gauss Map of Complete Minimal Surfaces	p. 190
Scholia	p. 200
Color Plates	p. 229
Plateau's Problem
The Plateau Problem and the Partially Free Boundary Problem	p. 239
Area Functional Versus Dirichlet Integral	p. 246
Rigorous Formulation of Plateau's Problem and of the Minimization Process	p. 251
Existence Proof, Part I: Solution of the Variational Problem	p. 255
The Courant-Lebesgue Lemma	p. 260
Existence Proof, Part II: Conformality of Minimizers of the Dirichlet Integral	p. 263
Variant of the Existence Proof. The Partially Free Boundary Problem	p. 275
Boundary Behavior of Minimal Surfaces with Rectifiable Boundaries	p. 282
Reflection Principles	p. 289
Uniqueness and Nonuniqueness Questions	p. 292
Another Solution of Plateau's Problem by Minimizing Area	p. 299
The Mapping Theorems of Riemann and Lichtenstein	p. 305
Solution of Plateau's Problem for Nonrectifiable Boundaries	p. 314
Plateau's Problem for Cartan Functionals	p. 320
Isoperimetric Inequalities	p. 327
Scholia	p. 335
Stable Minimal- and H-Surfaces	p. 365
H-Surfaces and Their Normals	p. 367
Bonnet's Mapping and Bonnet's Surface	p. 371
The Second Variation of F for H-Surfaces and Their Stability	p. 376
On µ-Stable Immersions of Constant Mean Curvature	p. 382
Curvature Estimates for Stable and Immersed cmc-Surfaces	p. 389
Nitsche's Uniqueness Theorem and Field-Immersions	p. 395
Some Finiteness Results for Plateau's Problem	p. 407
Scholia	p. 420
Unstable Minimal Surfaces	p. 425
Courant's Function ¿	p. 426
Courant's Mountain Pass Lemma	p. 438
Unstable Minimal Surfaces in a Polygon	p. 442
The Douglas Functional Convergence Theorems for Harmonic Mappings	p. 450
When Is the Limes Superior of a Sequence of Paths Again a Path?	p. 461
Unstable Minimal Surfaces in Rectifiable Boundaries	p. 463
Scholia	p. 472
Historical Remarks and References to the Literature	p. 472
The Theorem of the Wall for Minimal Surfaces in Textbooks	p. 473
Sources for This Chapter	p. 474
Multiply Connected Unstable Minimal Surfaces	p. 474
Quasi-Minimal Surfaces	p. 474
Graphs with Prescribed Mean Curvature	p. 493
H-Surfaces with a One-to-One Projection onto a Plane, and the Nonparametric Dirichlet Problem	p. 494
Unique Solvability of Plateau's Problem for Contours with a Nonconvex Projection onto a Plane	p. 508
Miscellaneous Estimates for Nonparametric H-Surfaces	p. 516
Scholia	p. 529
Introduction to the Douglas Problem	p. 531
The Douglas Problem Examples and Main Result	p. 532
Conformality of Minimizers of D in $$$(¿)	p. 538
Cohesive Sequences of Mappings	p. 552
Solution of the Douglas Problem	p. 561
Useful Modifications of Surfaces	p. 563
Douglas Condition and Douglas Problem	p. 568
Further Discussion of the Douglas Condition	p. 578
Examples	p. 581
Scholia	p. 584
Problems	p. 587
On Relative Minimizers of Area and Energy	p. 589
Minimal Surfaces in Heisenberg Groups	p. 597
Bibliography	p. 599
Index	p. 681
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