Global Analysis of Minimal Surfaces

by Dierkes, Ulrich; Hildebrandt, Stefan; Tromba, Anthony J.

• ISBN13: 9783642117053
• ISBN10: 3642117058
• Edition: 2nd
• Format: Hardcover
• Copyright: 2010-10-30
• Publisher: Springer Nature

Summary

Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary.The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived.The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau´s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.Of particular interest are the so-called "forced Jacobi fields", which play an important role both for the index theorems and for the branch point theory developed in Vol.2 (Nr. 340).

Table of Contents

Free Boundaries and Bernstein Theorems
Minimal Surfaces with Supporting Half-Planes	p. 3
An Experiment	p. 4
Examples of Minimal Surfaces with Cusps on the Supporting Surface	p. 7
Setup of the Problem Properties of Stationary Solutions	p. 11
Classification of the Contact Sets	p. 13
Nonparametric Representation, Uniqueness, and Symmetry of Solutions	p. 18
Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet's Integral	p. 21
Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II	p. 23
Final Results on the Shape of the Trace Absence of Cusps Optimal Boundary Regularity	p. 26
Proof of the Representation Theorem	p. 28
Scholia	p. 34
Embedded Minimal Surfaces with Partially Free Boundaries	p. 37
The Geometric Setup	p. 38
Inclusion and Monotonicity of the Free Boundary Values	p. 44
A Modification of the Kneser-Radó Theorem	p. 50
Properties of the Gauss Map, and Stable Surfaces	p. 52
Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface	p. 60
Uniqueness of Freely Stable Minimal Surfaces	p. 66
Asymptotic Expansions	p. 74
Edge Creeping	p. 86
Embedded Minimizers for Nonsmooth Supporting Surfaces	p. 96
A Bernstein 'Theorem for Minimal Surfaces in a Wedge	p. 108
Scholia	p. 126
Bernstein Theorems and Related Results	p. 135
Entire and Exterior Minimal Graphs of Controlled Growth	p. 137
Jörgens's Theorem	p. 137
Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser's Bernstein Theorem	p. 140
The Interior Gradient Estimate and Consequences	p. 144
First and Second Variation Formulae	p. 145
First and Second Variation of the Area Integral	p. 146
First and Second Variation Formulae for Singular Minimal Surfaces	p. 152
Some Geometric Identities	p. 156
Covariant Derivatives of Tensor Fields	p. 159
Simons's Identity and Jacobi's Field Equation	p. 161
Nonexistence of Stable Cones and Integral Curvature Estimates Further Bernstein Theorems	p. 163
Stability of Minimal Cones	p. 164
Nonexistence of Stable Cones	p. 172
Integral Curvature Estimates for Minimal and ¿-Minimal Hypersurfaces. Further Bernstein Theorems	p. 180
Monotonicity and Mean Value Formulae Michael-Simon Inequalities	p. 198
Pointwise Curvature Estimates	p. 217
Scholia	p. 236
References to the Literature on Bernstein's Theorem and Curvature Estimates for n = 2	p. 236
Bernstein Theorems and Curvature Estimates for n ≥ 3 dimensions	p. 238
Bernstein Theorems in Higher Codimensions	p. 242
Sobolev Inequalities	p. 245
Global Analysis of Minimal Surfaces
The General Problem of Plateau: Another Approach	p. 249
The General Problem of Plateau Formulation and Examples	p. 249
A Geometric Approach to Teichmüller Theory of Oriented Surfaces	p. 255
Symmetric Riemann Surfaces and Their Teichmüller Spaces	p. 263
The Mumford Compactness Theorem	p. 271
The Variational Problem	p. 276
Existence Results for the General Problem of Plateau in R3	p. 285
Scholia	p. 296
The Index Theorems for Minimal Surfaces of Zero and Higher Genus	p. 299
Introduction	p. 299
The Statement of the Index Theorem of Genus Zero	p. 302
Stratification of Harmonic Surfaces by Singularity Type	p. 304
Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type	p. 318
The Index Theorem for Classical Minimal Surfaces	p. 324
The Forced Jacobi Fields	p. 329
Some Theorems on the Linear Algebra of Fredholm Maps	p. 341
Generic Finiteness, Stability, and the Stratification of the Sets M¿0	p. 347
The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries	p. 353
Review of Some Basic Results in Riemann Surface Theory	p. 354
Vector Bundles over Teichmüller Space	p. 359
Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions	p. 364
Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators	p. 365
The Corank of the Partial Conformality Operators	p. 369
The Corank of the Complete Conformality Operators	p. 377
Manifolds of Harmonic Surfaces of Prescribed Branching Type	p. 380
The Proof of the Index Theorem	p. 385
Scholia	p. 399
Euler Characteristic and Morse Theory for Minimal Surfaces	p. 401
Fredholm Vector Fields	p. 402
The Gradient Vector Field Associated to Plateau's Problem	p. 405
The Euler Characteristic X(W¿) of W¿	p. 411
The Sard-Brown Theorem for Functionals	p. 423
The Morse Lemma	p. 424
The Normal Form of Dirichlet's Energy about a Generic Minimal Surface in R3	p. 436
The Local Winding Number of W¿ about a Generically Branched Minimal Surface in R3	p. 442
Scholia	p. 447
Historical Remarks and References to the Literature	p. 447
On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory	p. 449
Bibliography	p. 477
Index	p. 531
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