Global analysis of minimal surfaces

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Global analysis of minimal surfaces / Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba

版:

Rev. and enl. 2nd ed

資料種別:

図書

出版情報:

Heidelberg ; Berlin : Springer, c2010

形態:

xvi, 537 p. ; 25 cm

シリーズ名:

Die Grundlehren der mathematischen Wissenschaften ; 341 <BA00007422>

著者名:

ISBN:

9783642117053 [3642117058]

書誌ID:

BB03583212

子書誌情報

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書誌詳細

注記:

Bibliography: p. 477-529
Includes index

主題:

Boundary value problems; Minimal surfaces

言語:

英語

目次情報:

Free Boundaries and Bernstein Theorems / Part I：

Minimal Surfaces with Supporting Half-Planes / Chapter 1：

An Experiment / 1.1：

Examples of Minimal Surfaces with Cusps on the Supporting Surface / 1.2：

Setup of the Problem Properties of Stationary Solutions / 1.3：

Classification of the Contact Sets / 1.4：

Nonparametric Representation, Uniqueness, and Symmetry of Solutions / 1.5：

Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet's Integral / 1.6：

Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II / 1.7：

Final Results on the Shape of the Trace Absence of Cusps Optimal Boundary Regularity / 1.8：

Proof of the Representation Theorem / 1.9：

Scholia / 1.10：

Embedded Minimal Surfaces with Partially Free Boundaries / Chapter 2：

The Geometric Setup / 2.1：

Inclusion and Monotonicity of the Free Boundary Values / 2.2：

A Modification of the Kneser-RadÃ³ Theorem / 2.3：

Properties of the Gauss Map, and Stable Surfaces / 2.4：

Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface / 2.5：

Uniqueness of Freely Stable Minimal Surfaces / 2.6：

Asymptotic Expansions / 2.7：

Edge Creeping / 2.8：

Embedded Minimizers for Nonsmooth Supporting Surfaces / 2.9：

A Bernstein 'Theorem for Minimal Surfaces in a Wedge / 2.10：

Bernstein Theorems and Related Results / 2.11：

Entire and Exterior Minimal Graphs of Controlled Growth / 3.1：

JÃ¶rgens's Theorem / 3.1.1：

Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser's Bernstein Theorem / 3.1.2：

The Interior Gradient Estimate and Consequences / 3.1.3：

First and Second Variation Formulae / 3.2：

First and Second Variation of the Area Integral / 3.2.1：

First and Second Variation Formulae for Singular Minimal Surfaces / 3.2.2：

Some Geometric Identities / 3.3：

Covariant Derivatives of Tensor Fields / 3.3.1：

Simons's Identity and Jacobi's Field Equation / 3.3.2：

Nonexistence of Stable Cones and Integral Curvature Estimates Further Bernstein Theorems / 3.4：

Stability of Minimal Cones / 3.4.1：

Nonexistence of Stable Cones / 3.4.2：

Integral Curvature Estimates for Minimal and ?-Minimal Hypersurfaces. Further Bernstein Theorems / 3.4.3：

Monotonicity and Mean Value Formulae Michael-Simon Inequalities / 3.5：

Pointwise Curvature Estimates / 3.6：

References to the Literature on Bernstein's Theorem and Curvature Estimates for n = 2 / 3.7：

Bernstein Theorems and Curvature Estimates for n ? 3 dimensions / 3.7.2：

Bernstein Theorems in Higher Codimensions / 3.7.3：

Sobolev Inequalities / 3.7.4：

Global Analysis of Minimal Surfaces / Part II：

The General Problem of Plateau: Another Approach / Chapter 4：

The General Problem of Plateau Formulation and Examples / 4.1：

A Geometric Approach to TeichmÃ¼ller Theory of Oriented Surfaces / 4.2：

Symmetric Riemann Surfaces and Their TeichmÃ¼ller Spaces / 4.3：

The Mumford Compactness Theorem / 4.4：

The Variational Problem / 4.5：

The Index Theorems for Minimal Surfaces of Zero and Higher Genus / 4.6：

Introduction / 5.1：

The Statement of the Index Theorem of Genus Zero / 5.2：

Stratification of Harmonic Surfaces by Singularity Type / 5.3：

Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type / 5.4：

The Index Theorem for Classical Minimal Surfaces / 5.5：

The Forced Jacobi Fields / 5.6：

Some Theorems on the Linear Algebra of Fredholm Maps / 5.7：

The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries / 5.8：

Review of Some Basic Results in Riemann Surface Theory / 5.10：

Vector Bundles over TeichmÃ¼ller Space / 5.11：

Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions / 5.12：

Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators / 5.13：

The Corank of the Partial Conformality Operators / 5.14：

The Corank of the Complete Conformality Operators / 5.15：

Manifolds of Harmonic Surfaces of Prescribed Branching Type / 5.16：

The Proof of the Index Theorem / 5.17：

Euler Characteristic and Morse Theory for Minimal Surfaces / 5.18：

Fredholm Vector Fields / 6.1：

The Gradient Vector Field Associated to Plateau's Problem / 6.2：

The Sard-Brown Theorem for Functionals / 6.3：

The Morse Lemma / 6.5：

Historical Remarks and References to the Literature / 6.6：

On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory / 6.8.2：

Bibliography

Index

Free Boundaries and Bernstein Theorems / Part I：

Minimal Surfaces with Supporting Half-Planes / Chapter 1：

An Experiment / 1.1：

Examples of Minimal Surfaces with Cusps on the Supporting Surface / 1.2：

Setup of the Problem Properties of Stationary Solutions / 1.3：

Classification of the Contact Sets / 1.4：

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