Dierkes / Hildebrandt / Tromba Regularity of Minimal Surfaces

2. Auflage 2010
ISBN: 978-3-642-11700-8
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 623 Seiten

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau´s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau´s problem have no interior branch points.

Autoren/Hrsg.

Dierkes, Ulrich

Hildebrandt, Stefan

Tromba, Anthony

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Introduction;15
4;Part I. Boundary Behaviour of Minimal Surfaces;18
4.1;Minimal Surfaces with Free Boundaries;19
4.1.1;Surfaces of Class H12 and Homotopy Classes of Their Boundary Curves. Nonsolvability of the Free Boundary Problem with Fixed Homotopy Type of the Boundary Traces;21
4.1.2;Classes of Admissible Functions. Linking Condition;34
4.1.3;Existence of Minimizers for the Free Boundary Problem;37
4.1.4;Stationary Minimal Surfaces with Free or Partially Free Boundaries and the Transversality Condition;44
4.1.5;Necessary Conditions for Stationary Minimal Surfaces;51
4.1.6;Existence of Stationary Minimal Surfaces in a Simplex;55
4.1.7;Stationary Minimal Surfaces of Disk-Type in a Sphere;57
4.1.8;Report on the Existence of Stationary Minimal Surfaces in Convex Bodies;59
4.1.9;Nonuniqueness of Solutions to a Free Boundary Problem. Families of Solutions;61
4.1.10;Scholia;81
4.2;The Boundary Behaviour of Minimal Surfaces;90
4.2.1;Potential-Theoretic Preparations;91
4.2.2;Solutions of Differential Inequalities;105
4.2.3;The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs;117
4.2.4;The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs;127
4.2.5;Hölder Continuity for Minima;133
4.2.6;Hölder Continuity for Stationary Surfaces;145
4.2.7;C1,1/2-Regularity;168
4.2.8;Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary;189
4.2.9;A Different Approach to Boundary Regularity;196
4.2.10;Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences;204
4.2.11;The Gauss-Bonnet Formula for Branched Minimal Surfaces;208
4.2.12;Scholia;215
4.3;Singular Boundary Points of Minimal Surfaces;228
4.3.1;The Method of Hartman and Wintner, and Asymptotic Expansions at Boundary Branch Points;229
4.3.2;A Gradient Estimate at Singularities Corresponding to Corners of the Boundary;250
4.3.3;Minimal Surfaces with Piecewise Smooth Boundary Curves and Their Asymptotic Behaviour at Corners;260
4.3.4;An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem;274
4.3.5;Scholia;286
4.3.5.1;References;286
4.3.5.2;Hölder Continuity at Intersection Points;286
5;Part II. Geometric Properties of Minimal Surfaces and H-Surfaces;292
5.1;Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities;293
5.1.1;Applications of the Maximum Principle and Nonexistence of Multiply Connected Minimal Surfaces with Prescribed Boundaries;294
5.1.2;Touching H-Surfaces and Enclosure Theorems. Further Nonexistence Results;298
5.1.3;Minimal Submanifolds and Submanifolds of Bounded Mean Curvature. An Optimal Nonexistence Result;309
5.1.3.1;An Optimal Nonexistence Result for Minimal Submanifolds of Codimension One;325
5.1.4;Geometric Maximum Principles;328
5.1.4.1;The Barrier Principle for Submanifolds of Arbitrary Codimension;328
5.1.4.2;A Geometric Inclusion Principle for Strong Subsolutions;336
5.1.5;Isoperimetric Inequalities;346
5.1.6;Estimates for the Length of the Free Trace;360
5.1.7;Obstacle Problems and Existence Results for Surfaces of Prescribed Mean Curvature;385
5.1.8;Surfaces of Prescribed Mean Curvature in a Riemannian Manifold;421
5.1.8.1;Estimates for Jacobi Fields;422
5.1.8.2;Riemann Normal Coordinates;432
5.1.8.3;Surfaces of Prescribed Mean Curvature in a Riemannian Manifold;438
5.1.9;Scholia;445
5.1.9.1;Enclosure Theorems and Nonexistence;445
5.1.9.2;The Isoperimetric Problem. Historical Remarks and References to the Literature;447
5.1.9.3;Experimental Proof of the Isoperimetric Inequality;449
5.1.9.4;Estimates for the Length of the Free Trace;449
5.1.9.5;The Plateau Problem for H-Surfaces;451
5.2;The Thread Problem;454
5.2.1;Experiments and Examples. Mathematical Formulation of the Simplest Thread Problem;454
5.2.2;Existence of Solutions to the Thread Problem;459
5.2.3;Analyticity of the Movable Boundary;476
5.2.4;Scholia;496
5.3;Branch Points;499
5.3.1;The First Five Variations of Dirichlet's Integral, and Forced Jacobi Fields;500
5.3.2;The Theorem for n+1 Even and m+1 Odd;531
5.3.3;Boundary Branch Points;540
5.3.4;Scholia;566
6;Bibliography;573
7;Index;630