Regularity of Minimal Surfaces

by Dierkes, Ulrich; Hildebrandt, Stefan; Tromba, Anthony J.; Kuster, Albrecht (CON)

• ISBN13: 9783642116995
• ISBN10: 364211699X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2010-10-30
• Publisher: Springer Nature

Summary

"Regularity of Minimal Surfaces" begins with survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaceswith 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 deriva 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 Osserman-Gulliver-Alt theorem that area minimizing solutions of Plateau´s problem have no interior branch points.

Table of Contents

Boundary Behaviour of Minimal Surfaces
Minimal Surfaces with Free Boundaries	p. 3
Surfaces of Class H21 and Homotopy Classes of Their Boundary Curves. Nonsolvability of the Free Boundary Problem with Fixed Homotopy Type of the Boundary Traces	p. 5
Classes of Admissible Functions. Linking Condition	p. 18
Existence of Minimizers for the Free Boundary Problem	p. 21
Stationary Minimal Surfaces with Free or Partially Free Boundaries and the Transversality Condition	p. 28
Necessary Conditions for Stationary Minimal Surfaces	p. 35
Existence of Stationary Minimal Surfaces in a Simplex	p. 39
Stationary Minimal Surfaces of Disk-Type in a Sphere	p. 41
Report on the Existence of Stationary Minimal Surfaces in Convex Bodies	p. 43
Nonuniqueness of Solutions to a Free Boundary Problem. Families of Solutions	p. 45
Scholia	p. 65
The Boundary Behaviour of Minimal Surfaces	p. 75
Potential-Theoretic Preparations	p. 76
Solutions of Differential Inequalities	p. 90
The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs	p. 102
The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs	p. 112
Hölder Continuity for Minima	p. 118
Hölder Continuity for Stationary Surfaces	p. 130
C1,1/2-Regularity	p. 153
Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary	p. 174
A Different Approach to Boundary Regularity	p. 181
Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences	p. 189
The Gauss-Bonnet Formula for Branched Minimal Surfaces	p. 193
Scholia	p. 200
Singular Boundary Points of Minimal Surfaces	p. 213
The Method of Hartman and Wintner, and Asymptotic Expansions at Boundary Branch Points	p. 214
A Gradient Estimate at Singularities Corresponding to Corners of the Boundary	p. 235
Minimal Surfaces with Piecewise Smooth Boundary Curves and their Asymptotic Behaviour at Corners	p. 245
An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem	p. 259
Scholia	p. 271
References	p. 271
Hölder Continuity at Intersection Points	p. 271
Geometric Properties of Minimal Surfaces and H-Surfaces
Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities	p. 279
Applications of the Maximum Principle and Nonexistence of Multiply Connected Minimal Surfaces with Prescribed Boundaries	p. 280
Touching H-Surfaces and Enclosure Theorems. Further Nonexistence Results	p. 284
Minimal Submanifolds and Submanifolds of Bounded Mean Curvature. An Optimal Nonexistence Result	p. 295
An Optimal Nonexistence Result for Minimal Submanifolds of Codimension One	p. 311
Geometric Maximum Principles	p. 314
The Barrier Principle for Submanifolds of Arbitrary Codimension	p. 314
A Geometric Inclusion Principle for Strong Subsolutions	p. 322
Isoperimetric Inequalities	p. 332
Estimates for the Length of the Free Trace	p. 346
Obstacle Problems and Existence Results for Surfaces of Prescribed Mean Curvature	p. 371
Surfaces of Prescribed Mean Curvature in a Riemannian Manifold	p. 407
Estimates for Jacobi Fields	p. 408
Riemann Normal Coordinates	p. 418
Surfaces of Prescribed Mean Curvature in a Riemannian Manifold	p. 424
Scholia	p. 431
Enclosure Theorems and Nonexistence	p. 431
The Isoperimetric Problem. Historical Remarks and References to the Literature	p. 433
Experimental Proof of the Isoperimetric Inequality	p. 435
Estimates for the Length of the Free Trace	p. 435
The Plateau Problem for H-Surfaces	p. 437
The Thread Problem	p. 441
Experiments and Examples. Mathematical Formulation of the Simplest Thread Problem	p. 441
Existence of Solutions to the Thread Problem	p. 446
Analyticity of the Movable Boundary	p. 463
Scholia	p. 483
Branch Points	p. 487
The First Five Variations of Dirichlet's Integral, and Forced Jacobi Fields	p. 488
The Theorem for n + 1 Even and m + 1 Odd	p. 519
Boundary Branch Points	p. 528
Scholia	p. 554
Bibliography	p. 561
Index	p. 619
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