Regularity of Minimal Surfaces (eBook)

Ulrich Dierkes, Stefan Hildebrandt, Anthony Tromba (Autoren)

eBook Download: PDF

2010 | 2nd, rev. and enlarged ed. 2010
XVII, 623 Seiten
Springer Berlin (Verlag)
978-3-642-11700-8 (ISBN)

Infos

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.

Inhalt

Preface 6
Contents 8
Introduction 15
Part I. Boundary Behaviour of Minimal Surfaces 18
Minimal Surfaces with Free Boundaries 19
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
Classes of Admissible Functions. Linking Condition 34
Existence of Minimizers for the Free Boundary Problem 37
Stationary Minimal Surfaces with Free or Partially Free Boundaries and the Transversality Condition 44
Necessary Conditions for Stationary Minimal Surfaces 51
Existence of Stationary Minimal Surfaces in a Simplex 55
Stationary Minimal Surfaces of Disk-Type in a Sphere 57
Report on the Existence of Stationary Minimal Surfaces in Convex Bodies 59
Nonuniqueness of Solutions to a Free Boundary Problem. Families of Solutions 61
Scholia 81
The Boundary Behaviour of Minimal Surfaces 90
Potential-Theoretic Preparations 91
Solutions of Differential Inequalities 105
The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs 117
The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs 127
Hölder Continuity for Minima 133
Hölder Continuity for Stationary Surfaces 145
C1,1/2-Regularity 168
Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary 189
A Different Approach to Boundary Regularity 196
Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences 204
The Gauss-Bonnet Formula for Branched Minimal Surfaces 208
Scholia 215
Singular Boundary Points of Minimal Surfaces 228
The Method of Hartman and Wintner, and Asymptotic Expansions at Boundary Branch Points 229
A Gradient Estimate at Singularities Corresponding to Corners of the Boundary 250
Minimal Surfaces with Piecewise Smooth Boundary Curves and Their Asymptotic Behaviour at Corners 260
An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem 274
Scholia 286
References 286
Hölder Continuity at Intersection Points 286
Part II. Geometric Properties of Minimal Surfaces and H-Surfaces 292
Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities 293
Applications of the Maximum Principle and Nonexistence of Multiply Connected Minimal Surfaces with Prescribed Boundaries 294
Touching H-Surfaces and Enclosure Theorems. Further Nonexistence Results 298
Minimal Submanifolds and Submanifolds of Bounded Mean Curvature. An Optimal Nonexistence Result 309
An Optimal Nonexistence Result for Minimal Submanifolds of Codimension One 325
Geometric Maximum Principles 328
The Barrier Principle for Submanifolds of Arbitrary Codimension 328
A Geometric Inclusion Principle for Strong Subsolutions 336
Isoperimetric Inequalities 346
Estimates for the Length of the Free Trace 360
Obstacle Problems and Existence Results for Surfaces of Prescribed Mean Curvature 385
Surfaces of Prescribed Mean Curvature in a Riemannian Manifold 421
Estimates for Jacobi Fields 422
Riemann Normal Coordinates 432
Surfaces of Prescribed Mean Curvature in a Riemannian Manifold 438
Scholia 445
Enclosure Theorems and Nonexistence 445
The Isoperimetric Problem. Historical Remarks and References to the Literature 447
Experimental Proof of the Isoperimetric Inequality 449
Estimates for the Length of the Free Trace 449
The Plateau Problem for H-Surfaces 451
The Thread Problem 454
Experiments and Examples. Mathematical Formulation of the Simplest Thread Problem 454
Existence of Solutions to the Thread Problem 459
Analyticity of the Movable Boundary 476
Scholia 496
Branch Points 499
The First Five Variations of Dirichlet's Integral, and Forced Jacobi Fields 500
The Theorem for n+1 Even and m+1 Odd 531
Boundary Branch Points 540
Scholia 566
Bibliography 573
Index 630

Produkt-Details

• Erscheint lt. Verlag: 16.8.2010
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Co-Autor: Albrecht Küster
• Zusatzinfo: XVII, 623 p. 68 illus., 6 illus. in color.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: 49Q05,53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20 • Boundary value problem • Calculus of Variations • Conformal Mappings • Differential Geometry • manifold • minimal surface • minimal surfaces • Minimum • Partial differential equations • regularity theory • Riemannian manifold
• ISBN-10: 3-642-11700-7 / 3642117007
• ISBN-13: 978-3-642-11700-8 / 9783642117008