Part I. Boundary Behaviour of Minimal Surfaces
Chapter 1. Minimal Surfaces with Free Boundaries ................ 3
1.1 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 ..................................................... 5
1.2 Classes of Admissible Functions. Linking Condition ........ 18
1.3 Existence of Minimizers for the Free Boundary Problem ..... 21
1.4 Stationary Minimal Surfaces with Free or Partially Free
Boundaries and the Transversality Condition ............... 28
1.5 Necessary Conditions for Stationary Minimal Surfaces ...... 35
1.6 Existence of Stationary Minimal Surfaces in a Simplex ..... 39
1.7 Stationary Minimal Surfaces of Disk-Type in a Sphere ...... 41
1.8 Report on the Existence of Stationary Minimal Surfaces
in Convex Bodies .......................................... 43
1.9 Nonuniqueness of Solutions to a Free Boundary Problem.
Families of Solutions ..................................... 45
1.10 Scholia ................................................... 65
Chapter 2. The Boundary Behaviour of Minimal Surfaces .......... 75
2.1 Potential-Theoretic Preparations .......................... 76
2.2 Solutions of Differential Inequalities .................... 90
2.3 The Boundary Regularity of Minimal Surfaces Bounded by
Jordan Arcs .............................................. 102
2.4 The Boundary Behaviour of Minimal Surfaces at Their
Free Boundary: A Survey of the Results and an Outline
of Their Proofs .......................................... 112
2.5 Holder Continuity for Minima ............................. 118
2.6 Holder Continuity for Stationary Surfaces ................ 130
2.7 C¹,½ -Regularity ......................................... 153
2.8 Higher Regularity in Case of Support Surfaces with
Empty Boundaries. Analytic Continuation Across a Free
Boundary ................................................. 174
2.9 A Different Approach to Boundary Regularity .............. 181
2.10 Asymptotic Expansion of Minimal Surfaces at Boundary
Branch Points and Geometric Consequences ................. 189
2.11 The Gauss-Bonnet Formula for Branched Minimal Surfaces ... 193
2.12 Scholia .................................................. 200
Chapter 3. Singular Boundary Points of Minimal Surfaces ....... 213
3.1 The Method of Hartman and Wintner, and Asymptotic
Expansions at Boundary Branch Points ..................... 214
3.2 A Gradient Estimate at Singularities Corresponding to
Corners of the Boundary .................................. 235
3.3 Minimal Surfaces with Piecewise Smooth Boundary
Curves and Their Asymptotic Behaviour at Corners ......... 245
3.4 An Asymptotic Expansion for Solutions of the Partially
Free Boundary Problem .................................... 259
3.5 Scholia .................................................. 271
3.5.1 References ........................................ 271
3.5.2 Holder Continuity at Intersection Points .......... 271
Part II. Geometric Properties of Minimal Surfaces and H-
Surfaces
Chapter 4. Enclosure and Existence Theorems for Minimal
Surfaces and H-Surfaces. Isoperimetric
Inequalities ....................................... 279
4.1 Applications of the Maximum Principle and Nonexistence
of Multiply Connected Minimal Surfaces with Prescribed
Boundaries ............................................... 280
4.2 Touching H-Surfaces and Enclosure Theorems. Further
Nonexistence Results ..................................... 284
4.3 Minimal Submanifolds and Submanifolds of Bounded Mean
Curvature. An Optimal Nonexistence Result ................ 295
4.3.1 An Optimal Nonexistence Result for Minimal
Submanifolds of Codimension One ................... 311
4.4 Geometric Maximum Principles ............................. 314
4.4.1 The Barrier Principle for Submanifolds of
Arbitrary Codimension ............................. 314
4.4.2 A Geometric Inclusion Principle for Strong
Subsolutions ..................................... 322
4.5 Isoperimetric Inequalities ............................... 332
4.6 Estimates for the Length of the Free Trace ............... 346
4.7 Obstacle Problems and Existence Results for Surfaces of
Prescribed Mean Curvature ................................ 371
4.8 Surfaces of Prescribed Mean Curvature in a Riemannian
Manifold ................................................. 407
4.8.1 Estimates for Jacobi Fields ....................... 408
4.8.2 Riemann Normal Coordinates ........................ 418
4.8.3 Surfaces of Prescribed Mean Curvature in a
Riemannian Manifold ............................... 424
4.9 Scholia .................................................. 431
4.9.1 Enclosure Theorems and Nonexistence ............... 431
4.9.2 The Isoperimetric Problem. Historical Remarks
and References to the Literature .................. 433
4.9.3 Experimental Proof of the Isoperimetric
Inequality......................................... 435
4.9.4 Estimates for the Length of the Free Trace ........ 435
4.9.5 The Plateau Problem for H-Surfaces ................ 437
Chapter 5. The Thread Problem ................................. 441
5.1 Experiments and Examples. Mathematical Formulation of
the Simplest Thread Problem .............................. 441
5.2 Existence of Solutions to the Thread Problem ............. 446
5.3 Analyticity of the Movable Boundary ...................... 463
5.4 Scholia .................................................. 483
Chapter 6. Branch Points ...................................... 487
6.1 The First Five Variations of Dirichlet's Integral, and
Forced Jacobi Fields ..................................... 488
6.2 The Theorem for n + 1 Even and m + 1 Odd ................. 519
6.3 Boundary Branch Points ................................... 528
6.4 Scholia .................................................. 554
Bibliography .................................................. 561
Index ......................................................... 619
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