Part I. Free Boundaries and Bernstein Theorems
Chapter 1 Minimal Surfaces with Supporting Half-Planes ......... 3
1.1 An Experiment .............................................. 4
1.2 Examples of Minimal Surfaces with Cusps on the Supporting
Surface .................................................... 7
1.3 Setup of the Problem. Properties of Stationary
Solutions ................................................. 11
1.4 Classification of the Contact Sets ........................ 13
1.5 Nonparametric Representation, Uniqueness, and Symmetry
of Solutions .............................................. 18
1.6 Asymptotic Expansions for Surfaces of Cusp-Types I
and III. Minima of Dirichlet's Integral ................... 21
1.7 Asymptotic Expansions for Surfaces of the Tongue/Loop-
Type II ................................................... 23
1.8 Final Results on the Shape of the Trace. Absence of
Cusps. Optimal Boundary Regularity ........................ 26
1.9 Proof of the Representation Theorem ....................... 28
1.10 Scholia ................................................... 34
Chapter 2 Embedded Minimal Surfaces with Partially Free
Boundaries .......................................... 37
2.1 The Geometric Setup ....................................... 38
2.2 Inclusion and Monotonicity of the Free Boundary Values .... 44
2.3 A Modification of the Kneser-Radó Theorem ................. 50
2.4 Properties of the Gauss Map, and Stable Surfaces .......... 52
2.5 Uniqueness of Minimal Surfaces that Lie on One Side
of the Supporting Surface ................................. 60
2.6 Uniqueness of Freely Stable Minimal Surfaces .............. 66
2.7 Asymptotic Expansions ..................................... 74
2.8 Edge Creeping ............................................. 86
2.9 Embedded Minimizers for Nonsmooth Supporting Surfaces ..... 96
2.10 A Bernstein Theorem for Minimal Surfaces in a Wedge ...... 108
2.11 Scholia .................................................. 126
Chapter 3. Bernstein Theorems and Related Results ............. 135
3.1 Entire and Exterior Minimal Graphs of Controlled
Growth ................................................... 137
3.1.1 Jörgens's Theorem ................................. 137
3.1.2 Asymptotic Behaviour for Solutions of Linear and
Quasilinear Equations, Moser's Bernstein
Theorem ........................................... 140
3.1.3 The Interior Gradient Estimate and Consequences ... 144
3.2 First and Second Variation Formulae ...................... 145
3.2.1 First and Second Variation of the Area Integral ... 146
3.2.2 First and Second Variation Formulae for Singular
Minimal Surfaces .................................. 152
3.3 Some Geometric Identities ................................ 156
3.3.1 Covariant Derivatives of Tensor Fields ............ 159
3.3.2 Simons's Identity and Jacobi's Field Equation ..... 161
3.4 Nonexistence of Stable Cones and Integral Curvature
Estimates. Further Bernstein Theorems .................... 163
3.4.1 Stability of Minimal Cones ........................ 164
3.4.2 Nonexistence of Stable Cones ...................... 172
3.4.3 Integral Curvature Estimates for Minimal and
a-Minimal Hypersurfaces. Further Bernstein
Theorems .......................................... 180
3.5 Monotonicity and Mean Value Formulae. Michael-Simon
Inequalities ............................................. 198
3.6 Pointwise Curvature Estimates ............................ 217
3.7 Scholia .................................................. 236
3.7.1 References to the Literature on Bernstein's
Theorem and Curvature Estimates for n = 2 ......... 236
3.7.2 Bernstein Theorems and Curvature Estimates
for n ≥ 3 dimensions .............................. 238
3.7.3 Bernstein Theorems in Higher Codimensions ......... 242
3.7.4 Sobolev Inequalities .............................. 245
Part II. Global Analysis of Minimal Surfaces
Chapter 4 The General Problem of Plateau: Another Approach ... 249
4.1 The General Problem of Plateau. Formulation and
Examples ................................................. 249
4.2 A Geometric Approach to Teichmьller Theory of Oriented
Surfaces ................................................. 255
4.3 Symmetric Riemann Surfaces and Their Teichmüller
Spaces ................................................... 263
4.4 The Mumford Compactness Theorem .......................... 271
4.5 The Variational Problem .................................. 276
4.6 Existence Results for the General Problem of Plateau in
 3 ....................................................... 285
4.7 Scholia .................................................. 296
Chapter 5 The Index Theorems for Minimal Surfaces of Zero
and Higher Genus ................................... 299
5.1 Introduction ............................................. 299
5.2 The Statement of the Index Theorem of Genus Zero ......... 302
5.3 Stratification of Harmonic Surfaces by Singularity
Type ..................................................... 304
5.4 Stratification of Harmonic Surfaces with Regular
Boundaries by Singularity Type ........................... 318
5.5 The Index Theorem for Classical Minimal Surfaces ......... 324
5.6 The Forced Jacobi Fields ................................. 329
5.7 Some Theorems on the Linear Algebra of Fredholm Maps ..... 341
5.8 Generic Finiteness, Stability, and the Stratification
of the Sets  λ0 .......................................... 347
5.9 The Index Theorem for Higher Genus Minimal Surfaces
Statement and Preliminaries .............................. 353
5.10 Review of Some Basic Results in Riemann Surface Theory ... 354
5.11 Vector Bundles over Teichmüller Space .................... 359
5.12 Some Results on Maximal Ideals in Sobolev Algebras of
Holomorphic Functions .................................... 364
5.13 Minimal Surfaces as Zeros of a Vector Field, and the
Conformality Operators ................................... 365
5.14 The Corank of the Partial Conformality Operators ......... 369
5.15 The Corank of the Complete Conformality Operators ........ 377
5.16 Manifolds of Harmonic Surfaces of Prescribed Branching
Type ..................................................... 380
5.17 The Proof of the Index Theorem ........................... 385
5.18 Scholia .................................................. 399
Chapter 6 Euler Characteristic and Morse Theory for Minimal
Surfaces ........................................... 401
6.1 Fredholm Vector Fields ................................... 402
6.2 The Gradient Vector Field Associated to Plateau's
Problem .................................................. 405
6.3 The Euler Characteristic χ(Wα) of Wα ..................... 411
6.4 The Sard-Brown Theorem for Functional .................... 423
6.5 The Morse Lemma .......................................... 424
6.6 The Normal Form of Dirichlet's Energy about a Generic
Minimal Surface in 3 .................................... 436
6.7 The Local Winding Number of Wα about a Generically
Branched Minimal Surface in 3 ........................... 442
6.8 Scholia .................................................. 447
6.8.1 Historical Remarks and References to the
Literature ........................................ 447
6.8.2 On the Generic Nondegeneracy of Closed Minimal
Surfaces in Riemannian Manifolds and Morse
Theory ............................................ 449
Bibliography .................................................. 477
Index ......................................................... 531
|
