| Preface | 6 |
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| Contents | 8 |
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| Introduction | 12 |
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| Part I. Free Boundaries and Bernstein Theorems | 18 |
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| Minimal Surfaces with Supporting Half-Planes | 20 |
| An Experiment | 21 |
| Examples of Minimal Surfaces with Cusps on the Supporting Surface | 24 |
| Setup of the Problem. Properties of Stationary Solutions | 28 |
| Classification of the Contact Sets | 30 |
| Nonparametric Representation, Uniqueness, and Symmetry of Solutions | 35 |
| Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet's Integral | 38 |
| Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II | 40 |
| Final Results on the Shape of the Trace. Absence of Cusps. Optimal Boundary Regularity | 43 |
| Proof of the Representation Theorem | 45 |
| Scholia | 51 |
| Embedded Minimal Surfaces with Partially Free Boundaries | 54 |
| The Geometric Setup | 55 |
| Inclusion and Monotonicity of the Free Boundary Values | 61 |
| A Modification of the Kneser-Radó Theorem | 67 |
| Properties of the Gauss Map, and Stable Surfaces | 69 |
| Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface | 77 |
| Uniqueness of Freely Stable Minimal Surfaces | 83 |
| Asymptotic Expansions | 91 |
| Edge Creeping | 103 |
| Embedded Minimizers for Nonsmooth Supporting Surfaces | 113 |
| A Bernstein Theorem for Minimal Surfaces in a Wedge | 125 |
| Scholia | 143 |
| Bernstein Theorems and Related Results | 152 |
| Entire and Exterior Minimal Graphs of Controlled Growth | 154 |
| Jörgens's Theorem | 154 |
| Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser's Bernstein Theorem | 157 |
| The Interior Gradient Estimate and Consequences | 161 |
| First and Second Variation Formulae | 162 |
| First and Second Variation of the Area Integral | 163 |
| First and Second Variation Formulae for Singular Minimal Surfaces | 169 |
| Some Geometric Identities | 173 |
| Covariant Derivatives of Tensor Fields | 176 |
| Simons's Identity and Jacobi's Field Equation | 178 |
| Nonexistence of Stable Cones and Integral Curvature Estimates. Further Bernstein Theorems | 180 |
| Stability of Minimal Cones | 181 |
| Nonexistence of Stable Cones | 189 |
| Integral Curvature Estimates for Minimal and alpha-Minimal Hypersurfaces. Further Bernstein Theorems | 197 |
| Monotonicity and Mean Value Formulae. Michael-Simon Inequalities | 215 |
| Pointwise Curvature Estimates | 234 |
| Scholia | 253 |
| References to the Literature on Bernstein's Theorem and Curvature Estimates for n = 2 | 253 |
| Bernstein Theorems and Curvature Estimates for n | 253 |
| 255 | 253 |
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| Bernstein Theorems in Higher Codimensions | 259 |
| Sobolev Inequalities | 262 |
| Part II. Global Analysis of Minimal Surfaces | 265 |
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| The General Problem of Plateau: Another Approach | 266 |
| The General Problem of Plateau. Formulation and Examples | 266 |
| A Geometric Approach to Teichmüller Theory of Oriented Surfaces | 272 |
| Symmetric Riemann Surfaces and Their Teichmüller Spaces | 280 |
| The Mumford Compactness Theorem | 288 |
| The Variational Problem | 293 |
| Existence Results for the General Problem of Plateau in R3 | 302 |
| Scholia | 313 |
| The Index Theorems for Minimal Surfaces of Zero and Higher Genus | 316 |
| Introduction | 316 |
| The Statement of the Index Theorem of Genus Zero | 319 |
| Stratification of Harmonic Surfaces by Singularity Type | 321 |
| Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type | 335 |
| The Index Theorem for Classical Minimal Surfaces | 341 |
| The Forced Jacobi Fields | 346 |
| Some Theorems on the Linear Algebra of Fredholm Maps | 358 |
| Generic Finiteness, Stability, and the Stratification of the Sets Mlambda0 | 364 |
| The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries | 370 |
| Review of Some Basic Results in Riemann Surface Theory | 371 |
| Vector Bundles over Teichmüller Space | 376 |
| Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions | 381 |
| Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators | 382 |
| The Corank of the Partial Conformality Operators | 386 |
| The Corank of the Complete Conformality Operators | 394 |
| Manifolds of Harmonic Surfaces of Prescribed Branching Type | 397 |
| The Proof of the Index Theorem | 402 |
| Scholia | 416 |
| Euler Characteristic and Morse Theory for Minimal Surfaces | 418 |
| Fredholm Vector Fields | 419 |
| The Gradient Vector Field Associated to Plateau's Problem | 422 |
| The Euler Characteristic chi(Walpha) of Walpha | 428 |
| The Sard-Brown Theorem for Functionals | 440 |
| The Morse Lemma | 441 |
| The Normal Form of Dirichlet's Energy about a Generic Minimal Surface in R3 | 453 |
| The Local Winding Number of Walpha about a Generically Branched Minimal Surface in R3 | 459 |
| Scholia | 464 |
| Historical Remarks and References to the Literature | 464 |
| On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory | 466 |
| Bibliography | 494 |
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| Index | 548 |
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