Minimal surfaces, stratified multivarifolds, and the Plateau problem

Table of Contents......Page 4
Preface......Page 8
Introduction ......Page 12
1. The sources of multidimensional calculus of variations ......Page 32
2. The 19th century-the epoch of discovery of the main properties of minimal surfaces ......Page 41
3. Topological and physical properties of two-dimensional mimimal surfaces ......Page 54
4. Plateau's four experimental principles and their consequences for two-dimensional mimimal surfaces ......Page 73
5. Two-dimensional minimal surfaces in Euclidean space and in a Riemannian manifold ......Page 79
1. Groups of singular and cellular homology ......Page 106
2. Cohomology groups ......Page 108
1. Minimal surfaces and homology ......Page 110
2. Theory of currents and varifolds ......Page 140
3. The theory of minimal cones and the equivariant Plateau problem ......Page 149
1. The solution of the multidimensional Plateau problem in the class of spectra of maps of smooth manifolds with a fixed boundary ......Page 178
2. Some versions of Plateau's problem require for their statement the concepts of generalized homology and cohomology ......Page 189
3. In certain cases the Dirichlet problem for the equation of a minimal surface of large codimension does not have a solution ......Page 194
4. Some new methods for an effective construction of globally minimal surfaces in Riemannian manifolds ......Page 197
1. The multidimensional Dirichlet functional and harmonic maps ......Page 218
2. Connections between the topology of manifolds and properties of harmonic maps ......Page 228
3. Some unsolved problems ......Page 239
1. Classical formulations ......Page 244
2. Multidimensional variational problems ......Page 245
3. The functional language of multivarifolds ......Page 250
4. Statement of Problems B, B', and B" in the language of the theory of multivarifolds ......Page 260
1. The topology of the space of multivarifolds ......Page 264
2. Local characteristics of multivarifolds ......Page 271
3. Induced maps ......Page 277
1. Spaces of parametrizations and parametrized multivarifolds ......Page 284
2. The structure of spaces of parametrizations and parametrized multivarifolds ......Page 291
3. Exact parametrizations ......Page 301
4. Real and integral multivarifolds ......Page 306
1. A theorem on deformation ......Page 310
2. Isoperimetric inequalities ......Page 320
3. Statement of variational problems in classes of parametrizations and parametrized multivarifolds ......Page 325
4. Existence and properties of minimal parametrizations and parametrized multivarifolds ......Page 328
1. Statement of the problem in the functional language of currents ......Page 342
2. Generalized forms and their properties ......Page 345
3. Conditions for global minimality of currents ......Page 347
4. Globally minimal currents in symmetric problems ......Page 353
5. Specific examples of globally minimal currents and surfaces ......Page 361
1. Statement of the problem. Formulation of the main theorem ......Page 370
2. Necessary information from the theory of representations of the compact Lie groups ......Page 372
3. Topological structure of the space G/TG ......Page 376
4. A brief outline of the proof of the main theorem ......Page 379
Appendix. Volumes of Closed Minimal Surfaces and the Connection with the Tensor Curvature of the Ambient Riemannian Space ......Page 388
Bibliography ......Page 392
Subject Index ......Page 412