The Plateau Problem, Part I: Historical Survey (Studies in the Development of Modern Mathematics)

Cover......Page 1
Volume 1: II Present State of the Theory......Page 2
THE PLATEAU PROBLEM:The Present State of the Theory. Part II......Page 3
©......Page 4
CONTENTS......Page 5
List of Contents of Part I......Page 8
Preface......Page 11
1. Two-dimensional minimal surfaces in Euclidean space and in Riemannian manifolds.......Page 13
2. The multi-dimensional Riemannian volume functional and second fundamental form on a submanifold.......Page 21
3. Multidimensional locally minimal surfaces......Page 28
4. The global minimality of complex submanifolds......Page 30
5. The complex Plateau problem......Page 32
6. On various approaches to the concepts of surface and boundary of a surface......Page 33
7. The homology boundary of a surface and the role of the coefficient group......Page 36
8. Surprising examples of physical stable minimal surfaces, that nevertheless retract onto their boundaries......Page 40
9. When does a soap film spanning a frame not contain closed soap-bubbles?......Page 53
§2 Integral Currents\r......Page 60
1. De Rham currents. Basic notions.......Page 63
2. Rectifiable currents and flat chains......Page 66
3. Normal and integral currents......Page 67
4. Various formulations of the minimal current existence theorem.......Page 69
5. Varifolds and minimal surfaces.......Page 71
6. The interior regularity for minimal surfaces and the structure of their singular points.......Page 77
7. Regularity almost everywhere for the supports of ellipticintegrand-minimizing k-currents and k-varifolds.......Page 79
8. The interior regularity for volume-minimizing hypersurfaces and the existence of minimal cones of codimension one.......Page 81
9. Dimension estimates for the set of singular points of a minimal surface.......Page 83
10. Other problems of minimal surface regularity.......Page 86
1. Minimal cones associated with singular points of minimal surfaces.......Page 87
2. Multidimensional minimal cones.......Page 90
3. Minimal surfaces invariant with respect to the action of Lie groups.......Page 100
4. The Fermat principle, minimal cones, and light rays.......Page 105
5. S.N. Bernstein\'s problem.......Page 116
1. Bordant manifolds and the multidimensional Plateau problem.......Page 120
2. The properties of bordant manifold classes......Page 127
3. The statement of the existence theorem for globally minimal surfaces in the spectral bordant manifold class.......Page 135
1. The definition of generalized homology and cohomology.......Page 138
2. The coboundary and boundary of a pair of spaces (X,A)......Page 141
3. Surface variational classes.......Page 142
4. The general existence theorem for globally minimal surfaces in an arbitrary class determined by a generalized spectral homology or cohomology theory.......Page 144
5. A short sketch of the proof of Theorem 1.......Page 147
1 The functional multivarifold language.......Page 154
2. Multivarifolds and variational problems in classes of surfaces of fIxed topological type.......Page 162
3. Minimization problems for generalized integrands in the parametrization and parametrized multivarifold classes.......Page 166
4. Criteria for the global minimality of surfaces and currents.......Page 170
§7 Cases where a Solution of the Dirichlet Problem for the Equation of Minimal Surfaces of High Codimensions does not Exist......Page 177
§8 Example of a Smooth, Closed, Unknotted Curve inR3, Bounding Only Minimal Surfaces of Large Genus.......Page 187
1. The universal lower estimate of the volumes of topologically non-trivial minimal surfaces.......Page 194
2. The coefficient of deformation of a vector field.......Page 197
3. Surfaces of non-trivial topological type and of least volume.......Page 198
4 On the minimal volume of surfaces passing through the centre of a symmetric convex domain in Euclidean space.......Page 209
1. Totally geodesic submanifolds in Lie groups.......Page 213
3. When does a totally geodesic submanifold realize a non-trivial cycle?......Page 215
4. The classification theorem for totally geodesic submanifolds realizing non-trivial cycles in symmetric spaces.......Page 219
5. The classification of cocycles realizable by totally geodesic spheres in compact Lie groups.......Page 220
6. The classification of elements of homotopy groups realizable by totally geodesic spheres in symmetric spaces of type I.......Page 223
1. The explicit description of unitary Bott periodicity.......Page 226
2. Unitary Bott periodicity follows from the properties of the two-dimensional extremals of the Dirichlet functional.......Page 229
3. Orthogonal periodicity follows from the properties of the eight-dimensional extremals of the Dirichlet functional.......Page 231
§12 Survey of Some Recent Results in Harmonic Mapping Theory.......Page 233
References and Bibliography......Page 246
INDEX......Page 261
Back......Page 264