Plateau's problem is a scientific trend in modern mathematics that unites several different problems connected with the study of minimal surfaces. In its simplest version, Plateau's problem is concerned with finding a surface of least area that spans a given fixed one-dimensional contour in three-dimensional space--perhaps the best-known example of such surfaces is provided by soap films. From the mathematical point of view, such films are described as solutions of a second-order partial differential equation, so their behavior is quite complicated and has still not been thoroughly studied. Soap films, or, more generally, interfaces between physical media in equilibrium, arise in many applied problems in chemistry, physics, and also in nature. In applications, one finds not only two-dimensional but also multidimensional minimal surfaces that span fixed closed ``contours'' in some multidimensional Riemannian space. An exact mathematical statement of the problem of finding a surface of least area or volume requires the formulation of definitions of such fundamental concepts as a surface, its boundary, minimality of a surface, and so on. It turns out that there are several natural definitions of these concepts, which permit the study of minimal surfaces by different, and complementary, methods. In the framework of this comparatively small book it would be almost impossible to cover all aspects of the modern problem of Plateau, to which a vast literature has been devoted. However, this book makes a unique contribution to this literature, for the authors' guiding principle was to present the material with a maximum of clarity and a minimum of formalization. Chapter 1 contains historical background on Plateau's problem, referring to the period preceding the 1930s, and a description of its connections with the natural sciences. This part is intended for a very wide circle of readers and is accessible, for example, to first-year graduate students. The next part of the book, comprising Chapters 2-5, gives a fairly complete survey of various modern trends in Plateau's problem. This section is accessible to second- and third-year students specializing in physics and mathematics. The remaining chapters present a detailed exposition of one of these trends (the homotopic version of Plateau's problem in terms of stratified multivarifolds) and the Plateau problem in homogeneous symplectic spaces. This last part is intended for specialists interested in the modern theory of minimal surfaces and can be used for special courses; a command of the concepts of functional analysis is assumed.
Author(s): Dao Trong Thi and A. T. Fomenko
Series: TMM084
Publisher: American Mathematical Society
Year: 1991
Language: English
Pages: 417
Tags: Математика;Высшая геометрия;
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
