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Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem
Đào Trong Thi and A. T. Fomenko
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Translations of Mathematical Monographs
1991; 404 pp; hardcover
Volume: 84
ISBN-10: 0-8218-4536-5
ISBN-13: 978-0-8218-4536-3
List Price: US$135
Member Price: US$108
Order Code: MMONO/84
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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.

Table of Contents

  • Historical survey and introduction to the classical theory of minimal surfaces
  • Information about some topological facts used in the modern theory of minimal surfaces
  • The modern state of the theory of minimal surfaces
  • The multidimensional Plateau problem in the spectral class of all manifolds with a fixed boundary
  • Multidimensional minimal surfaces and harmonic maps
  • Multidimensional variational problems and multivarifolds. The solution of Plateau's problem in the homotopy class of a map of a multivarifold
  • The space of multivarifolds
  • Parametrizations and parametrized multivarifolds
  • Problems of minimizing generalized integrands in classes of parametrizations and parametrized multivarifolds. A criterion for global minimality
  • Criteria for global minimality
  • Globally minimal surfaces in regular orbits of the adjoint representation of the classical Lie groups
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