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Hardcover ISBN:  9780821845363 
Product Code:  MMONO/84 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470444976 
Product Code:  MMONO/84.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9780821845363 
eBook ISBN:  9781470444976 
Product Code:  MMONO/84.B 
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MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 84; 1991; 404 ppMSC: Primary 49; 53; Secondary 58
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 onedimensional contour in threedimensional space—perhaps the bestknown example of such surfaces is provided by soap films. From the mathematical point of view, such films are described as solutions of a secondorder 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 twodimensional 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 firstyear graduate students. The next part of the book, comprising Chapters 25, gives a fairly complete survey of various modern trends in Plateau's problem. This section is accessible to second and thirdyear 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

Chapters

Introduction

Chapter I. Historical survey and introduction to the classical theory of minimal surfaces

Chapter II. Information about some topological facts used in the modern theory of minimal surfaces

Chapter III. The modern state of the theory of minimal surfaces

Chapter IV. The multidimensional Plateau problem in the spectral class of all manifolds with a fixed boundary

Chapter V. Multidimensional minimal surfaces and harmonic maps

Chapter VI. Multidimensional variational problems and multivarifolds. The solution of Plateau’s problem in the homotopy class of a map of a multivarifold

Chapter VII. The space of multivarifolds

Chapter VIII. Parametrizations and parametrized multivarifolds

Chapter IX. Problems of minimizing generalized integrands in classes of parametrizations and parametrized multivarifolds. A criterion for global minimality

Chapter X. Criteria for global minimality

Chapter XI. Globally minimal surfaces in regular orbits of the adjoint representation of the classical Lie groups


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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 onedimensional contour in threedimensional space—perhaps the bestknown example of such surfaces is provided by soap films. From the mathematical point of view, such films are described as solutions of a secondorder 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 twodimensional 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 firstyear graduate students. The next part of the book, comprising Chapters 25, gives a fairly complete survey of various modern trends in Plateau's problem. This section is accessible to second and thirdyear 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.

Chapters

Introduction

Chapter I. Historical survey and introduction to the classical theory of minimal surfaces

Chapter II. Information about some topological facts used in the modern theory of minimal surfaces

Chapter III. The modern state of the theory of minimal surfaces

Chapter IV. The multidimensional Plateau problem in the spectral class of all manifolds with a fixed boundary

Chapter V. Multidimensional minimal surfaces and harmonic maps

Chapter VI. Multidimensional variational problems and multivarifolds. The solution of Plateau’s problem in the homotopy class of a map of a multivarifold

Chapter VII. The space of multivarifolds

Chapter VIII. Parametrizations and parametrized multivarifolds

Chapter IX. Problems of minimizing generalized integrands in classes of parametrizations and parametrized multivarifolds. A criterion for global minimality

Chapter X. Criteria for global minimality

Chapter XI. Globally minimal surfaces in regular orbits of the adjoint representation of the classical Lie groups