**Translations of Mathematical Monographs**

1991;
404 pp;
Hardcover

MSC: Primary 49; 53;
Secondary 58

**Print ISBN: 978-0-8218-4536-3
Product Code: MMONO/84**

List Price: $135.00

Individual Member Price: $108.00

# Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem

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*Dao Chang Tkhi; A. T. Fomenko*

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.