"The Plateau Problem, Part I: Historical Survey" by A. T. Fomenko
Studies in the Development of Modern Mathematics, volume 1. Faculty of Mathematics and Mechanics Moscow State University

Gordon and Breach Science Publishers | 1990 | ISBN: 2881247008 2881247024 9782881247019 | 234 pages | PDF/djvu | 14/3 MB
Charting the historical origins of the Plateau Problem, the author discusses substantial extracts from 18th, 19th and early 20th century works devoted to the investigation of minimal surfaces, including Plateau's famous physical experiments. The theories of homology and cohomology, necessary for an understanding of modern multidimensional variational problems, are elucidated.

Explores the history and current state of the theory of minimal surfaces containing original research data.
Its clear presentation and numerous illustrations make this topic accessible to both students and research workers in the fields of mathematics and physics.

The Plateau Problem
Part I Historical Survey
Part II Present State of the Theory

List of Contents of Part I
Preface
I HISTORICAL SURVEY AND INTRODUCTION TO THE THEORY OF MINIMAL SURFACES
§1. The Origins of Multidimensional Variational Calculus.
§2. The Nineteenth Century, the Era of the Discovery of Basic Minimal Surface Properties.
1. Plateau's physical experiments and methods of forming soap films.
2. Physical principles underlying the formation of soap films.
3. The extremal properties of soap films and minimality of their area. The properties of the surface of separation between two media.
4. The surface of separation between two media in equilibrium is a surface of constant mean curvature.
S. Soap films of constant positive curvature and constant zero curvature.
§3. Topological and Physical Properties of Minimal Surfaces.
1. Stable and unstable surfaces.
2. Plateau's experiments with stable columns of liquids.
3. The physical realization of a helicoid.
4. The physical realization of a catenoid and its restructuring as the boundary contour changes. The change of the topological type of minimal surface in accordance with its stability or instability.
§4. The Plateau Principles. Minimal Surfaces in Animate Nature.
1. Two-dimensional minimal surfaces in three-dimensional space and the first Plateau principle.
2. The area functional, Dirichlet functional, harmonic mappings and conformal coordinates.
3. Singular points of minimal surfaces and three Plateau principles.
4. The realization of minimal surfaces in animate nature.

II SURVEY OF SOME IMPORTANT PUBLICATIONS IN MINIMAL SURFACE THEORY FROM THE NINETEENTH TO THE EARLY TWENTIETH CENTURY
§1. Monge (1746-1818).
§2. Poisson (1781-1840).
§3. Plateau (1801-1883).
§4. Some works of the Early Twentieth Century Rad6, Douglas).
§s. Minimal Surfaces in the Large.

III SOME FACTS FROM ELEMENTARY TOPOLOGY
§1. Singular and Cellular Homology Groups.
1. Singular chains and homology groups.
2. Cellular complexes, barycentric subdivisions.
3. Cellular homology and computation of the singular homology of the sphere.
4. Theorem on the coincidence of the singular and cellular homology of a finite complex.
5. The geometric determination of cellular homology groups.
6. The simplest examples of cellular homology group computations.
§2. Cohomology Groups and Obstructions to the Extensions of Mappings.
1. Singular cochains and the coboundary operator.
2. The problem of the extension of a continuous mapping from a subspace to the whole space.
3. Obstructions to the extension of mappings.
4. The cases of the existence of the retraction of a space onto a subspace which is homeomorphic to the sphere.
References and Bibliography
Index