Plateau's Problem : An Invitation to Varifold Geometry

By: Frederick J. Almgren

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There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book-or by Fred Almgren himself. The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting newanalysis and many new results.Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.

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Foreword to the AMS Edition	p. ix
Editors' Foreword	p. xiii
Preface	p. xv
The Phenomena of Least Area Problems	p. 1
Integration of Differential Forms over Rectifiable Sets	p. 15
Notation	p. 15
Hausdorff measure	p. 16
The Grassmann algebra and its dual	p. 19
Differential forms	p. 22
The Grassmann manifolds associated with R[superscript 3]	p. 23
Integration of differential forms over manifolds	p. 25
Rectifiable sets	p. 31
Integration of differential forms over rectifiable sets	p. 35
Varifolds	p. 37
Rectifiable sets regarded as real valued functions on the space of differential forms	p. 37
Rectifiable geometry, current geometry, and varifold geometry	p. 39
Varifolds	p. 44
The weight of a varifold	p. 48
Elementary varifolds	p. 52
Mappings of varifolds	p. 53
Variational Problems Involving Varifolds	p. 55
Vector fields and deformations	p. 55
Variations	p. 56
Boundaries	p. 58
Curvature and mean curvature	p. 62
The compactness theorem for regular integral varifolds	p. 67
A solution to the existence portion of Plateau's problem	p. 69
Useful facts about varifolds	p. 71
References	p. 73
Additional References	p. 75
Index	p. 77
Table of Contents provided by Syndetics. All Rights Reserved.

Plateau's Problem : An Invitation to Varifold Geometry

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Paperback

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