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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 new analysis 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.

Cover 1
Title 6
Copyright 7
Contents 8
Foreword to the AMS Edition 10
Editors' Foreword 14
Preface 16
Chapter 1. The Phenomena of Least Area Problems 18
Chapter 2. Integration of Differential Forms over Rectifiable Sets 32
2–1. Notation 32
2–2. Hausdorff measure 33
2–3. The Grassmann algebra and its dual 36
2–4. Differential forms 39
2–5. The Grassmann manifolds associated with R[(sup)3] 40
2–6. Integration of differential forms over manifolds 42
2–7. Rectifiable sets 48
2–8. Integration of differential forms over rectifiable sets 52
Chapter 3. Varifolds 54
3–1. Rectifiable sets regarded as real valued functions on the space of differential forms 54
3–2. Rectifiable geometry, current geometry, and varifold geometry 56
3–3. Varifolds 61
3–4. The weight of a varifold 65
3–5. Elementary varifolds 69
3–6. Mappings of varifolds 70
Chapter 4. Variational Problems Involving Varifolds 72
4–1. Vector fields and deformations 72
4–2. Variations 73
4–3. Boundaries 75
4–4. Curvature and mean curvature 79
4–5. The compactness theorem for regular integral varifolds 84
4–6. A solution to the existence portion of Plateau's problem 86
4–7. Useful facts about varifolds 88
References 90
Additional References 92
Index 94
A 94
C 94
D 94
F 94
G 94
H 94
I 94
L 94
M 94
P 94
R 94
S 94
T 95
U 95
V 95
W 95
X 95
Back Cover 96
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