# Plateau’s Problem: An Invitation to Varifold Geometry, Revised Edition

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*Frederick J. Almgren, Jr.*

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.

#### Readership

Advanced undergraduates and graduate students interested in mathematics.

#### Table of Contents

# Table of Contents

## Plateau's Problem: An Invitation to Varifold Geometry, Revised Edition

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Foreword to the AMS Edition ix10 free
- Editors' Foreword xiii14 free
- Preface xv16 free
- Chapter 1. The Phenomena of Least Area Problems 118 free
- Chapter 2. Integration of Differential Forms over Rectifiable Sets 1532
- 2–1. Notation 1532
- 2–2. Hausdorff measure 1633
- 2–3. The Grassmann algebra and its dual 1936
- 2–4. Differential forms 2239
- 2–5. The Grassmann manifolds associated with R[(sup)3] 2340
- 2–6. Integration of differential forms over manifolds 2542
- 2–7. Rectifiable sets 3148
- 2–8. Integration of differential forms over rectifiable sets 3552

- Chapter 3. Varifolds 3754
- Chapter 4. Variational Problems Involving Varifolds 5572
- References 7390
- Additional References 7592
- Index 7794
- Back Cover Back Cover196