Plateau’s Problem: An Invitation to Varifold Geometry, Revised Edition
About this Title
Frederick J. Almgren Jr.
Publication: The Student Mathematical Library
Publication Year 2001: Volume 13
ISBNs: 978-0-8218-2747-5 (print); 978-1-4704-2129-8 (online)
MathSciNet review: MR1853442
MSC: Primary 49-01; Secondary 49Q20, 53C42
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.
Advanced undergraduates and graduate students interested in mathematics.
Table of Contents
- Chapter 1. The phenomena of least area problems
- Chapter 2. Integration of differential forms over rectifiable sets
- Chapter 3. Varifolds
- Chapter 4. Variational problems involving varifolds