Plateau'S Problem. An Invitation To Varifold Geometry

There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past to 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 imitation to the world of variational geometry. A central topic is Plateau's Problem, Which is concerned with surfaces that model the behavior of soap film. When trying to resolve the problem, however, one soon find, 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.