Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities

Abstract

This is the second central chapter of Vol. 2, collecting a wealth of information. It could have been labeled as geometric properties of minimal surfaces. First we derive inclusion theorems for minimal surfaces in dependence of their boundary data. Such results are more or less sophisticated versions of the maximum principle. They lead to interesting nonexistence results for connected minimal surfaces and H-surfaces whose boundaries consist of several disjoint components. Here even the situation for higher dimensional surfaces and for solutions of variational inequalities, obtained from obstacle problems, is discussed. Inclusion principles for such solutions are the fundament for results ensuring the existence of minimal surfaces and H-surfaces solving Plateau’s problem in Euclidean space or in a Riemannian manifold respectively. Of particular interest in this context are Jacobi field estimates.

In addition, isoperimetric inequalities for minimal surfaces solving either Plateau’s problem or a free boundary problem are derived.