Minimal Surfaces with Free Boundaries

Abstract

This chapter is centered on the proof of existence theorems for minimal surfaces with completely free boundaries. We approach the problem by applying the direct methods of the calculus of variations, thus establishing the existence of minimizers within a given supporting surface S. However, this method does not yield the existence of stationary minimal surfaces which are not area minimizing As certain kinds of supporting surfaces are not able to hold nontrivial minimizers, our method is restricted by serious topological limitations. For example, it does not furnish existence of nontrivial stationary minimal surfaces within a closed convex surface. It seems that the techniques of geometrical measure theory are best suited to handle this problem. Unfortunately they are beyond the scope of our lecture notes, but we shall at least present a survey of the pertinent results in Section 5.8 as well as an existence result for the particular case of S being a tetrahedron. There the reader will also find references to the literature.