Minimal Surfaces with Free Boundaries

Abstract

This chapter is centered on the proof of existence theorems for minimal surfaces with completely free boundaries. The problem is approached by applying the direct methods of the calculus of variations, thus establishing the existence of minimizers with a boundary on a given supporting surface S. However, this method does not yield the existence of stationary minimal surfaces which are not area minimizing. The remaining part of the chapter deals with additional properties of minimal surfaces with free boundaries. For instance, such a surface has to intersect the free boundary surface perpendicularly and in a balanced way. This fact implies nonexistence in certain cases. Finally an extensive report on the existence of stationary minimal surfaces with free or partially free boundaries is given.