Embedded Minimal Surfaces with Partially Free Boundaries

Abstract

As a generalization of the situation above, here the topic is the investigation of disk-type minimal surfaces X in ℝ³ that span boundary configurations 〈Γ,S〉 consisting of a cylinder surface S and a Jordan arc Γ with endpoints on S, and we ask the following question: When can such a minimal surface be viewed as a nonparametric surface over a planar domain? Any satisfactory answer yields the existence of embedded minimal surfaces that are stationary in 〈Γ,S〉. Simultaneously, such results lead to uniqueness theorems for the partially free boundary problem which generalize Radó’s uniqueness theorem for Plateau’s problem that was discussed in Vol. 1.

First, conditions on 〈Γ,S〉 are exhibited which allow it to verify these expectations when S is smooth. By approximation one can also tackle nonsmooth supporting surfaces with edges parallel to the cylinder axis. However, in this case the phenomenon of edge creeping may occur. Asymptotic expansions for X and its Gauss map N will show how X behaves in the edge creeping case and in the transversal case.

Finally a Bernstein theorem for nonparametric minimal surfaces in a wedge is proved.