Minimal Surfaces with Supporting Half-Planes

Abstract

In Vol. 3, the regularity of stationary minimal surfaces with a partially free boundary was discussed. It was shown that, for a uniformly smooth surface S with a smooth boundary ∂ S, the stationary surfaces X belong to the class C ^1,1/2(B∪I,ℝ³). One of the consequences of results proved in the present chapter will be that this regularity result is optimal. The nonoriented tangent of the free trace Σ X changes continuously which, in particular, means that the free trace cannot have corners at points where it attaches to the border of the supporting surface S. On the other hand, since isolated branch points of odd order cannot be excluded, there might exist cusps on the free trace. In fact, experimental evidence suggests that cusps do appear for certain shapes of the boundary configuration 〈Γ,S〉.

Soap film experiments as well as classical examples described in Vol. 1 demonstrate that cusps may indeed appear. This chapter is devoted to the study of the free trace in a special, somewhat symmetric situation where the experimental observations can be completely verified.