Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II

Abstract

In this article, the author proves that a simple closed polygon ⊂ R 3 can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles {η l } at its N + 3 vertices, has to be smaller than 6π.

Exclusion of boundary branch points for minimal surfaces

“…For example, the paper [13] shows that every polygon whose total curvature is smaller than 6π can bound only finitely many immersed (stable and unstable) minimal surfaces of disctype, under the a priori assumption that any such surface does not possess boundary branch points. In order to exclude boundary branch points it is widely assumed to consider only extremal curves , i.e.…”