Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves

Abstract

It is proved that for a simple, closed, extreme polygon$\Gamma \subset {\mathbb{R}}^3$ every immersed, stable minimal surface spanning Γ is an isolated point of the set of all minimal surfaces spanning Γ w.r.t. the $C^0$-topology. Since the subset of immersed, stable minimal surfaces spanning Γ is shown to be closed in the compact set of all minimal surfaces spanning Γ, this proves in particular that Γ can bound only finitely many immersed, stable minimal surfaces.