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

It is proved that for a simple, closed, extreme polygon Gamma subset of R-3 every immersed, stable minimal surface spanning Gamma is an isolated point of the set of all minimal surfaces spanning Gamma w.r.t. the C-0-topology. Since the subset of immersed, stable minimal surfaces spanning Gamma is shown to be closed in the compact set of all minimal surfaces spanning Gamma, this proves in particular that Gamma can bound only finitely many immersed, stable minimal surfaces. (c) 2006 Elsevier Masson SAS. All rights reserved.