Schwarz operators of minimal surfaces spanning polygonal boundary curves

Abstract

This paper examines the Schwarz operator A and its relatives Ȧ, Ā and Ǡ that are assigned to a minimal surface X which maps consequtive arcs of the boundary of its parameter domain onto the straight lines which are determined by pairs P _j , P _j+1 of two adjacent vertices of some simple closed polygon ${\Gamma\subset \mathbb{R}^3}$ . In this case X possesses singularities in those boundary points which are mapped onto the vertices of the polygon Γ. Nevertheless it is shown that A and its closure Ā have essentially the same properties as the Schwarz operator assigned to a minimal surface which spans a smooth boundary contour. This result is used by the author to prove in [Jakob, Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. I.H.P. Analyse Non-lineaire (in press)] the finiteness of the number of immersed stable minimal surfaces which span an extreme simple closed polygon Γ, and in [Jakob, Local boundedness of the set of solutions of Plateau’s problem for polygonal boundary curves (in press)] even the local boundedness of this number under sufficiently small perturbations of Γ.