Schwarz operators of minimal surfaces spanning polygonal boundary curves

This paper examines the Schwarz operator A and its relatives.angstrom, (A) over bar and.(angstrom) over bar 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, Pj+1 of two adjacent vertices of some simple closed polygon Gamma subset of R-3. In this case X possesses singularities in those boundary points which are mapped onto the vertices of the polygon Gamma. Nevertheless it is shown that A and its closure _ A 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 Gamma, 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 Gamma.