Schwarz operators of minimal surfaces spanning polygonal boundary curves

Jakob, Ruben
December 2007
Calculus of Variations & Partial Differential Equations;Dec2007, Vol. 30 Issue 4, p467
Academic Journal
This paper examines the Schwarz operator A and its relatives A?, A and ? 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 G. 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 G, 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 G.


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