Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

Details Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

2011-03-07

arxiv.org/abs/1103.1340v2

Mathematics

Differential Geometry

12 pages

Scientific paper

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas.

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