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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.

Immersed Solutions of Plateau’s Problem for Piecewise Smooth Boundary Curves with Small Total Curvature