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

Abstract

We provide a new proof of the classical result that any closed rectifiable Jordan curve $${\Gamma \subset \mathbb{R}^3}$$ 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 Γ is smaller than 6π. In contrast to the methods due to Osserman (Ann Math 91(2):550–569, 1970), Gulliver (Ann Math 97(2):275–305, 1973) and Alt (Math Z 127:333–362, 1972, Math Ann 201:33–35, 1973), 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 (The Plateau problem, Fuchsian equations and the Riemann–Hilbert problem. Memoires de la Soc. Math. Fr. (to appear) arXiv: 1003.0978) of Garnier’s ideas in (Annales scientifiques de l’E.N.S. 45:53–144, 1928).