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

TL;DR

This paper presents a new proof that any closed piecewise smooth curve with total curvature less than 6π bounds an immersed minimal disc, using polygonal approximation techniques instead of traditional methods.

Contribution

It introduces a novel polygonal approximation approach to Plateau's problem for curves with small total curvature, expanding the classical understanding.

Findings

Proof of existence of immersed minimal surfaces for curves with total curvature < 6π

Use of polygonal approximation techniques for boundary curves

Alternative proof method differing from Osserman, Gulliver, and Alt

Abstract

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.