The Plateau-Douglas problem for singular configurations and in general metric spaces

TL;DR

This paper extends the existence theory of minimal surfaces spanning complex, potentially singular curve configurations to general metric spaces, broadening classical results to more irregular and abstract settings.

Contribution

It proves the existence of minimal surfaces for singular configurations in general metric spaces, extending previous results from smooth manifolds.

Findings

Existence of minimal surfaces for singular configurations in metric spaces.

Extension of classical Plateau-Douglas results to non-disjoint, self-intersecting curves.

Generalization of minimal sequence approach to metric space setting.

Abstract

Assume you are given a finite configuration $\Gamma$ of disjoint rectifiable Jordan curves in $\mathbb{R}^n$. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most $p$ which span $\Gamma$. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $\Gamma$ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in $\mathbb{R}^n$ but in general proper metric spaces. Thus we are also able to extend previously known existence results of J\"urgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus $p$ bounding a given configuration $\Gamma$ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.