# Area minimizing surfaces of bounded genus in metric spaces

## TL;DR

This paper extends the classical Plateau-Douglas problem to proper metric spaces with a local quadratic isoperimetric inequality, establishing existence, boundary continuity, and interior regularity of area-minimizing surfaces of bounded genus.

## Contribution

It generalizes previous Riemannian results to a broader class of metric spaces, solving the Plateau-Douglas problem in this setting.

## Key findings

- Existence of area-minimizing surfaces of bounded genus in proper metric spaces.
- Proven boundary continuity and interior Hölder regularity of solutions.
- Extension of classical results to non-Riemannian metric space contexts.

## Abstract

The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior H\"older regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.