Area minimizing surfaces of bounded genus in metric spaces
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.
Keywords
Cite
@article{arxiv.1904.02618,
title = {Area minimizing surfaces of bounded genus in metric spaces},
author = {Martin Fitzi and Stefan Wenger},
journal= {arXiv preprint arXiv:1904.02618},
year = {2019}
}