Area minimizing discs in metric spaces

TL;DR

This paper extends the classical Plateau problem to proper metric spaces, proving the existence of area-minimizing discs with quasi-conformal parametrizations and regularity properties under certain conditions.

Contribution

It generalizes the existence and regularity results of minimal discs from Euclidean and Riemannian settings to proper metric spaces.

Findings

Existence of area-minimizing discs with prescribed boundary in proper metric spaces.

Such discs admit a quasi-conformal parametrization.

Regularity results: interior Hölder continuity and boundary continuity under isoperimetric conditions.

Abstract

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally H\"older continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.