Area minimizing discs in metric spaces
Differential Geometry
2016-11-23 v2 Analysis of PDEs
Metric Geometry
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.
Cite
@article{arxiv.1502.06571,
title = {Area minimizing discs in metric spaces},
author = {Alexander Lytchak and Stefan Wenger},
journal= {arXiv preprint arXiv:1502.06571},
year = {2016}
}
Comments
typos corrected, some comments added, minor changes to the exposition at some places