Area minimizing discs in metric spaces

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 Hölder 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.

typos corrected, some comments added, minor changes to the exposition at some places

References in corpus (1)

• Gromov hyperbolic spaces and the sharp isoperimetric constant

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• Polyhedral approximation of metric surfaces and applications to uniformization
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• Elastic membranes spanning deformable boundaries
• Short retractions of CAT(1) spaces
• CAT(0) Polygonal Complexes are 2-Median
• Isoperimetric inequalities vs. upper curvature bounds