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Archive for Rational Mechanics and Analysis

, Volume 223, Issue 3, pp 1123–1182 | Cite as

Area Minimizing Discs in Metric Spaces

  • Alexander Lytchak
  • Stefan Wenger
Article

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 Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität KölnKölnGermany
  2. 2.Department of MathematicsUniversity of FribourgFribourgSwitzerland

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