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Area Minimizing Discs in Metric Spaces

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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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Authors and Affiliations

  1. Mathematisches Institut, Universität Köln, Weyertal 86–90, 50931, Köln, Germany

    Alexander Lytchak

  2. Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700, Fribourg, Switzerland

    Stefan Wenger

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  1. Alexander Lytchak
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  2. Stefan Wenger
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Correspondence to Stefan Wenger.

Additional information

Communicated by C. De Lellis

S.Wenger was partially supported by Swiss National Science Foundation Grant 153599.

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Lytchak, A., Wenger, S. Area Minimizing Discs in Metric Spaces. Arch Rational Mech Anal 223, 1123–1182 (2017). https://doi.org/10.1007/s00205-016-1054-3

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