Area Minimizing Discs in Metric Spaces

By A. Lytchak, S. Wenger. 2015

70 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2cb83bb76500ecc44070a0bbbeeacf5f1ab32e3e

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In Applied Mathematics

1. S. Wenger
2. A. Lytchak
3. U. Lang
4. C. Sormani
5. Shin-ichi Ohta
6. Simeon Reich
7. Eva Kopecká

Citations in the corpus

• Intrinsic structure of minimal discs in metric spaces (2016, methodology): “All definitions of area coincide for all Sobolev maps with values in X if the space X has the so-called property (ET) discussed in [24].”