Area minimizing surfaces of bounded genus in metric spaces
Abstract
The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1904.02618
- arXiv:
- arXiv:1904.02618
- Bibcode:
- 2019arXiv190402618F
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- Mathematics - Metric Geometry;
- 49Q05;
- 53C23