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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Intrinsic structure of minimal discs in metric spaces

Alexander Lytchak and Stefan Wenger

Geometry & Topology 22 (2018) 591–644
Abstract

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces.

Keywords
minimal disc, Plateau problem
Mathematical Subject Classification 2010
Primary: 49Q05, 53A10, 53C23
References
Publication
Received: 1 December 2016
Revised: 7 April 2017
Accepted: 7 May 2017
Published: 31 October 2017
Proposed: John Lott
Seconded: Tobias H Colding, Bruce Kleiner
Authors
Alexander Lytchak
Mathematisches Institut
Universität Köln
Köln
Germany
Stefan Wenger
Department of Mathematics
University of Fribourg
Fribourg
Switzerland