Volume 22, issue 1 (2018)

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Intrinsic structure of minimal discs in metric spaces

Alexander Lytchak and Stefan Wenger

Geometry & Topology 22 (2018) 591–644

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.

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minimal disc, Plateau problem
Mathematical Subject Classification 2010
Primary: 49Q05, 53A10, 53C23
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
Alexander Lytchak
Mathematisches Institut
Universität Köln
Stefan Wenger
Department of Mathematics
University of Fribourg