Intrinsic structure of minimal discs in metric spaces

Lytchak, Alexander; Wenger, Stefan

doi:10.2140/gt.2018.22.591

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Intrinsic structure of minimal discs in metric spaces

Alexander Lytchak and Stefan Wenger

Geometry & Topology 22 (2018) 591–644

DOI: 10.2140/gt.2018.22.591

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

Volume 22, issue 1 (2018)