Isoperimetric inequalities vs upper curvature bounds

Stadler, Stephan; Wenger, Stefan

doi:10.2140/gt.2025.29.829

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Isoperimetric inequalities vs upper curvature bounds

Stephan Stadler and Stefan Wenger

Geometry & Topology 29 (2025) 829–862

DOI: 10.2140/gt.2025.29.829

Abstract

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by $κ$ in the sense of Alexandrov if and only if its Dehn function is bounded above by the Dehn function of the model surface of constant curvature $κ$ . This extends work of Lytchak and the second author (2018) from locally compact spaces to the general case. A key ingredient in the proof is the construction of minimal discs with suitable properties in certain ultralimits. Our arguments also yield quantitative local and stable versions of our main result. The latter has implications on the geometry of asymptotic cones.

Keywords

minimal surface, Plateau problem, CAT(0)

Mathematical Subject Classification

Primary: 49Q05, 53C23

References

Publication

Received: 29 November 2023

Revised: 19 February 2024

Accepted: 16 March 2024

Published: 12 March 2025

Proposed: Dmitri Burago

Seconded: Urs Lang, Bruce Kleiner

Authors

Stephan Stadler
Max Planck Institute for Mathematics Bonn Germany

Stefan Wenger
Department of Mathematics University of Fribourg Fribourg Switzerland

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Volume 29, issue 2 (2025)