Isoperimetric inequalities vs upper curvature bounds

Stadler, Stephan¹ ; Wenger, Stefan²

¹Max Planck Institute for Mathematics, Bonn, Germany
²Department of Mathematics, University of Fribourg, Fribourg, Switzerland

Geometry & topology, Tome 29 (2025) no. 2, pp. 829-862

Résumé

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.

Détail

DOI : 10.2140/gt.2025.29.829

Keywords: minimal surface, Plateau problem, CAT(0)

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