Isoperimetric inequalities vs. upper curvature bounds

TL;DR

This paper establishes a precise equivalence between upper curvature bounds in Alexandrov spaces and bounds on the Dehn function, extending previous results to more general spaces and providing quantitative and asymptotic insights.

Contribution

It proves that curvature bounds are characterized by Dehn function bounds in general metric spaces, extending prior work to non-locally compact spaces.

Findings

Curvature bounds are equivalent to Dehn function bounds in metric spaces.

Constructs minimal discs in ultralimits to analyze geometric properties.

Provides quantitative and stable versions of the main curvature-Dehn function relationship.

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 $\kappa$ 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 $\kappa$. This extends work of Lytchak and the second author 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.