Spaces with almost Euclidean Dehn function

TL;DR

This paper proves that proper geodesic metric spaces with Euclidean-like Dehn function growth have asymptotic cones that are CAT(0), linking isoperimetric properties to non-positive curvature in a large-scale geometric context.

Contribution

It establishes a new connection between Dehn function growth and CAT(0) geometry, including a large-scale characterization and a stability result near Euclidean Dehn functions.

Findings

Spaces with Euclidean Dehn function growth have CAT(0) asymptotic cones.

Proper geodesic spaces with near-Euclidean Dehn functions are close to CAT(0) spaces.

The results extend to a large-scale stability theorem for spaces near Euclidean Dehn functions.

Abstract

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper ${\rm CAT}(0)$ in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being ${\rm CAT}(0)$ up to that scale.