Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

TL;DR

This paper proves a new isoperimetric gap theorem for CAT(0) spaces, establishing thresholds for inequalities on 2-spheres that imply the space's geometric properties, and introduces minimal tetrahedra as a novel tool.

Contribution

It establishes the first higher-dimensional isoperimetric gap theorem in CAT(0) spaces and introduces minimal tetrahedra to analyze these inequalities.

Findings

A Euclidean inequality threshold of c_3=1/(6√π) implies an almost linear isoperimetric inequality.

Minimal tetrahedra satisfy a linear isoperimetric inequality.

A new Euclidean isoperimetric inequality for null-homotopies of 2-spheres was proved.

Abstract

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than $c_2=1/(4\pi)$ -- the optimal constant for the Euclidean plane -- implies that the underlying space is Gromov hyperbolic, and a linear inequality holds. We establish the first analogous gap theorem in higher dimensions: if a proper CAT(0) space satisfies a Euclidean inequality for 2-spheres with a constant below the sharp threshold $c_3=1/(6\sqrt{\pi})$, then the space also admits an inequality with an exponent arbitrarily close to 1. As a corollary we obtain a similar result for Lipschitz surfaces of higher genus. Towards our main theorem we prove a (non-sharp) Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature. A novelty in our approach is the introduction of minimal tetrahedra, which we demonstrate satisfy a linear inequality.