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arXiv · 2502.03389

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

Cornelia Druţu·Urs Lang·Panos Papasoglu·Stephan Stadler
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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.

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Cornelia Druţu, Urs Lang, Panos Papasoglu, Stephan Stadler. 2025-02-05. Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature. https://arxiv.org/abs/2502.03389

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