Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature
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arXiv
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| Format: | Preprint |
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2025
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| author | Druţu, Cornelia Lang, Urs Papasoglu, Panos Stadler, Stephan |
| author_facet | Druţu, Cornelia Lang, Urs Papasoglu, Panos Stadler, Stephan |
| contents | 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 $c_2=1/(4π)$ -- 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 $c_3=1/(6\sqrtπ)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature Druţu, Cornelia Lang, Urs Papasoglu, Panos Stadler, Stephan Metric Geometry Differential Geometry Group Theory 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 $c_2=1/(4π)$ -- 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 $c_3=1/(6\sqrtπ)$, 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. |
| title | Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature |
| topic | Metric Geometry Differential Geometry Group Theory |
| url | https://arxiv.org/abs/2502.03389 |