A sharp isoperimetric gap theorem in non-positive curvature |
Urs Lang (ETHZ) |
Montag, 27. Januar 2025 16:15Uhr |
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In joint work with Cornelia Drutu, Panos Papasoglu, and Stephan Stadler, we study isoperimetric inequalities for null-homotopies of Lipschitz 2-spheres in Hadamard manifolds or, more generally, proper CAT(0) spaces. In one dimension less, for fillings of circles by discs, it is known that a quadratic inequality with a constant smaller than the sharp threshold $1/(4\pi)$ implies that the underlying space is Gromov hyperbolic and satisfies a linear inequality. Our main result is a first analogous gap theorem in higher dimensions, yielding exponents arbitrarily close to 1. Towards this we prove a Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature, and introduce so-called minimal tetrahedra, which we demonstrate satisfy a linear inequality. |