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Linear bounds for constants in Gromov's systolic inequality and related results

Alexander Nabutovsky

Geometry & Topology 26 (2022) 3123–3142

Gromov’s systolic inequality asserts that the length, sys 1(Mn), of the shortest noncontractible curve in a closed essential Riemannian manifold Mn does not exceed c(n)vol 1n(Mn) for some constant c(n). (Essential manifolds is a class of non–simply connected manifolds that includes all non–simply connected closed surfaces, tori and projective spaces.)

Here we prove that all closed essential Riemannian manifolds satisfy sys 1(Mn) nvol 1n(Mn). (The best previously known upper bound for c(n) was exponential in n.)

We similarly improve a number of related inequalities. We also give a qualitative strengthening of Guth’s theorem (2011, 2017) asserting that if volumes of all metric balls of radius r in a closed Riemannian manifold Mn do not exceed (rc(n))n, then the (n1)–dimensional Urysohn width of the manifold does not exceed r. In our version the assumption of Guth’s theorem is relaxed to the assumption that for each x Mn there exists ρ(x) (0,r] such that the volume of the metric ball B(x,ρ(x)) does not exceed (ρ(x)c(n))n, where one can take c(n) = 1 2n.

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Hausdorff content, systole, systolic inequality, isoperimetric inequality, geometry of metric spaces, shortest periodic geodesic
Mathematical Subject Classification
Primary: 51F30, 53C20, 53C23
Received: 7 October 2020
Revised: 23 June 2021
Accepted: 10 August 2021
Published: 23 January 2023
Proposed: Urs Lang
Seconded: Bruce Kleiner, Mladen Bestvina
Alexander Nabutovsky
Department of Mathematics
University of Toronto
Toronto ON