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

Nabutovsky, Alexander

doi:10.2140/gt.2022.26.3123

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

Alexander Nabutovsky

Geometry & Topology 26 (2022) 3123–3142

DOI: 10.2140/gt.2022.26.3123

Abstract

Gromov’s systolic inequality asserts that the length, ${sys}_{1} (M^{n})$ , of the shortest noncontractible curve in a closed essential Riemannian manifold $M^{n}$ does not exceed $c (n) {vol}^{1 ∕ n} (M^{n})$ 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 ${sys}_{1} (M^{n}) \leq n {vol}^{1 ∕ n} (M^{n})$ . (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 $M^{n}$ do not exceed ${(r ∕ c (n))}^{n}$ , then the $(n - 1)$ –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 $x \in M^{n}$ there exists $ρ (x) \in (0, r]$ such that the volume of the metric ball $B (x, ρ (x))$ does not exceed ${(ρ (x) ∕ c (n))}^{n}$ , where one can take $c (n) = \frac{1}{2} n$ .

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Keywords

Hausdorff content, systole, systolic inequality, isoperimetric inequality, geometry of metric spaces, shortest periodic geodesic

Mathematical Subject Classification

Primary: 51F30, 53C20, 53C23

References

Publication

Received: 7 October 2020

Revised: 23 June 2021

Accepted: 10 August 2021

Published: 23 January 2023

Proposed: Urs Lang

Seconded: Bruce Kleiner, Mladen Bestvina

Authors

Alexander Nabutovsky
Department of Mathematics University of Toronto Toronto ON Canada

Volume 26, issue 7 (2022)