Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Linear bounds for constants in Gromov's systolic inequality and related results

Alexander Nabutovsky

Geometry & Topology 26 (2022) 3123–3142
Abstract

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.

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