Geom. Topol. 26 (7), 3123-3142, (2022) DOI: 10.2140/gt.2022.26.3123
KEYWORDS: Hausdorff content, systole, systolic inequality, Isoperimetric inequality, geometry of metric spaces, shortest periodic geodesic, 51F30, 53C20, 53C23
Gromov’s systolic inequality asserts that the length, , of the shortest noncontractible curve in a closed essential Riemannian manifold does not exceed for some constant . (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 . (The best previously known upper bound for was exponential in .)
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 in a closed Riemannian manifold do not exceed , then the –dimensional Urysohn width of the manifold does not exceed . In our version the assumption of Guth’s theorem is relaxed to the assumption that for each there exists such that the volume of the metric ball does not exceed , where one can take .