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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, ${\mathrm{sys}}_{1}\left({M}^{n}\right)$, of the shortest noncontractible curve in a closed essential Riemannian manifold ${M}^{n}$ does not exceed $c\left(n\right){\mathrm{vol}}^{1∕n}\left({M}^{n}\right)$ for some constant $c\left(n\right)$. (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 ${\mathrm{sys}}_{1}\left({M}^{n}\right)\le n{\mathrm{vol}}^{1∕n}\left({M}^{n}\right)$. (The best previously known upper bound for $c\left(n\right)$ 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 ${\left(r∕c\left(n\right)\right)}^{n}$, then the $\left(n-1\right)$–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 $\rho \left(x\right)\in \left(0,r\right]$ such that the volume of the metric ball $B\left(x,\rho \left(x\right)\right)$ does not exceed ${\left(\rho \left(x\right)∕c\left(n\right)\right)}^{n}$, where one can take $c\left(n\right)=\frac{1}{2}n$.

Keywords
Hausdorff content, systole, systolic inequality, isoperimetric inequality, geometry of metric spaces, shortest periodic geodesic
Mathematical Subject Classification
Primary: 51F30, 53C20, 53C23