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

Abstract

Gromov’s systolic inequality asserts that the length, ${\text{sys}}_1(M^n)$, of the shortest noncontractible curve in a closed essential Riemannian manifold $M^n$ does not exceed $c(n){ \text{vol}}^{1∕n}(M^n)$ 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 ${\text{sys}}_1(M^n)\le n{\text{vol} }^{1∕n}(M^n)$. (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 $\varrho (x)\in (0,r]$ such that the volume of the metric ball $B(x,\varrho (x))$ does not exceed ${(\varrho (x)∕c(n))}^n$, where one can take $c(n)=\frac{1}{2}n$.