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

Abstract

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $(\frac{r}{c(n)}{)}^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $FillRad(M^n)\le c(n)vol(M^n{)}^{\frac{1}{n}}$ and, if $M^n$ is essential, then also $sy{s}_1(M^n)\le 6c(n)vol(M^n{)}^{\frac{1}{n}}$ with the same constant $c(n)$. Here $sy{s}_1(M^n)$ denotes the length of a shortest non-contractible closed curve in $M^n$. We prove that these results hold with $c(n)=(\frac{n!}{2}{)}^{\frac{1}{n}}\le \frac{n}{2}$. We demonstrate that for essential Riemannian manifolds $sy{s}_1(M^n)\le nvo{l}^{\frac{1}{n}}(M^n)$. All previously known upper bounds for $c(n)$ were exponential in $n$. Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius $r$ is less than $(\frac{r}{c(n)}{)}^n$ can be replaced by a weaker assumption that for every point $x\in M^n$ there exists a positive $\rho (x)\le r$ such that the volume of the metric ball of radius $\rho (x)$ centered at $x$ is less than $(\frac{\rho (x)}{c(n)}{)}^n$ (for $c(n)=(\frac{n!}{2}{)}^{\frac{1}{n}}$). Also, if $X$ is a boundedly compact metric space such that for some $r>0$ and an integer $n\ge 1$ the $n$-dimensional Hausdorff content of each metric ball of radius $r$ in $X$ is less than $(\frac{r}{4n}{)}^n$, then there exists a continuous map from $X$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$.

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