Volumes of balls in Riemannian manifolds and Uryson width

TL;DR

This paper establishes a relationship between the volume of unit balls in a Riemannian manifold and its Uryson width, showing that small volume bounds imply a bounded Uryson width.

Contribution

It proves that small volume conditions on unit balls in a Riemannian manifold imply a uniform bound on the manifold's Uryson width, linking local volume constraints to global geometric properties.

Findings

Small volume of unit balls implies bounded Uryson width

Uryson width is at most 1 under volume constraints

Provides a quantitative geometric relationship

Abstract

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $\epsilon_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.