Uryson Width, Asymptotic Dimension and Ricci Curvature

TL;DR

This paper proves that lower bounds on Ricci and scalar curvature imply bounds on the Uryson width of Riemannian manifolds, confirming a conjecture of Gromov and linking curvature to geometric complexity.

Contribution

It establishes a new connection between curvature bounds and Uryson width, confirming Gromov's conjecture for scalar curvature.

Findings

Lower Ricci curvature bounds imply bounded Uryson width.

Scalar curvature bounds imply bounded Uryson width.

Results depend only on curvature lower bounds, not on manifold specifics.

Abstract

A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters bounded by c. We prove that an n-dimensional Riemannian manifold M with at least n-k eigenvalues of the Ricci curvature bounded below by a positive constant (n-1)b has k-dimensional Uryson width bounded by a constant c >0. The constant c depends only on b. In particular, it follows that a Riemannian n-manifold M with scalar curvature S bounded below by a positive constant n (n-1) s has (n-1)-dimensional Uryson width bounded by a constant c >0 depending only on s. This result confirms a conjecture of M. Gromov.