Uryson Width, Asymptotic Dimension and Ricci Curvature
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.
Cite
@article{arxiv.2001.02644,
title = {Uryson Width, Asymptotic Dimension and Ricci Curvature},
author = {Jon Wolfson},
journal= {arXiv preprint arXiv:2001.02644},
year = {2020}
}
Comments
The paper uses a modification of the techniques of "The Fundamental Group of Closed Manifolds with Two-positive Ricci Curvature". The same error in that paper occurs here. In particular, the proofs of Theorem 2.12 and 2.14 contain errors. Therefore Corollary 2.15 is not correct. The limiting procedure does not converge to a geodesic