Uryson Width, Asymptotic Dimension and Ricci Curvature
Jon Wolfson
备注：
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
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.
Some rigidity theorems for spectral curvature bounds
Central China Normal University（华中师范大学）；POSTECH（韩国科学技术院）；Henan University（河南大学）；Center for Applied Mathematics of Henan Province, Henan University（河南省应用数学中心）
