Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

Abstract

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.