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

TL;DR

This paper establishes a uniform diameter bound on level sets of Morse functions in three-dimensional manifolds with nonnegative Ricci curvature and mean convex boundary, providing an upper bound for the Uryson 1-width.

Contribution

It introduces a method to bound the Uryson 1-width in such manifolds by constructing Morse functions with controlled level set diameters.

Findings

Existence of Morse functions with level sets of bounded diameter

Upper bound for Uryson 1-width depending on mean curvature

Application to three-dimensional manifolds with boundary

Abstract

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary, 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.