A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

TL;DR

This paper establishes a sharp comparison theorem for compact Riemannian manifolds with nonnegative Ricci curvature and mean convex boundary, characterizing when the manifold is isometric to a Euclidean ball based on boundary mean curvature.

Contribution

It proves a sharp upper bound for the distance to the boundary in manifolds with specified curvature conditions, and characterizes the equality case as a Euclidean ball.

Findings

Distance to boundary is at most 1/k.

Equality case occurs iff the manifold is a Euclidean ball.

Provides a geometric rigidity result.

Abstract

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the boundary $\partial M$ is bounded from above by $\frac{1}{k}$ and the upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional Euclidean ball of radius $\frac{1}{k}$.