Comparison Theorems for Manifold with Mean Convex Boundary

TL;DR

This paper establishes sharp geometric bounds for manifolds with boundary based on Ricci curvature and mean curvature, characterizes equality cases, and extends results to Kähler manifolds including eigenvalue estimates.

Contribution

It provides new sharp comparison theorems for manifolds with boundary under Ricci curvature bounds, including a Kähler version and eigenvalue estimates.

Findings

Sharp upper bounds for distance to boundary in manifolds with Ricci curvature bounds.

Characterization of equality cases as isometric disks in space forms.

Laplace comparison and eigenvalue estimates for Kähler manifolds.

Abstract

Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\in \RR$, we give a sharp estimate of the upper bound of $\rho(x)=\dis(x, \partial M)$, in terms of the mean curvature bound of the boundary. When $\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of the real Laplacian.