Comparison theorems for manifolds with mean convex boundary

Abstract

Let Mn be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d (x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.