%0 Journal Article
%T Comparison Theorems for Manifold with Mean Convex Boundary
%A Jian Ge
%J Mathematics
%D 2013
%I arXiv
%R 10.1142/S0219199715500108
%X 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.
%U http://arxiv.org/abs/1306.5079v1