Manifolds with Nonnegative Ricci Curvature and Mean Convex Boundary

Details Manifolds with Nonnegative Ricci Curvature and Mean Convex Boundary Manifolds with Nonnegative Ricci Curvature and Mean Convex Boundary

2012-04-08

arxiv.org/abs/1204.1695v1

Mathematics

Differential Geometry

6 pages

Scientific paper

Let $M$ be an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty 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 $\frac{1}{k}$.

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