Sharp diameter estimates for compact manifold with boundary

Abstract

Let $(N,g)$ be an $n$-dimensional complete Riemannian manifold with nonempty boundary $\pt N$. Assume that the Ricci curvature of $N$ has a negative lower bound $Ric\ge -(n-1)c^2$ for some $c>0$, and the mean curvature of the boundary $\pt N$ satisfies $H\ge (n-1)c_0>(n-1)c$ for some $c_0>c>0$. Then a known result (see \cite{LN}) says that $\sup_{x\in N}d(x,\pt N)\leq \frac 1c\coth^{-1}\frac{c_0}c$. In this paper, we prove that if the boundary $\pt N$ is compact, then the equality holds if and only if $N$ is isometric to the geodesic ball of radius $\frac{1}{c}{\coth }^{-1}\frac{c_0}{c}$ in an $n$-dimensional hyperbolic space ${\mathbb{H}}^n(-c^2)$ of constant sectional curvature $-c^2$. Moreover, we also prove an analogous result for manifold with nonempty boundary and with $m$-Bakry-Émery Ricci curvature bounded below by a negative constant.

15 pages, comments are welcome

References in corpus (3)

• $f$-minimal surface and manifold with positive $m$-Bakry-Émery Ricci curvature
• A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound
• Response to a question of Yanyan Li and Luc Nguyen in their paper "A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound", arXiv:1212.0460

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• Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound