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A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

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Abstract

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M. Assume that the mean curvature H of the boundary ∂M satisfies H≥(n−1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂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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  1. Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Martin Man-chun Li

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  1. Martin Man-chun Li
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Correspondence to Martin Man-chun Li.

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Communicated by Jiaping Wang.

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Li, M.Mc. A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary. J Geom Anal 24, 1490–1496 (2014). https://doi.org/10.1007/s12220-012-9381-6

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  • DOI: https://doi.org/10.1007/s12220-012-9381-6

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