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Comparison theorems for manifolds with mean convex boundary

Jian Ge

Beijing International Center for Mathematical Research, Beijing University, Beijing 100871, P. R. China

https://doi.org/10.1142/S0219199715500108Cited by:5 (Source: Crossref)

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.

Keywords:

AMSC: 53C20

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