Note on a Weighted Geometric Inequality in Hyperbolic Space.

Abstract

In this paper, we prove that the solution to a locally constrained flow in hyperbolic space introduced by Hu et al. [14] exists for a long time and converges exponentially fast to a geodesic sphere centered at the origin if the initial hypersurface is strictly horo-convex. Unlike previous results, we do not require the initial hypersurface to be star-shaped. As a consequence, we refine an Alexandrov-Fenchel type weighted geometric inequality proved by them so that both sides are isometry-invariant.