A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

Abstract

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space H n + 1 . We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the ( n − 1 )-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.