Symmetry of Constant Mean Curvature Hypersurfaces in Hyperbolic Space

Abstract

In a recent paper, M. Do Carmo and B. Lawson studied hypersurfaces M of constant mean curvature in hyperbolic space [2]. They use the Alexandrov reflection technique to study M given the asymptotic boundary ðM. For example, one of their theorems says M is a horoshpere when ðM reduces to a point. They also prove a Bernstein type theorem for minimal graphs. In this paper we shall extend their results to other boundary conditions. We prove an embedded M, of constant mean curvature, with ðM a subset of a codimension one sphere S, either is invariant by reflection in the hyperbolic hyperplane containg S or is a hypersphere. In the former case M is a "bigraph" over H: it meets any geodesic orthogonal to H either not at all or transversaly in two points (one on each side of H) or tangentially on H.