Some Remarks on H-Surfaces in H^n with Planar Boundary

We consider embedded compact hypersurfaces M in a halfspace of hyperbolic space with boundary ∂M in the boundary geodesic hyperplane P of the halfspace and with non-zero constant mean curvature. We prove the following. Let {Mn} be a sequence of such hypersurfaces with ∂Mn contained in a disk of radius rn centered at a point σ ∈ P such that rn → 0 and that each Mn is a large H-hypersurface, H → 1. Then there exists a subsequence of {Mn} converging to the sphere of mean curvature H tangent to P at σ. In the case of small H-hypersurfaces or H ≤ 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {Mn} which are graphs. The convergence is smooth on compact subset of ℍ3\ σ