Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurface Σn immersed with constant mean curvature H in the anti-de Sitter space ℍ1n+1, when its boundary ∂Σis contained into an umbilical spacelike hypersurfaceof this spacetime which is isometric to the hyperbolic space ℍn. Our estimate depends only on the value of H and on the geometryof ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of ℍ1n+1 and nonexistence results in connection with such types of hypersurfaces