Isometric immersions in the hyperbolic space with their image contained in a horoball

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.