An Estimate for the Mean Curvature of Submanifolds Contained in a Horoball

We obtain the Omori-Yau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate on the mean curvature of properly immersed submanifolds in ℍn × ℝl with the image under the projection on ℍn contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.