  We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for
submanifolds confined into either a horocylinder of N X L or a horoball of N,
where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these
submanifolds behave in many respects like submanifolds immersed into compact
balls and into cylinders over compact balls. The proofs rely on the Hessian
comparison theorem for the Busemann function.
