Curvature estimates for submanifolds immersed into horoballs and horocylinders

We prove mean and sectional curvature estimates for submanifolds confined into either a horocylinder of N × 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.