Higher order mean curvature estimates for complete hypersurfaces into horoballs

Abstract

We consider properly immersed two-sided hypersurfaces \({\varphi}:{M \rightarrow N}\) such that \({\varphi(M)}\) is contained in a horoball of N, where N satisfies fairly weak curvature bounds and we prove higher order mean curvature estimates that are natural extensions of the estimates obtained by Alias, Dajczer and Rigoli in [3] and Albanese, Alias and Rigoli in [1]. We show that these ambient curvature bounds in the presence of the properness of \({\varphi}\) guarantees that M satisfies a general version of the weak maximum principle established by Albanese, Alias and Rigoli in [1].