A new open form of the weak maximum principle and geometric applications

The aim of this paper is to give various height estimates and to establish some geometrical constraints for non-compact hypersurfaces with constant mean curvature or, more generally, constant higher order mean curvature into warped product manifolds. Results are sharp and agree with those in the compact case already considered in the literature. The main technical tool of the paper is a new form of the weak maximum principle for a very large class of differential operators that, despite of its simplicity, reveals very interesting for further applications.