Height estimates and topology at infinity of hypersurfaces immersed in a certain class of warped products

Abstract

Height estimates are given for hypersurfaces immersed in a class of warped products of the type $$\mathbb {R}\times _{\rho } M^n$$ , under the assumption that some higher order mean curvatures are linearly related. When the fiber $$M^n$$ is compact and such a hypersurface $$\Sigma ^n$$ is noncompact, two-sided and properly immersed, we apply our height estimates in order to get information concerning the topology at infinity of $$\Sigma ^n$$ . Furthermore, when $$M^n$$ is not necessarily compact, using a generalized version of the Omori–Yau maximum principle we establish new half-space theorems for these hypersurfaces.