New characterizations of spacelike hyperplanes in the steady state space

Abstract

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space Sn+11 which is known as the steady state space Hn+1. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of Hn+1. Furthermore, through the analysis of the hyperbolic cylinders of Hn+1, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.