Rigidity and nonexistence of complete spacelike hypersurfaces in the steady state space

We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space ${\mathbb{S}}_1^{n+1}$ which is known as the steady state space ${\mathcal{H}}^{n+1}$. In this setting, under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we prove that they must be spacelike hyperplanes of ${\mathcal{H}}^{n+1}$. Nonexistence results concerning these spacelike hypersurfaces are also given. Our approach is based on a suitable extension of the Omori-Yau’s generalized maximum principle due to Alías, Impera and Rigoli in [5].