We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb{ S}^{ n+1}_{ 1}$ 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 Al\'{ \i}as, Impera and Rigoli in [5].
Categories: math.DG