In this manuscript, we deal with complete space-like hypersurfaces immersed in the \((n+1)\) -dimensional steady state space \(\mathcal {H}^{n+1}\) , which is a Lorentzian manifold isometric to an open region of the de Sitter space \(\mathbb {S}^{n+1}_{1}\) . Assuming suitable restrictions on the higher order mean curvatures of these space-like hypersurfaces and applying an appropriate version of the Omori–Yau’s generalized maximum principle due to Alías et al. [4], we establish new uniqueness and nonexistence results concerning these space-like hypersurfaces.
