Characterization results and a sharp integral inequality for LW spacelike hypersurfaces in locally symmetric Lorentzian spaces

Our aim is to study the geometry of linear Weingarten (LW) spacelike hypersurfaces immersed in a locally symmetric Lorentzian space obeying some standard curvature conditions. Under appropriate constraints on the scalar curvature function, we use a suitable extension of the generalized maximum principle of Omori–Yau to show that a complete LW spacelike hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, where one of them is simple. Furthermore, we deal with the parabolicity of complete LW hypersurfaces with respect to a Cheng–Yau modified operator and we also establish a sharp integral inequality concerning compact LW hypersurfaces (without boundary) in a locally symmetric Einstein spacetime. Applications to the de Sitter space are also given.