Rigidity of spacelike hypersurfaces in spatially weighted generalized Robertson-Walker spacetimes

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.