f-stability of spacelike hypersurfaces in weighted spacetimes

We establish the notions of f -stability and strong f -stability concerning closed spacelike hypersurfaces immersed with constant f -mean curvature in a conformally stationary spacetime endowed with a conformal timelike vector field V and a weight function f . When V is closed, with the aid of the f -Laplacian of a suitable support function, we characterize f -stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem. Furthermore, we obtain sufficient conditions which assure that a strongly f -stable closed spacelike hypersurface must be either f -maximal or isometric to a leaf orthogonal to V .