Abstract
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.
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