On the Stability of $f$-Maximal Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

Our purpose in this paper is to study the stability of $f$-maximal spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime $-I\times _{\rho }M^n_f$, where $M^n_f$ is a weighted Riemannian manifold endowed with a weight function $f$. In this setting, we obtain sufficient conditions to guarantee that an $f$-maximal hypersurface be $L_f$-stable, where $L_f$ stands for the weighted Jacobi operator.