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In this article, we establish the notion of strong stability related to closed linear Weingarten hypersurfaces immersed in the hyperbolic space. In this setting, initially we show that geodesic spheres are strongly stable. Afterwards, under a suitable restriction on the mean and scalar curvatures, we prove that if a closed linear Weingarten hypersurface into the hyperbolic space is strongly stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in a chronological future (or past) of the de Sitter space.Mathematics Subject Classification. Primary 53C42; Secondary 53B30, 53C50, 53Z05, 83C99.

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In this paper, our aim is to establish optimal upper estimates for the first positive eigenvalue of a Jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed either in spherical or in hyperbolical spaces.Mathematics Subject Classification. Primary 53C42; Secondary 53B30.

In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.