A new stability notion of closed hypersurfaces in the hyperbolic space

In this article, we establish the notion of strong (r, k, a, b)-stability related to closed hypersurfaces immersed in the hyperbolic space Hn+1, where r and k are nonnegative integers satisfying the inequality 0 <= k < r <= n - 2 and a and b are real numbers (at least one nonzero). In this setting, considering some appropriate restrictions on the constants a and b, we show that geodesic spheres are strongly (r, k, a, b)-stable. Afterwards, under a suitable restriction on the higher order mean curvatures Hr+1 and Hk+1, we prove that if a closed hypersurface into the hyperbolic space Hn+1 is strongly (r, k, a, b)-stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in the chronological future (or past) of an equator of the de Sitter space.