A new stability notion of closed hypersurfaces in the hyperbolic space

Abstract

In this article, we establish the notion of strong $(r,k,a,b)$-stability related to closed hypersurfaces immersed in the hyperbolic space $\mathbb{H}^{n+1}$, where $r$ and $k$ are nonnegative integers satisfying the inequality $0 \leq k<r \leq n-2$ 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 $H_{r+1}$ and $H_{k+1}$, we prove that if a closed hypersurface into the hyperbolic space $\mathbb{H}^{n+1}$ 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.