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 ℍ^𝑛+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 𝐻_𝑟+1 and 𝐻_𝑘+1, we prove that if a closed hypersurface into the hyperbolic space 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.