Strongly Stable Linear Weingarten Hypersurfaces Immersed in the Hyperbolic Space

Abstract

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.