The stability of hypersurfaces revisited

Abstract

In this paper, we revisit the problem of characterize (r, s)-stable closed hypersurfaces immersed in a Riemannian space form, which was firstly established in Velasquez et al. (J Math Anal Appl 406:134–146, 2013). With a different approach of that used in the proof of the main theorem of Velasquez et al. (J Math Anal Appl 406:134–146, 2013), we complete its program showing that a closed hypersurface contained in the Euclidian space \({\mathbb {R}}^{n+1}\) and having higher order mean curvatures linearly related is (r, s)-stable if, and only if, it is a geodesic sphere of \({\mathbb {R}}^{n+1}\).