On the stability of hypersurfaces in space forms

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We define the notion of (r,s)-stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a Riemannian space form. By supposing that such a hypersurface Mn is contained either in an open hemisphere of the Euclidean sphere or in the hyperbolic space, we are able to show that Mn is (r,s)-stable if, and only if, Mn is a geodesic sphere. Moreover, we obtain a suitable characterization of the (r,s)-stability through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem.