On the stability of hypersurfaces in space forms

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 M n is contained either in an open hemisphere of the Euclidean sphere or in the hyperbolic space, we are able to show that M n is ( r , s ) -stable if, and only if, M n 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.