(r, k, a, b)-Stability of hypersurfaces in space forms

Abstract

In a Riemannian space form, we define the (r, k, a, b)-stability concerning closed hypersurfaces, where r and k are entire numbers satisfying the inequality $$0\le k<r\le n-2$$ and a and b are real numbers (at least one nonzero). In this context, when $$b=0$$ , we provide a characterization of the geodesic spheres as critical points of the Jacobi functional associated with the notion of (r, k, a, 0)-stability. Moreover, in the case $$b\not =0$$ , by supposing that a hypersurface $$\Sigma ^n$$ is contained either in an open hemisphere of the Euclidean sphere or in the Euclidean space or in the hyperbolic space, and considering some appropriate restrictions on the constants a and b, we are able to show that $$\Sigma ^n$$ is (r, k, a, b)-stable if, and only if, $$\Sigma ^n$$ is a geodesic sphere.