Stable constant mean curvature hypersurfaces in the real projective space

In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in RPn+1 and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in RPn+1; and (iii) the quotient to RPn+1 of the hypersurface S-k (r) x Sn-k (root 1-r(2)) hooked right arrow Sn+1 obtained as the product of two spheres of dimensions k and n-k, with k = 1,..., n-1, and radii r and root 1-r(2), respectively, with root k/(n+ 2) <= r <= root(k+2)/(n+2).