On the stability index of hypersurfaces with constant mean curvature in spheres

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface $M^n$ in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to $n+2$, with equality if and only if $M$ is a constant mean curvature Clifford torus $\mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^2})$ with radius $\sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}$.