The stability index of hypersurfaces with constant scalar curvature in spheres

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n - 1)r, r > 1, in an (n + 1)-dimensional sphere Sn+1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H-3 are constant. In this paper, we prove the same results, without the assumption that H-3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n - 1)r, r > 1, in Sn+1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.