Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion ${i:M^nto mathbb S^{n+1}}$ with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus ${mathbb S^1(sqrt{1-r^2})timesmathbb S^{n-1} (r)}$ . For R > (n ? 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in ${mathbb S^{n+1}}$ with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.