Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i:M^n\to \mathbb S^{n+1}}$$\end{document} 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 Cn(R) depending on R and n so that R ≥ 1 and sup |A|2 = Cn(R) imply that the hypersurface is a H(r)-torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}$$\end{document}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > Cn(R) there is a complete hypersurface in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb S^{n+1}}$$\end{document} with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng.