On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature 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}. Denote by H and S the mean curvature and the squared norm of the second fundamental form of M, respectively. We verify that there exists a positive constant γ(n) depending only on n such that if ∣H∣ ⩽ γ(n) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \left( {n,H} \right)\,\, \le \,\,S\,\, \le \,\,\beta \left( {n,H} \right) + {n \over {18}}$$\end{document}, then S ≡ β(n, H) and M is a Clifford torus. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \left( {n,H} \right) = n + {{{n^3}} \over {2\left( {n - 1} \right)}}{H^2} + {{n\left( {n - 2} \right)} \over {2\left( {n - 1} \right)}}\sqrt {{n^2}{H^4} + 4\left( {n - 1} \right){H^2}} $$\end{document}.