Rigidity theorems for hypersurfaces with constant mean curvature

Let Mn be a compact oriented hypersurface of a unit sphere \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}(1) with constant mean curvature H. Given an integer k between 2 and n − 1, we introduce a tensor ⌽ related to H and to the second fundamental form A of M, and show that if |⌽|2 ≤ BH,k and tr(⌽3) ≤ Cn,k |⌽|3, where BH,k and Cn,k are numbers depending only on H, n and k, then either |⌽|2 ≡ 0 or |⌽|2 ≡ BH,k. We characterize all Mn with |⌽|2 ≡ BH,k. We also prove that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}$$\end{document} and tr(⌽3) ≤ Cn,k |⌽|3 then |A|2 is constant and characterize all Mn with |A|2 in the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] $$\end{document}.