A new characterization of the Clifford torus via scalar curvature pinching

Let Mn be a compact hypersurface with constant mean curvature H in Sn+1. Denote by S the squared norm of the second fundamental form of M. We prove that there exists an explicit positive constant γ(n) depending only on n such that if |H|≤γ(n) and β(n,H)≤S≤β(n,H)+n23, then S≡β(n,H) and M is one of the following cases: (i) Sk(kn)×Sn−k(n−kn), 1≤k≤n−1; (ii) S1(11+μ2)×Sn−1(μ1+μ2). Here β(n,H)=n+n32(n−1)H2+n(n−2)2(n−1)n2H4+4(n−1)H2 and μ=n|H|+n2H2+4(n−1)2. For n∈{6,12,24}, we give examples to show that the assumption |H|≤γ(n) cannot be removed.