A new characterization of the Clifford torus via scalar curvature pinching

Abstract

Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $\mathbb{S}^{n+1}$. Denote by $S$ the squared norm of the second fundamental form of $M$. We prove that there exists a positive constant $\gamma(n)$ depending only on $n$ such that if $|H|\leq\gamma(n)$ and $\beta(n,H)\leq S\leq\beta(n,H)+\frac{n}{23}$, then $S\equiv\beta(n,H)$ and $M$ is one of the following cases: (i) $\mathbb{S}^{k}(\sqrt{\frac{k}{n}})\times \mathbb{S}^{n-k}(\sqrt{\frac{n-k}{n}})$, $\,1\le k\le n-1$; (ii) $\mathbb{S}^{1}(\frac{1}{\sqrt{1+\mu^2}})\times \mathbb{S}^{n-1}(\frac{\mu}{\sqrt{1+\mu^2}})$. Here $\beta(n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$ and $\mu=\frac{n|H|+\sqrt{n^2H^2+4(n-1)}}{2}$. This provides a new characterization of the Clifford torus.