The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

Abstract

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. 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 exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$, 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 $\beta (n,H)=n+\frac{n^3}{2(n-1)}{H}^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2{H}^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2{H}^2+4(n-1)}}{2}$.

13 pages

References in corpus (1)

• Some results on Chern's problem