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

TL;DR

This paper extends the second pinching theorem for minimal hypersurfaces in spheres to those with small constant mean curvature, characterizing the shape of hypersurfaces under specific curvature bounds.

Contribution

It generalizes the second pinching theorem to hypersurfaces with small constant mean curvature, providing a classification under curvature pinching conditions.

Findings

Hypersurfaces with small constant mean curvature satisfying the pinching condition are classified.

The second fundamental form's squared norm is constant and explicitly determined.

The results include specific product of spheres as the only solutions under the given conditions.

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 $|H|\leq\gamma(n)$ and $\beta(n,H)\leq S\leq\beta(n,H)+\delta(n)$, 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 $\beta(n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$ and $\mu=\frac{n|H|+\sqrt{n^2H^2+4(n-1)}}{2}$.