Scalar curvature pinching for CMC hypersurfaces in a sphere

We consider an $n$-dimensional closed hypersurface $M$ with constant mean curvature $H$ in $S^{n+1}, 3 \leq n \leq 8$. Denote by $S$ and $\beta(n,H)$ the squared norm of the second fundamental form of $M$ and $S^1(\frac{1}{\sqrt{1 + \mu^2}}) \times S^{n-1}(\frac{\mu}{\sqrt{1 + \mu^2}})$ respectively, where $\mu = \frac{n | H | + \sqrt{n^2H^2 + 4(n-1)}}{2}$. We prove that there exist two positive constants $\gamma(n)$ and $\epsilon(n)$ such that if $| H | \leq \gamma(n)$ and $\beta(n,H) \leq S \lt \beta(n,H) + \epsilon(n)$, then $S \equiv \beta(n,H)$ and $M$ is one of the following cases:

(i) $S^k(\sqrt{\frac{k}{n}}) \times S^{n-k}(\sqrt{\frac{n-k}{n}}), k = 1, 2, \ldots, n-1$;

(ii) $S^1(\frac{1}{\sqrt{1+\mu^2}}) \times S^{n-1}(\frac{1}{\sqrt{1+\mu^2}})$

This result extends the scalar curvature pinching theorem for minimal hypersurfaces due to Peng-Terng, Wei-Xu and Zhang.

Keywords

hypersurfaces with constant mean curvature, rigidity, scalar curvature, Clifford torus

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Published 13 November 2013