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 \(S^{n+1}\). 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) \(S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )\), \(\,1\le k\le n-1\); (ii) \(S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )\). 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 \(\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}\).
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Acknowledgments
We would like to thank Dr. En-Tao Zhao for his helpful discussions. Thanks also to Professor Y. L. Xin for sending us the manuscript of [7].
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Research supported by the Chinese NSF, Grant No. 11071211, 10771187; the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.
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Xu, Hw., Xu, Zy. The second pinching theorem for hypersurfaces with constant mean curvature in a sphere. Math. Ann. 356, 869–883 (2013). https://doi.org/10.1007/s00208-012-0875-0
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DOI: https://doi.org/10.1007/s00208-012-0875-0