Abstract
Let M be a compact hypersurface with constant mean curvature in \({\mathbb{S}^{n + 1}}\). Denote by H and S the mean curvature and the squared norm of the second fundamental form of M, respectively. We verify that there exists a positive constant γ(n) depending only on n such that if ∣H∣ ⩽ γ(n) and \(\beta \left( {n,H} \right)\,\, \le \,\,S\,\, \le \,\,\beta \left( {n,H} \right) + {n \over {18}}\), then S ≡ β(n, H) and M is a Clifford torus. Here, \(\beta \left( {n,H} \right) = n + {{{n^3}} \over {2\left( {n - 1} \right)}}{H^2} + {{n\left( {n - 2} \right)} \over {2\left( {n - 1} \right)}}\sqrt {{n^2}{H^4} + 4\left( {n - 1} \right){H^2}} \).
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11531012), China Postdoctoral Science Foundation (Grant No. BX20180274) and Natural Science Foundation of Zhejiang Province (Grant No. LY20A010024).
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In Memory of Professor Zhengguo Bai (1916–2015)
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Lei, L., Xu, H. & Xu, Z. On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere. Sci. China Math. 64, 1493–1504 (2021). https://doi.org/10.1007/s11425-020-1841-y
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DOI: https://doi.org/10.1007/s11425-020-1841-y
Keywords
- generalized Chern conjecture
- hypersurfaces with constant mean curvature
- rigidity theorem
- scalar curvature
- the second fundamental form