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CONSTANT MEAN CURVATURE HYPERSURFACES IN SPHERES

Published online by Cambridge University Press:  02 August 2011

QIN-TAO DENG ,
HUI-LING GU  and
YAN-HUI SU
QIN-TAO DENG
Affiliation:
Laboratory of Nonlinear Analysis, Huazhong Normal University, Wuhan 430079, P. R. China e-mail: qintaodeng@yahoo.com.cn
HUI-LING GU
Affiliation:
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China e-mail: guhuil@mail.sysu.edu.cn
YAN-HUI SU
Affiliation:
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China e-mail: r314272@yahoo.com.cn
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Abstract

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In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere n+1 with n ≤ 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51(2) (2009), 413–423). In order to be precise, we prove that if |H| ≤ ϵ(n), then there exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S = S0 and M is isometric to the Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n.

Keywords

53C4053C42
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 77 - 86
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

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