[1] Wang Peijun, Chao Xiaoli,. Hypersurfaces with constant mean curvature in unit sphere [J]. Journal of Southeast University (English Edition), 2016, 32 (1): 132-134. [doi:10.3969/j.issn.1003-7985.2016.01.022]

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Hypersurfaces with constant mean curvature in unit sphere()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

32

Issue:

2016 1

Page:

132-134

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2016-03-20

Info

Title:

Hypersurfaces with constant mean curvature in unit sphere

Author(s):

Wang Peijun;  Chao Xiaoli

Department of Mathematics, Southeast University, Nanjing 210096, China

Keywords:

hypersurface with constant mean curvature;  unit sphere;  pinching

PACS:

O186.1

DOI:

10.3969/j.issn.1003-7985.2016.01.022

Abstract:

The pinching of n-dimensional closed hypersurface M with constant mean curvature H in unit sphere Sⁿ⁺¹1(1)is considered. Let(~overA)=∑_{i, j, k}h²_ijk(λ_i+nH)², (~overB)=∑_{i, j, k}h²_ijk(λ_i+nH)·(λ_j+nH), (~overS)=∑_i(λ_i+nH)², where h_ij=λ_iδ_ij. Utilizing Lagrange’s method, a sharper pointwise estimation of 3((~overA)-2(~overB))in terms of (~overS) and ∇h² is obtained, here ∇h²=∑_{i, j, k}h^2_ijk2. Then, with the help of this, it is proved that M is isometric to the Clifford hypersurface if the square norm of the second fundamental form of M satisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.

References:

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[7] Zhang Q. Scalar curvature of hypersurfaces with constant mean curvature in spheres [J]. Glasg Math J, 2012, 54(1): 67-75.
[8] Xu H W, Xu Z Y. The second pinching theorem for hypersurfaces with constant mean curvature in a sphere [J]. Math Ann, 2013, 356(3): 869-883.

Memo

Memo:

Biographies: Wang Peijun(1989—), male, graduate; Chao Xiaoli(corresponding author), male, doctor, professor, xlchao@seu.edu.cn.
Citation: Wang Peijun, Chao Xiaoli. Hypersurfaces with constant mean curvature in unit sphere[J].Journal of Southeast University(English Edition), 2016, 32(1):132-134. DOI:10.3969/j.issn.1003-7985.2016.01.022.

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