Hostname: page-component-784d4fb959-4tbfc Total loading time: 0 Render date: 2025-07-16T22:09:26.776Z Has data issue: false hasContentIssue false
  • English
  • Français

SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

Published online by Cambridge University Press:  02 August 2011

QIN ZHANG
QIN ZHANG*
Affiliation:
Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China e-mail: zhangdiligence@126.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. 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 a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

Keywords

Primary 53C4253B25

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 67 - 75
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Alencar, H. and do Carmo, M., Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120(1994), 1223–1229.CrossRefGoogle Scholar
2.Cheng, Q. M., The classification of complete hypersurfaces with constant mean curvature of space form of dimension 4, Mem. Fac. Sci. Kyushu Univ. 47 (1993), 79–102.Google Scholar
3.Cheng, Q. M., The rigidity of Clifford torus , Comment. Math. Helvetici 71 (1996), 60–69.CrossRefGoogle Scholar
4.Cheng, Q. M., He, Y. and Li, H., Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasgow Math. J. 51 (2009), 413–423.CrossRefGoogle Scholar
5.Cheng, Q. M. and Ishikawa, S., A characterization of the Clifford torus, Proc. Amer. Math. Soc. 127 (3) (1999), 819–828.CrossRefGoogle Scholar
6.Cheng, Q. M. and Yang, H. C., Chern's conjecture on minimal hypersurfaces, Math. Z. 227 (3) (1998), 377–390.Google Scholar
7.Cheng, S. Y. and Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann. 225 (3) (1977), 195–204.CrossRefGoogle Scholar
8.Chern, S. S., do Carmo, M. and Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional analysis and related fields (Springer, New York, 1970), pp. 59–75.Google Scholar
9.Hasanis, T., Vlachos, T., A pinching theorem for minimal hypersurfaces in a sphere, Arch. Math. 75 (2000), 469–471.CrossRefGoogle Scholar
10.Lawson, H. B., Local rigidity theorems for minimal hypersurfaces, Ann. Math. 89 (1969), 179–185.CrossRefGoogle Scholar
11.Li, H., Scalar curvature of hypersurfaces with constant mean curvature in spheres, Tsinghua Sci. Technol. 1 (1996), 266–269.Google Scholar
12.Li, A. M. and Li, J. M., An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel) 58 (6) (1992), 582–594.Google Scholar
13.Okumura, M., Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213.CrossRefGoogle Scholar
14.Peng, C. K. and Terng, C. L., Minimal hypersurfaces of sphere with constant scalar curvature, Ann. Math. Stud. 103 (1983), 179–198.Google Scholar
15.Peng, C. K. and Terng, C. L., The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983), 105–113.CrossRefGoogle Scholar
16.Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math. 88 (1968), 62–105.CrossRefGoogle Scholar
17.Wei, S. M. and Xu, H. W., Scalar curvature of minimal hypersurfaces in spheres, Math. Res. Lett. 14 (2007), 423–432.CrossRefGoogle Scholar
18.Xu, H. W., A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Arch. Math. (Basel) 61 (5) (1993), 489–496.CrossRefGoogle Scholar
19.Xu, H. W., On closed minimal submanifolds in pinched Riemannian manifolds, Trans. Amer. Math. Soc. 347 (5) (1995), 1743–1751.CrossRefGoogle Scholar
20.Xu, H. W., Fang, W. and Xiang, F., A generalization of Gauchman's rigidity theorem, Pac. J. Math. 228 (1) (2006), 185–199.CrossRefGoogle Scholar
21.Yau, S. T., Submanifolds with constant mean curvature, I, II, Amer. J. Math. 96 (1974), 346–366; 96 (1975), 76–100.CrossRefGoogle Scholar
22.Zhang, Qin, The pinching constant of minimal hypersurfaces in the unit spheres, Proc. Amer. Math. Soc. 138 (5) (2010), 1833–1841.CrossRefGoogle Scholar