Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T07:36:20.198Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
HUI-LING GU
Affiliation:
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China e-mail: [email protected]
YAN-HUI SU
Affiliation:
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China e-mail: [email protected]
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.

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

REFERENCES

1.Almeida, S. and Brito, F., Closed 3-dimensional hypersurfaces with constant with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), 195–206.CrossRefGoogle Scholar
2.Almeida, S., Brito, F. and Sousa, L. A. M. Jr., Closed hypersurfaces of S 4 with two constant curvature functions, Results Math. 50 (2007), 17–26.CrossRefGoogle Scholar
3.Cartan, E., Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939), 335–367.CrossRefGoogle Scholar
4.Chang, S. P., A closed hypersurface with constant scalar curvature and constant mean curvature in 4 is isoparametric, Comm. Anal. Geom. 1 (1993), 71–100.CrossRefGoogle Scholar
5.Chang, S. P., On minimal hypersurfaces with constant scalar curvatures in S 4, J. Diff. Geom. 37 (1993), 523–534.Google Scholar
6.Cheng, Q. M., He, Y. J. and Li, H. Z., Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51 (2) (2009), 413–423.CrossRefGoogle Scholar
7.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 (Browder, F. E., Editor) (Springer-Verlag Berlin, 1970), pp. 59–75.Google Scholar
8.Hsiang, W. Y., Remarks on closed minimal submanifolds in the standard riemannian m-Sphere, J. Diff. Geom. 1 (1967), 257–267.Google Scholar
9.Lawson, H. B. Jr., Local rigidity theorems for minimal hypersurfaces, Ann. Math. 89 (1969), 167–179.CrossRefGoogle Scholar
10.Lusala, T., Scherfner, M. and Sousa, L. A. M. Jr., Closed minimal Willmore hypersurfaces of S 5(1) with constant scalar curvature, Asian J. Math. 9 (1) (2005), 65–78.CrossRefGoogle Scholar
11.Peng, C. K. and Terng, C. L., Minimal hypersurfaces of spheres with constant scalar curvature. Seminar on minimal submanifolds, Ann. Math. Stud. 103 (1983), 177–198 (Princeton University Press, Princeton, NJ).Google Scholar
12.Peng, C. K. and Terng, C. L., The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1) (1983), 105–113.CrossRefGoogle Scholar
13.Scherfner, M. and Weiss, S., Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres, Proceedings of 33rd South German Colloquium on Differential Geometry, TU, Vienna (2008), 1–33.Google Scholar
14.Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math. 88 (1968), 62–105.CrossRefGoogle Scholar
15.Suh, Y. J. and Yang, H. Y., The scalar curvature of minimal hypersurfaces in a unit sphere, Commun. Contemp. Math. 9 (2) (2007), 183–200.CrossRefGoogle Scholar
16.Wei, S. M. and Xu, H. W., Scalar curvature of minimal hypersurfaces in a sphere, Math. Res. Lett. 14 (3) (2007), 423–432.CrossRefGoogle Scholar
17.Yang, H. C. and Cheng, Q. M., A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Chinese Sci. Bull. 36 (1991), 1–6.Google Scholar
18.Yang, H. C. and Cheng, Q. M., An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Manuscripta Math. 82 (1994), 89–100.CrossRefGoogle Scholar
19.Yang, H. C. and Cheng, Q. M., Chern's conjecture on minimal hypersurfaces, Math. Z. 227 (1998), 377–390.CrossRefGoogle Scholar
20.Zhang, Q., The pinching constant of minimal hypersurfaces in the unit spheres, Proc. Amer. Math. Soc. 138 (5) (2010), 1833–1841.CrossRefGoogle Scholar