Abstract
Let M be an \(n(\ge \)3)-dimensional closed hypersurface in a unit sphere with constant m-th order mean curvature and with two distinct principal curvatures. We obtain a sharp curvature integral for M in terms of Ricci curvature, which gives a characterization of a Clifford hypersurface. Moreover we give a generalization of Simons’ integral inequality for closed hypersurface with vanishing m-th order mean curvature by making use of the Laplacian of the function of principal curvatures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alencar, H., do Carmo, M.: Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120(4), 1223-1229 (1994)
Cartan, E.: Familles de surfaces isoparametriques dans les espaces a courvure constante. Ann. Mat. Pura Appl. 17(1), 177-191 (1938)
Cheng, Q.M.: Hypersurfaces in a unit sphere $S^{n+1}(1)$ with constant scalar curvature. J. Lond. Math. Soc. (2) 64(3), 755-768 (2001)
Cheng, Q.M., Ishikawa, S.: A characterization of the Clifford torus. Proc. Am. Math. Soc. 127(3), 819-828 (1999)
Hasanis, T., Savas-Halilaj, A., Vlachos, T.: Complete minimal hypersurfaces in a sphere. Monatsh. Math. 145(4), 301-305 (2005)
Hasanis, T., Vlachos, T.: A pinching theorem for minimal hypersurfaces in a sphere. Arch. Math. (Basel) 75(6), 469-471 (2000)
Hasanis, T., Vlachos, T.: Ricci curvature and minimal submanifolds. Pac. J. Math. 197(1), 13-24 (2001)
Lawson, H.B., Jr.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. (2) 89, 187-197 (1969)
Chern, S.S., do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In: Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., vol. 1970, pp. 59-75. Springer, New York (1968)
Li, H.: A characterization of Clifford minimal hypersurfaces in $S^4$. Proc. Am. Math. Soc. 123(10), 3183-3187 (1995)
Li, H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305(4), 665-672 (1996)
Otsuki, T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature. Am. J. Math. 92, 145-173 (1970)
Perdomo, O.: Ridigity of minimal hypersurfaces of spheres with two principal curvatures. Arch. Math. (Basel) 82(2), 180-184 (2004)
Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. (2) 88, 62-105 (1968)
Wang, Q.: Rigidity of Clifford minimal hypersurfaces. Monatsh. Math. 140(2), 163-167 (2003)
Wei, G.: Complete hypersurfaces with constant mean curvature in a unit sphere. Monatsh. Math. 149(3), 251-258 (2006)
Wei, G.: Rigidity theorem for hypersurfaces in a unit sphere. Monatsh. Math. 149(4), 343-350 (2006)
Wei, G.: Complete hypersurfaces with $H_k=0$ in a unit sphere. Differ. Geom. Appl. 25(5), 500-505 (2007)
Wei, G.: J. Simons’ type integral formula for hypersurfaces in a unit sphere. J. Math. Anal. Appl. 340(2), 1371-1379 (2008)
Wu, B.Y.: On hypersurfaces with two distinct principal curvatures in a unit sphere. Differ. Geom. Appl. 27(5), 623-634 (2009)
Zhang, Y.T.: Pinching theorems of hypersurfaces in a unit sphere. Differ. Geom. Appl. 29(6), 730-736 (2011)
Acknowledgments
The authors would like to thank the referee for the careful review and the valuable comments. The second author was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013R1A1A1A05006277).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Cap.
Rights and permissions
About this article
Cite this article
Min, SH., Seo, K. Characterizations of a Clifford hypersurface in a unit sphere via Simons’ integral inequalities. Monatsh Math 181, 437–450 (2016). https://doi.org/10.1007/s00605-015-0842-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0842-4