A characterization of the Clifford torus

Cheng, Qing-Ming; Ishikawa, Susumu

doi:10.1090/S0002-9939-99-05088-1

A characterization of the Clifford torus

Authors: Qing-Ming Cheng and Susumu Ishikawa
Journal: Proc. Amer. Math. Soc. 127 (1999), 819-828
MSC (1991): Primary 53C20, 53C42
DOI: https://doi.org/10.1090/S0002-9939-99-05088-1
MathSciNet review: 1636934
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that an -dimensional closed minimal hypersurface with Ricci curvature $Ric(M) \geq \dfrac{n}{2}$ of a unit sphere $S^{n+1}(1)$ is isometric to a Clifford torus if $n\leq S\leq n+\frac{14(n+4)}{9n+30}$ , where is the squared norm of the second fundamental form of .

1. Qing Ming Cheng, The classification of complete hypersurfaces with nonzero constant mean curvature of space form of dimension 4, Mem. Fac. Sci. Kyushu Univ. Ser. A 47 (1993), no. 1, 79–102. MR 1222356, https://doi.org/10.2206/kyushumfs.47.79
2. Qing-Ming Cheng, The rigidity of Clifford torus 𝑆¹(√1/𝑛)×𝑆ⁿ⁻¹(√(𝑛-1)/𝑛), Comment. Math. Helv. 71 (1996), no. 1, 60–69. MR 1371678, https://doi.org/10.1007/BF02566409
3. S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR 0273546
4. H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, https://doi.org/10.2307/1970816
5. Chia Kuei Peng and Chuu-Lian Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann. 266 (1983), no. 1, 105–113. MR 722930, https://doi.org/10.1007/BF01458707
6. Hong Cang Yang and Qing Ming Cheng, An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Manuscripta Math. 84 (1994), no. 1, 89–100. MR 1283329, https://doi.org/10.1007/BF02567445
7. Yang, H.C. and Cheng, Q.M., Chern's conjecture on minimal hypersurfaces, Math. Z. 227 (1998), 377-390. CMP 98:11

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Additional Information

Qing-Ming Cheng
Affiliation: Department of Mathematics, Faculty of Science, Josai University, Sakado, Saitama 350-0295, Japan
Email: cheng@math.josai.ac.jp

Susumu Ishikawa
Affiliation: Department of Mathematics, Saga University, Saga 840-0027, Japan

DOI: https://doi.org/10.1090/S0002-9939-99-05088-1
Keywords: Minimal hypersurfaces, scalar curvature, Ricci curvature, Clifford torus
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: November 1, 1996
Additional Notes: The first author’s research was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture and by a Grant-in-Aid for Scientific Research from Josai University.
The second author’s research was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society