A characterization of the Clifford torus
HTML articles powered by AMS MathViewer
- by Qing-Ming Cheng and Susumu Ishikawa
- Proc. Amer. Math. Soc. 127 (1999), 819-828
- DOI: https://doi.org/10.1090/S0002-9939-99-05088-1
- PDF | Request permission
Abstract:
In this paper, we prove that an $n$-dimensional closed minimal hypersurface $M$ 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 $S$ is the squared norm of the second fundamental form of $M$.References
- 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, DOI 10.2206/kyushumfs.47.79
- Qing-Ming Cheng, The rigidity of Clifford torus $S^1(\sqrt {1/n})\times S^{n-1}(\sqrt {(n-1)/n})$, Comment. Math. Helv. 71 (1996), no. 1, 60–69. MR 1371678, DOI 10.1007/BF02566409
- 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
- H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
- 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, DOI 10.1007/BF01458707
- 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, DOI 10.1007/BF02567445
- Yang, H.C. and Cheng, Q.M., Chern’s conjecture on minimal hypersurfaces, Math. Z. 227 (1998), 377-390.
Bibliographic 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
- 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
- © Copyright 1999 American Mathematical Society
- 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