The pinching constant of minimal hypersurfaces in the unit spheres

Author:
Qin Zhang

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1833-1841

MSC (2000):
Primary 53C40

DOI:
https://doi.org/10.1090/S0002-9939-09-10251-4

Published electronically:
December 31, 2009

MathSciNet review:
2587468

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that if () is a closed minimal hypersurface in a unit sphere , then there exists a positive constant depending only on such that if , then is isometric to a Clifford torus, where is the squared norm of the second fundamental form of .

**1.**J. Simons,*Minimal varieties in Riemannian manifolds*, Ann. of Math. (2),**88**(1968), 62-105. MR**0233295 (38:1617)****2.**H. B. Lawson Jr.,*Local rigidity theorems for minimal hypersurfaces*, Ann. of Math. (2),**89**(1969), 187-197. MR**0238229 (38:6505)****3.**S. S. Chern, M. do Carmo, S. Kobayashi,*Minimal submanifolds of a sphere with second fundamental form of constant length*, Functional analysis and related fields, Springer, New York, 1970, pp. 59-75. MR**0273546 (42:8424)****4.**C. K. Peng, C. L. Terng,*Minimal hypersurfaces of spheres with constant scalar curvature*, Ann. of Math. Stud., 103, Princeton University Press, 1983, 179-198. MR**795235 (87k:53143)****5.**C. K. Peng, C. L. Terng,*The scalar curvature of minimal hypersurfaces in spheres*, Math. Ann.,**266**(1983), 105-113. MR**722930 (85c:53099)****6.**Q. M. Cheng, S. Ishikawa,*A characterization of the Clifford torus*, Proc. Amer. Math. Soc.,**127**(1999), 819-828. MR**1636934 (99g:53064)****7.**Q. M. Cheng,*The classification of complete hypersurfaces with nonzero constant mean curvature of space form of dimension*, Mem. Fac. Sci. Kyushu Univ. Ser. A,**47**(1993), 79-102. MR**1222356 (94h:53067)****8.**S. Y. Cheng, S. T. Yau,*Hypersurfaces with constant scalar curvature*, Math. Ann.,**225**(1977), no. 3, 195-204. MR**0431043 (55:4045)****9.**A. M. Li, J. M. Li,*An intrinsic rigidity theorem for minimal submanifolds in a sphere*, Arch. Math. (Basel),**58**(1992), no. 6, 582-594. MR**1161925 (93b:53050)****10.**H. W. Xu,*A rigidity theorem for submanifolds with parallel mean curvature in a sphere*, Arch. Math. (Basel),**61**(1993), no. 5, 489-496. MR**1241055 (94m:53084)****11.**H. W. Xu,*On closed minimal submanifolds in pinched Riemannian manifolds*, Trans. Amer. Math. Soc.,**347**(1995), no. 5, 1743-1751. MR**1243175 (95h:53088)****12.**H. W. Xu, W. Fang, F. Xiang,*A generalization of Gauchman's rigidity theorem*, Pacific J. Math.,**228**(2006), no. 1, 185-199. MR**2263029 (2007j:53048)****13.**S. T. Yau,*Submanifolds with constant mean curvature. I, II*, Amer. J. Math.,**96**(1974), 346-366; ibid.,**97**(1975), 76-100. MR**0370443 (51:6670)****14.**Q. M. Cheng, H. C. Yang,*Chern's conjecture on minimal hypersurfaces*, Math. Z.,**227**(1998), no. 3, 377-390. MR**1612653 (99c:53070)****15.**Q. M. Cheng,*The rigidity of Clifford torus*, Comment. Math. Helvetici,**71**(1996), 60-69. MR**1371678 (97a:53094)****16.**Y. T. Zhang, S. L. Xu,*Rigidity of the Clifford torus*, Acta Mathematica Scientia Ser. A Chin. Ed.,**28**(2008), 128-132. MR**2392041 (2008m:53145)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
53C40

Retrieve articles in all journals with MSC (2000): 53C40

Additional Information

**Qin Zhang**

Affiliation:
Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, People’s Republic of China

Email:
zhangdiligence@126.com

DOI:
https://doi.org/10.1090/S0002-9939-09-10251-4

Keywords:
Minimal hypersurface,
Clifford torus,
second fundamental form

Received by editor(s):
June 7, 2009

Received by editor(s) in revised form:
August 18, 2009

Published electronically:
December 31, 2009

Communicated by:
Richard A. Wentworth

Article copyright:
© Copyright 2009
American Mathematical Society