Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hypersurfaces in a sphere
with constant mean curvature


Author: Zhong Hua Hou
Journal: Proc. Amer. Math. Soc. 125 (1997), 1193-1196
MSC (1991): Primary 53C42, 53A10
DOI: https://doi.org/10.1090/S0002-9939-97-03668-X
MathSciNet review: 1363169
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $M^n$ be a closed hypersurface of constant mean curvature immersed in the unit sphere $S^{n+1}$. Denote by $S$ the square of the length of its second fundamental form. If $S<2\sqrt {n-1}$, $M$ is a small hypersphere in $S^{n+1}$. We also characterize all $M^n$ with $S=2\sqrt {n-1}$.


References [Enhancements On Off] (What's this?)

  • 1. H. Alencar and M. P. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), pp. 1223-1229. MR 94f:53108
  • 2. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), pp. 195-204. MR 55:4045
  • 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), Springer-Verlag, New York (1970), pp. 59-75. MR 42:8424
  • 4. H. B. Lawson, Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), pp. 187-197. MR 38:6505
  • 5. K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces of constant mean curvature, J. Diff. Geom. 3 (1969), pp. 367-378. MR 42:1018
  • 6. M. Okumura, Submanifolds and a pinching problem on the second fundamental tensors, Trans. Amer. Math. Soc. 178 (1973), pp. 285-291. MR 47:5793
  • 7. -, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), pp. 207-213. MR 50:5701
  • 8. W. Santos, Submanifolds with parallel mean curvature vector spheres, Tôhoku Math. J. 46 (1994), pp. 403-415. MR 95f:53109
  • 9. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), pp. 62-105. MR 38:1617
  • 10. H. W. Xu, A pinching constant of Simon's type and isometric immersion, Chinese Ann. of Math. Ser. A 12 (1991), No. 3, pp. 261-269. MR 92h:53077
  • 11. -, A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Arch. Math. 61 (1993), pp. 489-496. MR 94m:53084
  • 12. S. T. Yau, Submanifolds with constant mean curvature II, Amer. J. Math. 97 (1975), No. 1, pp. 76-100. MR 51:6670

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C42, 53A10

Retrieve articles in all journals with MSC (1991): 53C42, 53A10


Additional Information

Zhong Hua Hou
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Japan; Department of Applied Mathematics, Dalian University of Technology, People’s Republic of China
Email: hou@math.titech.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-97-03668-X
Received by editor(s): July 27, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society