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DOI QR Code

HYPERSURFACES WITH CONSTANT k-TH MEAN CURVATURE AND TWO DISTINCT PRINCIPAL CURVATURES IN SPHERES

  • Liu, Jiancheng (College of Mathematics and Information Science Northwest Normal University) ;
  • Wei, Yan (College of Mathematics and Information Science Northwest Normal University)
  • Received : 2010.10.22
  • Published : 2012.03.31

Abstract

In this paper, we investigate the hypersurface M in a unit sphere with constant k-th mean curvature and two distinct principal curvatures, and characterize such a hypersurface.

Keywords

References

  1. H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1223-1229. https://doi.org/10.1090/S0002-9939-1994-1172943-2
  2. J. N. Barbosa, Hypersurfaces of $S^{n+1}$ with two distinct principal curvatures, Glasg. Math. J. 47 (2005), no. 1, 149-153. https://doi.org/10.1017/S0017089504002137
  3. E. Cartan, Familles de surfaces isoparametriques dans les espaces a courvure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177-191. https://doi.org/10.1007/BF02410700
  4. Q. M. Cheng, Hypersurfaces in a unit sphere $S^{n+1}$(1) with constant scalar curvature, J. London Math. Soc. (2) 64 (2001), no. 3, 755-768. https://doi.org/10.1112/S0024610701002587
  5. Q. M. Cheng and S. Ishikawa, A characterization of the Clifford torus, Proc. Amer. Math. Soc. 127 (1999), no. 3, 819-828. https://doi.org/10.1090/S0002-9939-99-05088-1
  6. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. https://doi.org/10.1007/BF01425237
  7. S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, In Functional Analysis and Related Fields, pp. 59-75, Springer, 1970.
  8. Jr. H. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 167-179.
  9. H. Li, Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), no. 2, 327-351. https://doi.org/10.1007/BF02559973
  10. T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970), 145-173. https://doi.org/10.2307/2373502
  11. G. Wei, Complete hypersurfaces with constant mean curvature in a unit sphere, Monatsh. Math. 149 (2006), no. 3, 251-258. https://doi.org/10.1007/s00605-005-0377-1
  12. G. Wei, Rigidity theorems of hypersurfaces with constant scalar curvature in a unit sphere, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 6, 1075-1082. https://doi.org/10.1007/s10114-005-0858-0
  13. G. Wei, Complete hypersurfaces with $H_{\kappa}$ = 0 in a unit sphere, Differential Geom. Appl. 25 (2007), no. 5, 500-505. https://doi.org/10.1016/j.difgeo.2007.06.001