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Hypersurfaces of the hyperbolic space with constant scalar curvature

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Abstract

We classify hypersurfaces of the hyperbolic space ℍn+1(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mn is a complete hypersurfaces with constant scalar curvature n(n − 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n− 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.

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References

  1. E. Cartan: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45, 335–367 (1939).

    Article  MathSciNet  Google Scholar 

  2. Q.M. Cheng: Hypersurfaces in a unit sphere Sn+1(l) with constant scalar curvature, J. London Math. Soc. 64, 755–768 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  3. Q.M. Cheng: Complete hypersurfaces in a Euclidean space ℝn+1 with constant scalar curvature, Indiana Univ. Math. J. 51, 53–68 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. S.Y. Cheng and S.T. Yau: Hypersurfaces with constant scalar curvature, Math. Ann. 225, 195–204 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Do Carmo and M. Dajczer: Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277, 685–709 (1983).

    MathSciNet  MATH  Google Scholar 

  6. M. Do Carmo, M. Dajczer and F. Mercuri: Compact conformally flat hypersurfaces, Trans. Amer. Math. Soc. 288, 189–203 (1985).

    MathSciNet  MATH  Google Scholar 

  7. Z.J. Hu: Isometric immersions of the hyperbolic space Hn(−1) into Hn+1(−1), Coll. Math. 79, 17–23 (1999).

    MATH  Google Scholar 

  8. Z.J. Hu and S.J. Zhai: On spacelike hypersurfaces with constant scalar curvature in the de Sitter space, Preprint (2004).

  9. M.L. Leite: Rotational hypersurfaces of space forms with constant scalar curvature, Manuscripta Math. 67, 285–304 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Li: Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305, 665–672 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  11. X. Liu and W. Su: Hypersurfaces with constant scalar curvature in a hyperbolic space form, Balkan J. Geom. Appl. 7, 121–132 (2001).

    MathSciNet  Google Scholar 

  12. T. Otsuki: Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92, 145–173 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  13. U. Pinkall: Compact conformally flat hypersurfaces, in Conformai Geometry (Eds. R.S. Kulkarni and U. Pinkall), Aspect of Math., Vol. E12, Max-Planck Inst. Math. 217–236 (1988).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, PRC

    Zejun Hu & Shujie Zhai

Authors
  1. Zejun Hu
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  2. Shujie Zhai
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Correspondence to Zejun Hu.

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Partially supported by DAAD, TU Berlin and SRF for ROCS, SEM.

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Hu, Z., Zhai, S. Hypersurfaces of the hyperbolic space with constant scalar curvature. Results. Math. 48, 65–88 (2005). https://doi.org/10.1007/BF03322897

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  • DOI: https://doi.org/10.1007/BF03322897

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