Journal of the Mathematical Society of Japan

Complete linear Weingarten hypersurfaces immersed in the hyperbolic space

Henrique Fernandes DE LIMA

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Abstract

In this paper, we apply the Hopf's strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb H^{n+1}$.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 2 (2014), 415-423.

Dates
First available in Project Euclid: 23 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1398258177

Digital Object Identifier
doi:10.2969/jmsj/06620415

Mathematical Reviews number (MathSciNet)
MR3201819

Zentralblatt MATH identifier
1303.53080

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
hyperbolic space linear Weingarten hypersurfaces totally umbilical hypersurfaces hyperbolic cylinders

Citation

DE LIMA, Henrique Fernandes. Complete linear Weingarten hypersurfaces immersed in the hyperbolic space. J. Math. Soc. Japan 66 (2014), no. 2, 415--423. doi:10.2969/jmsj/06620415. https://projecteuclid.org/euclid.jmsj/1398258177


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References

  • N. Abe, N. Koike and S. Yamaguchi, Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form, Yokohama Math. J., 35 (1987), 123–136.
  • H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc., 120 (1994), 1223–1229.
  • É. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177–191.
  • S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195–204.
  • H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann., 305 (1996), 665–672.
  • H. Li, Global rigidity theorems of hypersurfaces, Ark. Mat., 35 (1997), 327–351.
  • H. Li, Y. J. Suh and G. Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc., 46 (2009), 321–329.
  • M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math., 96 (1974), 207–213.
  • H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19 (1967), 205–214.
  • P. J. Ryan, Hypersurfaces with parallel Ricci tensor, Osaka J. Math., 8 (1971), 251–259.
  • S. Shu, Complete hypersurfaces with constant scalar curvature in a hyperbolic space, Balkan J. Geom. Appl., 12 (2007), 107–115.
  • S. Shu, Linear Weingarten hypersurfaces in a real space form, Glasg. Math. J., 52 (2010), 635–648.
  • S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201–228.