Rocky Mountain Journal of Mathematics

Characterizations of linear Weingarten spacelike hypersurfaces in Lorentz space forms

Cícero P. Aquino, Henrique F. de Lima, and Marco Antonio L. Velásquez

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In this article, we deal with complete linear Weingarten spacelike hypersurfaces (that is, complete spacelike hypersurfaces whose mean and scalar curvatures are linearly related) immersed in a Lorentz space form. By assuming that the mean curvature attains its maximum and supposing appropriated restrictions on the norm of the traceless part of the second fundamental form, we apply Hopf's strong maximum principle in order to prove that such a spacelike hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space.

Article information

Rocky Mountain J. Math., Volume 45, Number 1 (2015), 13-27.

First available in Project Euclid: 7 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Lorentz space forms linear Winegarten spacelike hypersurfaces second fundamental form totally umbilical hypersurfaces hyperbolic cylinders


Aquino, Cícero P.; Lima, Henrique F. de; Velásquez, Marco Antonio L. Characterizations of linear Weingarten spacelike hypersurfaces in Lorentz space forms. Rocky Mountain J. Math. 45 (2015), no. 1, 13--27. doi:10.1216/RMJ-2015-45-1-13.

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  • 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.
  • K. Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13–19.
  • H. Alencar \and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), 1223–1229.
  • A. Brasil, Jr., A.G. Colares and O. Palmas, A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space, J. Geom. Phys. 37 (2001), 237–250.
  • E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Sympos. Pure Math. 15 (1970), 223–230.
  • F.E.C. Camargo, R.M.B. Chaves and L.A.M. Sousa, Jr., Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in de Sitter space, Diff. Geom. Appl. 26 (2008), 592–599.
  • A. Caminha, A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds, Diff. Geom. Appl. 24 (2006), 652–659.
  • 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, Maximal spacelike hypersurfaces in the Lorentz-Minkowski space, Ann. Math. 104 (1976), 407–419.
  • ––––, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.
  • A.J. Goddard, Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Cambr. Phil. Soc. 82 (1977), 489–495.
  • Z.H. Hou and D. Yang, Linear Weingarten spacelike hypersurfaces in de Sitter space, Bull. Belgian Math. Soc. Simon Stevin 17 (2010), 769–780.
  • Z.-J. Hu, M. Scherfner and S.-J. Zhai, On spacelike hypersurfaces with constant scalar curvature in the de Sitter space, Diff. Geom. Appl. 25 (2007), 594–611.
  • H. Li, Y.J. Suh and G. Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Kor. Math. Soc. 46 (2009), 321–329.
  • S. Montiel, An integral inequality for compact spacelike hypersurfaces in the de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909–917.
  • S. Nishikawa, On spacelike hypersurfaces in a Lorentzian manifold, Nagoya Math. J. 95 (1984), 117–124.
  • M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213.