## Characterizations of linear Weingarten spacelike hypersurfaces in Lorentz space forms.(English)Zbl 1314.53104

Summary: In this article, we deal with complete linear Weingarten space-like hypersurfaces (that is, complete space-like 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 appropriate 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 space-like hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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### References:

 [1] 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. · Zbl 0645.53010 [2] K. Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space , Math. Z. 196 (1987), 13-19. · Zbl 0611.53047 [3] H. Alencar \and M. do Carmo, Hypersurfaces with constant mean curvature in spheres , Proc. Amer. Math. Soc. 120 (1994), 1223-1229. · Zbl 0802.53017 [4] 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. · Zbl 1027.53065 [5] E. Calabi, Examples of Bernstein problems for some nonlinear equations , Proc. Sympos. Pure Math. 15 (1970), 223-230. · Zbl 0211.12801 [6] 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. · Zbl 1160.53361 [7] A. Caminha, A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds , Diff. Geom. Appl. 24 (2006), 652-659. · Zbl 1107.53040 [8] É. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante , Ann. Mat. Pura Appl. 17 (1938), 177-191. · Zbl 0020.06505 [9] S.Y. Cheng and S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski space , Ann. Math. 104 (1976), 407-419. · Zbl 0352.53021 [10] —-, Hypersurfaces with constant scalar curvature , Math. Ann. 225 (1977), 195-204. · Zbl 0349.53041 [11] A.J. Goddard, Some remarks on the existence of spacelike hypersurfaces of constant mean curvature , Math. Proc. Cambr. Phil. Soc. 82 (1977), 489-495. · Zbl 0386.53042 [12] Z.H. Hou and D. Yang, Linear Weingarten spacelike hypersurfaces in de Sitter space , Bull. Belgian Math. Soc. Simon Stevin 17 (2010), 769-780. · Zbl 1210.53060 [13] 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. · Zbl 1134.53034 [14] H. Li, Y.J. Suh and G. Wei, Linear Weingarten hypersurfaces in a unit sphere , Bull. Kor. Math. Soc. 46 (2009), 321-329. · Zbl 1165.53361 [15] 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. · Zbl 0677.53067 [16] S. Nishikawa, On spacelike hypersurfaces in a Lorentzian manifold , Nagoya Math. J. 95 (1984), 117-124. · Zbl 0544.53050 [17] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor , Amer. J. Math. 96 (1974), 207-213. · Zbl 0302.53028
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