Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CLOSED CONVEX SPACELIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACES

  • Sun, Zhongyang (School of Mathematics Science Huaibei Normal University)
  • Received : 2016.08.19
  • Accepted : 2017.02.24
  • Published : 2017.11.30
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Abstract

In 1997, H. Li [12] proposed a conjecture: if $M^n(n{\geqslant}3)$ is a complete spacelike hypersurface in de Sitter space $S^{n+1}_1(1)$ with constant normalized scalar curvature R satisfying $\frac{n-2}{n}{\leqslant}R{\leqslant}1$, then is $M^n$ totally umbilical? Recently, F. E. C. Camargo et al. ([5]) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^n$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^n$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ satisfies $K(M^n)$ > 0, then $M^n$ is totally umbilical, i.e., Theorem 2.

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References

  1. K. Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), no. 1, 12-19.
  2. 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
  3. J. O. Baek, Q. M. Cheng, and Y. J. Suh, Complete space-like hypersurface in locally symmetric Lorentz spaces, J. Geom. Phys. 49 (2004), no. 2, 231-247. https://doi.org/10.1016/S0393-0440(03)00090-1
  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), no. 3, 237-250. https://doi.org/10.1016/S0393-0440(00)00046-2
  5. 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, Differential Geom. Appl. 26 (2008), no. 6, 592-599. https://doi.org/10.1016/j.difgeo.2008.04.020
  6. Q. M. Cheng and S. Ishikawa, Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (1998), no. 4, 499-505. https://doi.org/10.1007/BF02678045
  7. S. S. Chern, M. do Carmo, and S. Koboyashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) pp. 59-75 Springer, New York, 1970.
  8. S. M. Choi, S. M. Lyu, and Y. J. Suh, Complete space-like hypersurfaces in a Lorentz manifold, Math. J. Toyama Univ. 22 (1999), 53-76.
  9. A. J. Goddard, Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 3, 489-495. https://doi.org/10.1017/S0305004100054153
  10. Z. Hu, M. Scherfner, and S. Zhai, On spacelike hypersurfaces with constant scalar cur- vature in the de Sitter space, Differential Geom. Appl. 25 (2007), no. 6, 594-611. https://doi.org/10.1016/j.difgeo.2007.06.008
  11. U. H. Ki, H. J. Kim, and H. Nakagawa, On space-like hypersurfaces with constant mean curvature of a Lorentz space form, Tokyo J. Math. 14 (1991), no. 1, 205-216. https://doi.org/10.3836/tjm/1270130500
  12. H. Li, Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), no. 2, 327-351. https://doi.org/10.1007/BF02559973
  13. J. C. Liu and Z. Y. Sun, On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces, J. Math. Anal. Appl. 364 (2010), no. 1, 195-203. https://doi.org/10.1016/j.jmaa.2009.10.029
  14. S. Montiel, An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), no. 4, 909-917. https://doi.org/10.1512/iumj.1988.37.37045
  15. V. Oliker, A priori estimates of the principal curvatures of spacelike hypersurfaces in the de Sitter space with applications to hypersurfaces in hyperbolic space, Amer. J. Math. 114 (1992), no. 3, 605-626. https://doi.org/10.2307/2374771
  16. J. Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (1987), no. 2, 349-359. https://doi.org/10.1512/iumj.1987.36.36020
  17. Y. J. Suh, Y. S. Choi, and H. Y. Yang, On space-like hypersurfaces with constant mean curvature in a Lorentz manifold, Houston J. Math. 28 (2002), no. 1, 47-70.
  18. Z. Y. Sun, Partial generalizations of some conjectures in Lorentzian manifolds, Kodai Math. J. 36 (2013), no. 3, 508-526. https://doi.org/10.2996/kmj/1383660696
  19. Q. L. Wang and C. Y. Xia, Rigidity theorems for closed hypersurfaces in a unit sphere, J. Geom. Phys. 55 (2005), no. 3, 227-240. https://doi.org/10.1016/j.geomphys.2004.12.005