Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Maximal Hypersurfaces of (m + 2)-Dimensional Lorentzian Space Forms

  • Dursun, Ugur (Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics)
  • Received : 2006.11.08
  • Published : 2008.03.31
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Abstract

We determine maximal space-like hypersurfaces which are the images of subbundles of the normal bundle of m-dimensional totally geodesic space-like submanifolds of an (m + 2)-dimensional Lorentzian space form $\tilde{M}_1^{m+2}$(c) under the normal exponential map. Then we construct examples of maximal space-like hypersurfaces of $\tilde{M}_1^{m+2}$(c).

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References

  1. K. Akutagawa, On space-like hypersurfaces with constant mean curvature in the de Sitter space, Math. Z., 196(1)(1987), 13-19. https://doi.org/10.1007/BF01179263
  2. L. J. Alias, R. M. B. Chaves, and P. Mira, Bjorling problem for maximal surfaces in Lorentz-Minkowski space, Math. Proc. Camb. Phil. Soc., 134(2003), 289-326. https://doi.org/10.1017/S0305004102006503
  3. J. O. Baek, Q. M. Cheng, and Y. J. Suh, Complete space-like hypersurfaces in locally symmetric Lorentz space, J. Geometry and Physics, 49(2004), 231-247. https://doi.org/10.1016/S0393-0440(03)00090-1
  4. E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math., 15(1970), 223-230.
  5. S. Y. Chehg and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104(1976), 407-419. https://doi.org/10.2307/1970963
  6. U. Dursun, On Minimal Hypersurfaces of Lorentzian Space Forms, Turk Matematik Dernegi XI. Ulusal Matematik Sempozyumu, Suleyman Demirel Universitesi, Isparta, Turkey, pp. 105-114, 1998.
  7. U. Dursun, On minimal and Chen immersions in space forms, J. Geom., 66(1999), 104-111. https://doi.org/10.1007/BF01225674
  8. M. Kimura, Minimal hypersurfaces foliated by geodesics of 4-dimensional space forms, Tokyo J. Math., 16(1993), 241-260. https://doi.org/10.3836/tjm/1270128995
  9. O. Kobayashi, Maximal surfaces in 3-dimensional Minkowski space $L^{3}$, Tokyo J. Math., 6(1983), 297-309. https://doi.org/10.3836/tjm/1270213872
  10. H. Li, On complete maximal space-like hypersurfaces in a Lorentz manifold, Soochow J. Math., 23(1)(1997), 79-89.
  11. J. J. Lopez, R. Lopez, and R. Souam, Maximal surfaces of Riemann type in Lorentz-Minkowski space $L^{3}$, Michigan Math. J., 47(3)(2000), 469-497. https://doi.org/10.1307/mmj/1030132590
  12. J. E. Marsden and F. J. Tipler,Maximal hypersurfaces and foliations of constant mean curvature in relativity, Phys. Rep., 66(1980), 109-139. https://doi.org/10.1016/0370-1573(80)90154-4
  13. P. Mira and J. A. PastorR, Helicoidal maximal surfaces Lorentz-Minkowski space, Monatsh. Math., 140(2003), 315-334. https://doi.org/10.1007/s00605-003-0111-9
  14. S. Montiel, An integral inequality for compact space-like hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J., 37(4)(1988), 909-917. https://doi.org/10.1512/iumj.1988.37.37045
  15. H. Nishikawa, On maximal space-like hypersurfaces in a Lorentzian manifold, Nagoya Math. J., 95(1984), 117-124.
  16. B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.
  17. S. H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math., 32(2002), 1019-1044. https://doi.org/10.1216/rmjm/1034968429
  18. 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(1)(2002), 47-70.