DOI QR코드

DOI QR Code

ON THE STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER ORDER MEAN CURVATURE IN A DE SITTER SPACE

  • Zhang, Shicheng (School of Mathematics and Statistics Jiangsu Normal University)
  • Received : 2013.08.31
  • Published : 2014.09.30

Abstract

The closed spacelike hypersurfaces with higher order mean curvature is discussed in a de Sitter space. The hypersurface is proved stable if and only if it is totally umbilical.

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. H. Alencar, M. do Carmo, and A. G. Colares, Stable hypersurfaces with constant scalar curvature, Math. Z. 213 (1993), no. 1, 117-131. https://doi.org/10.1007/BF03025712
  2. J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339-353. https://doi.org/10.1007/BF01215045
  3. J. L. Barbosa, M. do Carmo, and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123-138. https://doi.org/10.1007/BF01161634
  4. J. L. Barbosa and A. G. Colares, Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277-297. https://doi.org/10.1023/A:1006514303828
  5. J. L. M. Barbosa and V. Oliker, Stable spacelike hypersurfaces with constant mean curvature in Lorentz spaces, Geometry and global analysis (Sendai, 1993), 161-164, Tohoku Univ., Sendai, 1993.
  6. A. Brasil Jr. and A. G. Colares. Stability of spacelike hypersurfaces with constant r-mean curvature in de Sitter space, Proceedings of the XII Fall Workshop on Geometry and Physics, 139-145, Publ. R. Soc. Mat. Esp., 7, R. Soc. Mat. Esp., Madrid, 2004.
  7. F. Camargo, A. Caminha, M. da Silva, and H. de Lima, On the r-stability of spacelike hypersurfaces, J. Geom. Phys. 60 (2010), no. 10, 1402-1410. https://doi.org/10.1016/j.geomphys.2010.05.004
  8. Q. M. Cheng, Complete space-like hypersurfaces of a de Sitter space with r = aH, Mem. Fac. Sci. Kyushu Univ. Ser. A 44 (1990), no. 2, 67-77.
  9. H. Cheng and X. Wang, Stability and eigenvalue estimates of linear weingarten hyper-surfaces in a sphere, J. Math. Anal. Appl. 397 (2013), no. 2, 658-670. https://doi.org/10.1016/j.jmaa.2012.08.003
  10. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. https://doi.org/10.1007/BF01425237
  11. X. Liu and J. Deng, Stable space-like hypersurfaces in the de Sitter space, Arc. Math. 40 (2004), no. 2, 111-117.
  12. M. A. Velasquez, A. F. de Sousa, and H. F. de Lima, On the stability of hypersurfaces in space forms, J. Math. Anal. Appl. 406 (2013), no. 1, 134-146. https://doi.org/10.1016/j.jmaa.2013.04.045
  13. H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211-239.