First eigenvalue of Schrodinger operator of space-like hypersurfaces

We introduce two Schrodinger operators of compact space-like hypersurfaces in a de Sitter space. If the hypersurfaces have constant mean curvature or constant scalar curvature, we obtain some spectral characterisations of totally umbilical space-like hypersurfaces by the first eigenvalue of the Schrodinger operators. References H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Pro. of the Amer. Math. Soc., 120(4) (1994), 1223--1229. http://www.jstor.org/pss/2160241 H. Alencar, M. do Carmo and A. G. Colares, Stable hypersurfaces with constant scalar curvature, Math. Z., 213 (1993), 117--131. doi:10.1007/BF03025712 L. J. Alias, A. Barros, and A. Jr. Brasil, A spectral characterization of $H(r)$-torus by the first stability eigenvalue, Proc. Amer. Math. Soc., 133 (2005), 875--884. doi:10.1090/S0002-9939-04-07559-8 J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z., 197 (1988), 123--138. doi:10.1007/BF01161634 J. L. Barbosa and V. Oliker, Stable space-like hypersurfaces with constant mean curvature in Lorentz space, Geometry and global Analysis, Tohoku Univ., Sendai, (1993), 161--164. J. L. Barbosa and V. Oliker, Space-like hypersurfaces with constant mean curvature in Lorentz space, Mat. Contemp., 4 (1993), 27--44. A. Brasil. Jr, A. G. Colares and O. Palmas, Complete space-like hypersurfaces with constant mean curvature in the de Sitter space: A gap Theorem. Illinois Journal of Mathematics, 47 (2003), 847--866. http://projecteuclid.org/euclid.ijm/1258138197 A. A. Barros, A. C. Brasil Jr. and L. A. M. Sousa Jr., A new characterization of submanifolds with parallel mean curvature vector in $S^{n+p}$, Kodai Math. J., 27 (2004), 45--56. doi:10.2996/kmj/1085143788 Q.-M. Cheng, First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature, Proc. Amer. Math. Soc., 136 (2008), 3309--3318. doi:10.1090/S0002-9939-08-09304-0 Q.-M. Cheng and S. Ishikawa, Space-like hypersurfaces in de Sitter spaces with constant scalar curvature, Manuscripta Math., 95 (1998), 499--505. doi:10.1007/BF02678045 S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature,Math. Ann., 225 (1977), 195--204. doi:10.1007/BF01425237 X. Liu and J. Deng, Stable space-like hypersurfaces in the de Sitter space, Archivum Math. (Brno), 40(2004), 111--117. http://dml.cz/dmlcz/107895 X. Liu, Complete space-like hypersurfaces with constant scalar curvature, Manuscripta Math., 105 (2001), 367--377. doi:10.1007/s002290100187 M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math., 96 (1974), 207--213. http://www.jstor.org/pss/2373587 S. C. Shu and S. Y. Liu, Complete space-like submanifolds with constant scalar curvature in a de Sitter space, Balkan J. of Geomety and its Appli., 9 (2004), 82--91. C. Wu, New characterizations of the Clifford tori and the Veronese surface, Arch. Math., 61 (1993), 277--284. doi:10.1007/BF0119872