Previous |  Up |  Next


Lorentzian space; spacelike hypersurface; the first eigenvalue; Gauss map
In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.
[1] Cheung L. F., Leung P. F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236 (2001), 525–530. MR 1821303 | Zbl 0990.53029
[2] Cheng S. Y., Yau S. T.: Differential equations on Riemannian manifolds and geometric applications. Comm. Pure Appl. Math. 28 (1975), 333–354. MR 0385749
[3] Kobayashi S., Nomizu K.: Foundations of Differential Geometry. vol II, Interscience, New York, 1969. MR 0238225 | Zbl 0175.48504
[4] Mckean H. P.: An upper bound for the spectrum of $\Delta $ on a manifold of negative curvature. J. Differential Geometry 4 (1970), 359–366. MR 0266100
[5] Pacellibessa G., Montenegro J. F.: Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Glob. Anal. Geom. 24 (2003), 279–290. MR 1996771
[6] Schoen R., Yau S. T.: Lectures on differential geometry. Lecture Notes in Geom. Topo. 1 (1994). MR 1333601 | Zbl 0830.53001
Partner of
EuDML logo