On the first eigenvalue of spacelike hypersurfaces in Lorentzian space

Wu Bing-Ye

Department of Mathematics, Minjiang University, Fuzhou, Fujian, 350108, P. R. China

E-mail.   bingyewu@yahoo.com.cn

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.

AMSclassification. Primary 53C42; Secondary 53B30.

Keywords.   Lorentzian space, spacelike hypersurface, the first eigenvalue, Gauss map.