Journal of the
Korean Mathematical Society JKMS

In this paper, we study complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space ${\bf H}^{4}_1(-1) $. It is proved that complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space ${\bf H}^{4}_1(-1) $ are isometric to the hyperbolic cylinder ${\bf H}^{2}(c_1)\times {\bf H}^{1}(c_2)$ with $S=3$ or they satisfy $S\leq 2$, where $S$ denotes the squared norm of the second fundamental form.

Keywords: maximal space-like hypersurface, zero Gauss-Kronecker curvature, anti-de Sitter space

MSC numbers: Primary 53C42; Secondary 53C20