New characterizations for hyperbolic cylinders in anti-de Sitter spaces

In this paper we study complete maximal spacelike hypersurfaces in anti-de Sitter space H 1 n + 1 with either constant scalar curvature or constant non-zero Gauss–Kronecker curvature. We characterize the hyperbolic cylinders H m ( c 1 ) × H n − m ( c 2 ) , 1 ≤ m ≤ n − 1 , as the only such hypersurfaces with ( n − 1 ) principal curvatures with the same sign everywhere. In particular we prove that a complete maximal spacelike hypersurface in H 1 5 with negative constant Gauss–Kronecker curvature is isometric to H 1 ( c 1 ) × H 3 ( c 2 ) .