New characterizations for hyperbolic cylinders in anti-de Sitter spaces

In this paper we study complete maximal spacelike hypersurfaces in anti-de Sitter space H1n+1 with either constant scalar curvature or constant non-zero Gauss–Kronecker curvature. We characterize the hyperbolic cylinders Hm(c1)×Hn−m(c2),1≤m≤n−1Hm(c1)×Hn−m(c2),1≤m≤n−1, as the only such hypersurfaces with (n−1n−1) principal curvatures with the same sign everywhere. In particular we prove that a complete maximal spacelike hypersurface in H15 with negative constant Gauss–Kronecker curvature is isometric to H1(c1)×H3(c2)H1(c1)×H3(c2).