Complete maximal spacelike hypersurfaces of \(H_ 1^ 4(c)\)
The author proves the following theorems: Theorem 1. Let \(M\) be a maximal complete 3-dimensional spacelike hypersurface of \(H^ 4_ 1(c)\) [4-dimensional anti-de Sitter space of constant curvature] with non-zero constant Gauss-Kronecker curvature. Then \(M\) is the hyperbolic cylinder \(H^ 1(c_ 1)\times H^ 2(c_ 2)\). Theorem 2. Let \(M\) be a 3-dimensional complete maximal spacelike hypersurface of \(H^ 4_ 1(c)\) with constant scalar curvature and \(\inf k^ 2>0\), where \(k\) is the Gauss-Kronecker curvature. Then \(M\) is the cylinder \(H^ 1(c_ 1)\times H^ 2(c_ 2)\). Theorem 3. There are no complete maximal spacelike hypersurfaces in \(H^ 4_ 1(c)\) with constant scalar and \(\sup k^ 2<{S^ 3\over S^ 4}\), where \(k\) is the Gauss-Kronecker curvature and \(S\) is the squared norm of the second fundamental form.
scientific article; zbMATH DE number 599655