Hypersurfaces of a Lorentz space form
In this paper, we completely classify complete spacelike hypersurfaces in an anti-de Sitter space \(H^ 4_ 1(-c)\) with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature and give some characterizations on the complete spacelike hypersurfaces in a de Sitter space \(S^{n + 1}_ 1(c)\) with \(h^ 2 = 4(n - 1)c\). Namely we prove the following: Let \(M\) be a 3-dimensional complete spacelike hypersurface of \(H^ 4_ 1 (-c)\) with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature. Then \(M\) is totally umbilic or the hyperbolic cylinder \(H^ 1(c_ 1) \times H^ 2(c_ 2)\). Theorem 4.2. Let \(M\) be an \(n\)-dimensional complete and noncompact spacelike hypersurface of a de Sitter space \(S^{n + 1}_ 1 (c)\) with \(h^ 2 = 4(n - 1)c\). Then \(M\) is the hyperbolic cylinder \(H^ 1(c_ 1) \times S^{n - 1}(c_ 2)\) or \(M\) has unbounded volume and positive Ricci curvature, and the squared norm \(S\) of the second fundamental form is a subharmonic function with \(\sup S = nc\) and \(S < nc\).
constant quasi-Gauss-Kronecker curvature
scientific article; zbMATH DE number 687046