Complete space-like hypersurfaces with constant mean curvature in de Sitter spaces (Q2732400)

Let \(M\) be a complete space-like hypersurface with constant mean curvature in a de Sitter space \(S^{n+1}_1(c)\). It is proved that if the square \(S\) of the norm of the second fundamental form of \(M\) satisfies \(S<2c\sqrt{n-1}\), then \(M\) is totally umbilical and is isometric to the sphere \(S^n(c-\frac 1n S)\) of constant curvature \(c-\frac 1n S\). If \(S=2c\sqrt{n-1}\), then \(n-2\) and \(M\) is totally umbilical and flat.