On spacelike hypersurfaces with constant mean curvature in the de Sitter space
In this paper, we shall prove the following Theorems 1 and 2 and show that our estimate (i) in Theorem 1 is optimal. Theorem 1. Let \(N=(N^{n+1}(c),g)\) be an \((n+1)\)-dimensional Lorentzian manifold of positive constant curvature c. Let M be a complete spacelike hypersurface with constant mean curvature H in N. Suppose that (i) \(| H| \leq c^{1/2}\) when \(n=2,\) (ii) \(| H| <2[(n-1)c]^{1/2}/n\) when \(n\geq 3.\) Then M is totally umbilical. Theorem 2. Let M be a compact spacelike surface with constant mean curvature in the 3-dimensional de Sitter space. Then M is totally umbilical. The fact that our estimate (i) in Theorem 1 is optimal is a consequence of the following. We can construct, for any constant H satisfying \(| H| >c^{1/2}\), complete noncompact spacelike surfaces embedded in the 3-dimensional de Sitter space of positive constant curvature c, which have constant mean curvature H and which are not totally umbilical.
scientific article; zbMATH DE number 3988183