On Space-Like Hypersurfaces with Constant Mean Curvature of a Lorentz Space Form

We study complete spaoe-like hypersurfaces with constant mean curvature of a Lorentz space form.Introduction.Let $R_{1}^{m}$ be an m-dimensional Minkowski space and $S_{1}^{m}(c)$ (resp.$H_{1}^{m}(c)$ ) be an m-dimensional de Sitter space (resp.an anti-de Sitter space) of constant curvature $c$ and of index 1.The class of these indefinite Riemannian manifolds of index 1 is called a Lorentz space form, which is denoted by $M_{1}^{m}(c)$ .A hypersurface $M$ of a Lorentz space form is said to be space-like if the induced metric on $M$ from that of the ambient space is positive definite.Now, let $M$ be an entire space-like hypersurface with constant mean curvature of a Minkowski space $R_{1}^{n+1}$ .Then Cheng and Yau [5] estimated the norm of the second fundamental form of $M$ , by which the Bernstein-type problem in the Lorentz version is affirmatively solved.On the other hand, it is pointed out by Marsden and Tipler [10] that space-like hypersurfaces with constant mean curvature of arbitrary spacetimes get interested in relativity theory.An entire space-like hypersurface with constant mean curvature of a Minkowski space is investigated by Goddard [8] and Treibergs [15].As standard models of space-like hypersurfaces with constant mean curvature of a Lorentz space form $M''+1(c),n\geq 3$ , it is seen that there are four classes ofhypersurfaces $H^{k}(c_{1})\times S^{n-k}(c_{2})$ and $R^{n},$ $H^{k}(c_{1})\times R^{n-k}$ and $H^{k}(c_{1})\times H^{n-k}(c_{2})$ , where $k=0,1,$ $\cdots,$ $n$ , according as $c>0$ ,