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A hypersurface in a Lorentzian manifold is said to be space-likeif the induced metric on the hypersurface is positive definite. On a space-like hypersurface, the firstfundamental form, the second fundamental form and the mean curvature are defined in the same way as those on a hypersurface in a Riemannian manifold [§1]. It has been proved by Bernstein and others that the only entire minimal hypersurface in a Euclidean space Rn+1 is a linear hyperplane for n^7, but there are other examples for n>7. So, Calabi proposed to study a Lorentzian analoge, called the Bernstein type problem, in Minkowki space i??+1, and this was solved by Cheng and Yau [4] for every n. More precisely, a space-like hypersurface in a Lorentzian manifold is said to be maximal, if the mean curvature is zero. The Bernstein type problem has led to the conclusion that the only entire maximal space-like hypersurface in i? +1 is a linear hyperplane. In order to prove this, Cheng and Yau [4] established the following result: (*) // an entire space-like hypersurface M in R"+l has a constant mean curvature H, then the induced Lorentzian metric on M is a complete Riemannian metric and the length of second fundamental form of M is bounded from above by n\H\.

On complete space-like surfaces with constant mean curvature in a Lorentzian 3-space form