On complete space-like surfaces with constant mean curvature in a Lorentzian 3-space form (Q808451)

Let M be a complete space-like surface with constant mean curvature H in a Lorentzian 3-space form \(M^ 3_ 1(c)\). Let \(\alpha\) be the second fundamental form of M. The main result of the present paper is the following: If c is non-positive, then \(| \alpha |^ 2\leq 4H^ 2-2c\), and if c is positive, then M is totally umbilic or \(| \alpha |^ 2\leq 4H^ 2-2c\). Applying this result, the author shows: The hyperbolic cylinder is the only complete space-like surface in \(M^ 3_ 1(c)\) with non-zero constant mean curvature whose principal curvatures \(\lambda\) and \(\mu\) satisfy \(\inf (\lambda -\mu)^ 2>0\).