Characterization of hyperbolic cylinders in a Lorentzian space form

We give a characterization of the $n$-dimensional ($n\geq3$) hyperbolic cylinders in a Lorentzian space form. We show that the hyperbolic cylinders are the only complete space-like hypersurfaces in an $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$ with non-zero constant mean curvature $H$ whose two distinct principal curvatures $\lambda$ and $\mu$ satisfy $\inf(\lambda-\mu)^2>0$ for $c\leq 0$ or $\inf(\lambda-\mu)^2>0$, $H^2\geq c$, for $c> 0$, where $\lambda$ is of multiplicity $n-1$ and $\mu$ of multiplicity $1$ and $\lambda<\mu$.