Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

We deal with complete constant mean curvature spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality, which is a weaker hypothesis than to assume that the spacelike hypersurface has two distinct principal curvatures. In this setting, we obtain estimates for the norm of the traceless part of the second fundamental form and we show that these estimates are sharp in the sense that the hyperbolic cylinders realize them.