  We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight
lines or cone points in a pair of parallel spacelike planes are part of either
a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the
same boundary condition, the same conclusion holds even when the maximal annuli
have a planar end. Moreover, we extend Shiffman's convexity result to maximal
annuli but by using Perron's method we construct a maximal annulus with a
planar end where Shiffman type result fails.
