Maximal annuli with parallel planar boundaries in the 3-dimensional Lorentz-Minkowski space

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.