Asymptotic behavior of Moncrief Lines in constant curvature space-times

We study the asymptotic behavior of Moncrief lines on $2+1$ maximal globally hyperbolic spatially compact space-time $M$ of non-negative constant curvature. We show that when the unique geodesic lamination associated with $M$ is either maximal uniquely ergodic or simplicial, the Moncrief line converges, as time goes to zero, to a unique point in the Thurston boundary of the Teichm\"uller space.