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.
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Belraout et al. (2024) studied this question.
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