Asymptotic behavior of Moncrief Lines in constant curvature space-times

TL;DR

This paper investigates the long-term behavior of Moncrief lines in certain 2+1-dimensional space-times with constant curvature, revealing convergence properties related to the Thurston boundary in specific lamination cases.

Contribution

It establishes the asymptotic convergence of Moncrief lines to a unique point in the Thurston boundary for space-times with maximal uniquely ergodic or simplicial laminations.

Findings

Moncrief lines converge to a point in the Thurston boundary as time approaches zero.

Convergence occurs when the associated lamination is either maximal uniquely ergodic or simplicial.

The results connect geometric structures in space-time to properties of Teichmüller space.

Abstract

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.