Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space.

This article investigates the asymptotic behavior of vacuum solutions to the Einstein equations in "2 + 1" dimensional classical general relativity on spacetimes of the form Σₚ × ℝ, where Σₚ is a closed Riemann surface of genus p > 1. The configuration space of the gauge-fixed dynamics is identified with the Teichmüller space TΣₚ, and the study focuses on the behavior of solution curves near the big-bang singularity (τ → −∞) in the constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. The main results prove Moncrief's conjecture that every non-trivial solution curve leaves every compact subset of TΣₚ at the big bang and converges to the Thurston boundary of TΣₚ, which is the space of projective measured laminations or foliations (PML, PMF). This characterization links the degeneration of conformal geometry at the singularity to a well-studied compactification of Teichmüller space, providing a geometric classification of big-bang singularities in this setting. The article also establishes precise asymptotic relations between hyperbolic lengths of closed geodesics and transverse measures associated with the holomorphic quadratic differentials arising from the transverse-traceless part of the second fundamental form, showing that the transverse measure with respect to the vertical foliation scales proportionally to hyperbolic length, while the horizontal measure collapses to zero near the singularity.