We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type Σp×R, p > 1, where Σp is a closed Riemann surface of genus p, in the regime of 2 + 1 dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space (TΣpR6p6) of Σp. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations (PMLPMF), the Thurston boundary of the Teichmüller space.

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