We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type $\Sigma_{ p}\times \mathbb{ R}$, $p>1$, where $\Sigma_{ p}$ 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 Teichm\"uller space ($\mathcal{ T}\Sigma_{ p}\approx \mathbb{ R}^{ 6p-6}$) of $\Sigma_{ p}$. 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 Teichm\"uller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations ($\mathcal{ PML}$ $\mathcal{ PMF}$), the Thurston boundary of the Teichm\"uller space.
Categories: gr-qc math-ph math.DG math.MP