Big-bang limit of $2+1$ gravity and Thurston boundary of Teichm\"uller space

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type ${\mathrm{\Sigma }}_p\times \mathbb{R}$, $p>1$, where ${\mathrm{\Sigma }}_p$ 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 ($\mathcal{T}{\mathrm{\Sigma }}_p\approx {\mathbb{R}}^{6p-6}$) of ${\mathrm{\Sigma }}_p$. 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 ($\mathcal{P}\mathcal{M}\mathcal{L}$ $\mathcal{P}\mathcal{M}\mathcal{F}$), the Thurston boundary of the Teichm"uller space.