2+1维引力的宇宙大爆炸极限与Teichmüller空间的Thurston边界
Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space
AI总结：
本文研究2+1维广义相对论中亏格p>1时空的动力学，证明所有非平凡解曲线在大爆炸奇点极限处趋于Teichmüller空间的Thurston边界，即投影可测叶状结构空间。
摘要：
我们研究拓扑类型为$Σ_{p}\times \mathbb{R}$（$p>1$）的时空动力学解曲线在$2+1$维经典广义相对论区域中的渐近行为，其中$Σ_{p}$是亏格为$p$的闭黎曼曲面。规范固定动力学的位形空间被等同于$Σ_{p}$的Teichmüller空间（$\mathcal{T}Σ_{p}\approx \mathbb{R}^{6p-6}$）。利用某些调和映射的Dirichlet能量性质、由相关椭圆方程导出的估计以及紧黎曼曲面理论中的若干标准结果，我们证明每条非平凡解曲线都在宇宙大爆炸奇点极限处跑出Teichmüller空间的边缘，并趋近于投影可测叶状结构/叶状结构空间（$\mathcal{PML}$ $\mathcal{PMF}$），即Teichmüller空间的Thurston边界。
英文摘要：
We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type $Σ_{p}\times \mathbb{R}$, $p>1$, where $Σ_{p}$ 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 Teichmüller space ($\mathcal{T}Σ_{p}\approx \mathbb{R}^{6p-6}$) of $Σ_{p}$. 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 Teichmüller 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 Teichmüller space.
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