Thurston boundary of the Teichmüller space is the space of big bang singularities of 2+1 gravity
Year: 2020
Type: preprint
Abstract: The ADM formalism of vacuum general relativity in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge is used to study the asymptotic behaviour of the solution curves of the dynamics of spacetimes of the topological type $\Sigma_{g}\times \mathbb{R}$, $g>1$, where $\Sigma_{g}$ is a closed Riemann surface of genus $g$. The configuration space of the gauge fixed dynamics is identified with the Teichmuller space ($\mathcal{T}\Sigma_{g}\approx \mathbb{R}^{6g-6}$) of $\Sigma_{g}$. 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 surface theory, we show that every non-trivial solution curve runs off the edge of the Teichmuller 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 Teichmuller space. This result which identifies the complete solution space of the Einstein equations on flat spacetimes of the type $\Sigma_{g}\times \mathbb{R}$, also yields yet another way to compactify the Teichmuller space.
Source: arXiv (Cornell University)
Author: Puskar Mondal
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3005251287