Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910296351703040 |
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| author | Mondal, Puskar |
| author_facet | Mondal, Puskar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2002_03551 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space Mondal, Puskar General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry 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. |
| title | Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2002.03551 |