Almost strict domination and anti-de Sitter 3-manifolds
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914686497193984 |
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| author | Sagman, Nathaniel |
| author_facet | Sagman, Nathaniel |
| contents | We define a condition called almost strict domination for pairs of representations $ρ_1:π_1(S_{g,n})\to \textrm{PSL}(2,\mathbb{R})$, $ρ_2:π_1(S_{g,n})\to G$, where $G$ is the isometry group of a Hadamard manifold $(X,ν)$, and prove it holds if and only if one can find a $(ρ_1,ρ_2)$-equivariant spacelike maximal surface in a certain pseudo-Riemannian manifold, unique up to fixing some parameters. The proof amounts to setting up and solving an interesting variational problem that involves infinite energy harmonic maps. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space.
When $(X,ν)=(\mathbb{H},σ)$, an almost strictly dominating pair is equivalent to the data of an anti-de Sitter 3-manifold with specific properties. The results on maximal surfaces provide a parametrization of the deformation space of such $3$-manifolds as a union of components in a $\textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ relative representation variety. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_12886 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Almost strict domination and anti-de Sitter 3-manifolds Sagman, Nathaniel Differential Geometry Geometric Topology We define a condition called almost strict domination for pairs of representations $ρ_1:π_1(S_{g,n})\to \textrm{PSL}(2,\mathbb{R})$, $ρ_2:π_1(S_{g,n})\to G$, where $G$ is the isometry group of a Hadamard manifold $(X,ν)$, and prove it holds if and only if one can find a $(ρ_1,ρ_2)$-equivariant spacelike maximal surface in a certain pseudo-Riemannian manifold, unique up to fixing some parameters. The proof amounts to setting up and solving an interesting variational problem that involves infinite energy harmonic maps. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space. When $(X,ν)=(\mathbb{H},σ)$, an almost strictly dominating pair is equivalent to the data of an anti-de Sitter 3-manifold with specific properties. The results on maximal surfaces provide a parametrization of the deformation space of such $3$-manifolds as a union of components in a $\textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ relative representation variety. |
| title | Almost strict domination and anti-de Sitter 3-manifolds |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2105.12886 |