Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds
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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917995676172288 |
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| author | Mazzoli, Filippo Seppi, Andrea Tamburelli, Andrea |
| author_facet | Mazzoli, Filippo Seppi, Andrea Tamburelli, Andrea |
| contents | In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on $Σ\times\mathbb R$, for $Σ$ a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on $K$-surfaces, the identification with the cotangent bundle $T^*\mathcal{T}(Σ)$, and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the $\mathrm{PSL}(2,\mathbb{B})$-character variety, where $\mathbb{B}$ is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_10363 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds Mazzoli, Filippo Seppi, Andrea Tamburelli, Andrea Differential Geometry Geometric Topology In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on $Σ\times\mathbb R$, for $Σ$ a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on $K$-surfaces, the identification with the cotangent bundle $T^*\mathcal{T}(Σ)$, and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the $\mathrm{PSL}(2,\mathbb{B})$-character variety, where $\mathbb{B}$ is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form. |
| title | Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2107.10363 |