ParahyperKähler geometry of the deformation space of maximal globally hyperbolic antide Sitter threemanifolds
Abstract
In this paper we study the parahyperKähler geometry of the deformation space of MGHC antide Sitter structures on $\Sigma\times\mathbb R$, for $\Sigma$ a closed oriented surface. We show that a neutral pseudoRiemannian metric and three symplectic structures coexist with an integrable complex structure and two paracomplex structures, satisfying the relations of paraquaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of KrasnovSchlenker by the induced metric on $K$surfaces, the identification with the cotangent bundle $T^*\mathcal{T}(\Sigma)$, and the circle action that arises from this identification. Finally, we study the relation to the natural paracomplex 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 paracomplex numbers, and the symplectic geometry deriving from Goldman symplectic form.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.10363
 arXiv:
 arXiv:2107.10363
 Bibcode:
 2021arXiv210710363M
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Geometric Topology
 EPrint:
 111 pages