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arXiv · 2107.10363

Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

Filippo Mazzoli·Andrea Seppi·Andrea Tamburelli
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Abstract

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.

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Filippo Mazzoli, Andrea Seppi, Andrea Tamburelli. 2021-07-21. Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds. https://arxiv.org/abs/2107.10363

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