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

Abstract

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on $\Sigma \times \mathbb{R}$, for $\Sigma$ 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 $T^{\ast }\mathcal{T}(\Sigma )$, 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.

111 pages

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