Para-hyper-Kähler Geometry of the Deformation Space of Maximal Globally Hyperbolic Anti-de Sitter Three-Manifolds

In this paper we study the para-hyper-Kähler geometry of the deformation space of MGHC anti-de Sitter structures on ∑ × R, for ∑ 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∗T p∑q, 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 PSLp2, Bq-character variety, where B is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.