UC Riverside

In this paper we study the para-hyper-Kähler geometry of the deformation space of MGHC anti-de Sitter structures on Σ × R \Sigma \times \mathbb R , for Σ \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 K -surfaces, the identification with the cotangent bundle T ∗ T ( Σ ) T^*\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 P S L ( 2 , B ) \mathbb {P}\mathrm {SL}(2,\mathbb {B}) -character variety, where B \mathbb {B} is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.