Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds - Archive ouverte HAL
Article Dans Une Revue Memoirs of the American Mathematical Society Année : 2022

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

Résumé

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on ΣxR, 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˚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 PSL(2,B)-character variety, where B is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.
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Dates et versions

hal-03421117 , version 1 (09-11-2021)

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Filippo Mazzoli, Andrea Seppi, Andrea Tamburelli. Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds. Memoirs of the American Mathematical Society, 2022. ⟨hal-03421117⟩
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