Para-hyperK\"ahler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

TL;DR

This paper explores the rich para-hyperK"ahler geometric structure of the deformation space of maximal globally hyperbolic anti-de Sitter 3-manifolds, revealing deep connections with Teichm"uller theory, character varieties, and symplectic geometry.

Contribution

It introduces a novel para-hyperK"ahler framework for the deformation space, linking it to various geometric and algebraic structures in anti-de Sitter geometry.

Findings

Identification of para-hyperK"ahler structures with MGHC anti-de Sitter geometry

Relation between the deformation space and cotangent bundle of Teichm"uller space

Connection to character varieties and Goldman symplectic form

Abstract

In this paper we study the para-hyperK\"ahler 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 $T^*\mathcal{T}(\Sigma)$, 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.