The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

Abstract

Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that $ϕ_Σ$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$.