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

Given a smooth spacelike surface $\mathrm{\Sigma }$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $\rho :{\pi }_1(S)\to {\mathrm{P}\mathrm{S}\mathrm{L}}_2\mathbb{R}\times {\mathrm{P}\mathrm{S}\mathrm{L}}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\ge 2$, a canonical construction associates to $\mathrm{\Sigma }$ a diffeomorphism ${\varphi }_{\mathrm{\Sigma }}$ of $S$. It turns out that ${\varphi }_{\mathrm{\Sigma }}$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h^{\mathrm{\prime }}$ on $S$ induced by the action of $\rho$ 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 ${\varphi }_{\mathrm{\Sigma }}$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h^{\mathrm{\prime }})$.