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

Abstract

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

20 pages

Cited by in corpus (1)

• Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space