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

TL;DR

This paper explores the flux homomorphism on closed hyperbolic surfaces within Anti-de Sitter 3D geometry, providing a new proof that certain surface diffeomorphisms decompose into Hamiltonian and minimal Lagrangian components.

Contribution

It introduces an algebraic approach using the flux homomorphism to analyze surface diffeomorphisms in Anti-de Sitter space, offering a novel proof of their decomposition.

Findings

entity of tual surface diffeomorphisms with Hamiltonian and minimal Lagrangian components

New algebraic proof of the decomposition of surface maps

Enhanced understanding of symplectomorphisms in Anti-de Sitter 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 $\geq 2$, a canonical construction associates to $\Sigma$ a diffeomorphism $\phi_\Sigma$ of $S$. It turns out that $\phi_\Sigma$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ 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 $\phi_\Sigma$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$.