Equivariant maps into Anti-de Sitter space and the symplectic geometry of $\mathbb H^2\times \mathbb H^2$

Abstract

Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sitter space, where $ρ=(ρ_l,ρ_r)$ is the corresponding representation into $\mathrm{PSL}_2\mathbb R\times \mathrm{PSL}_2\mathbb R$. It is known that, if $σ$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold $Λ_{\mathrm{ML}}$ of $M_ρ$. We show that, given any $ρ$-equivariant embedding $σ$, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$. Conversely, any Lagrangian submanifold Hamiltonian isotopic to $Λ_{\mathrm{ML}}$ is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.