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

Abstract

Given two Fuchsian representations ${\rho }_l$ and ${\rho }_r$ of the fundamental group of a closed oriented surface $S$ of genus $\ge 2$, we study the relation between Lagrangian submanifolds of $M_{\rho }=({\mathbb{H}}^2/{\rho }_l({\pi }_1(S)))\times ({\mathbb{H}}^2/{\rho }_r({\pi }_1(S)))$ and $\rho$-equivariant embeddings $\sigma$ of $\tilde{S}$ into Anti-de Sitter space, where $\rho =({\rho }_l,{\rho }_r)$ is the corresponding representation into ${\mathrm{PSL}}_2\mathbb{R}\times {\mathrm{PSL}}_2\mathbb{R}$. It is known that, if $\sigma$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold ${\Lambda }_{\mathrm{ML}}$ of $M_{\rho }$. We show that, given any $\rho$-equivariant embedding $\sigma$, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to ${\Lambda }_{\mathrm{ML}}$. Conversely, any Lagrangian submanifold Hamiltonian isotopic to ${\Lambda }_{\mathrm{ML}}$ is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.

24 pages. Several corrections and improvements. Final version. To appear in Transactions of the AMS

References in corpus (2)

• The Calabi homomorphism, Lagrangian paths and special Lagrangians
• Lorentzian Kleinian Groups