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Given two Fuchsian representations $\rho_l$ and $\rho_r$ of the fundamentalgroup of a closed oriented surface $S$ of genus $\geq 2$, we study the relationbetween 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 $\widetilde S$ into Anti-deSitter space, where $\rho=(\rho_l,\rho_r)$ is the corresponding representationinto $\mathrm{PSL}_2\mathbb R\times \mathrm{PSL}_2\mathbb R$. It is known that,if $\sigma$ is a maximal embedding, then its Gauss map takes values in theunique minimal Lagrangian submanifold $\Lambda_{\mathrm{ML}}$ of $M_\rho$.  We show that, given any $\rho$-equivariant embedding $\sigma$, its Gauss mapgives 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 thefuture unit tangent bundle of the universal cover of Anti-de Sitter space.

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