Transactions of the American Mathematical Society

Abstract:

Given two Fuchsian representations $\rho _l$ and $\rho _r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, 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 $\widetilde S$ 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.