Given two Fuchsian representations ρ l and ρ r of the fundamental group of a closed oriented surface S of genus ≥ 2, we study the relation between Lagrangian submanifolds of Mρ = (ℍ 2 /ρ l (π 1 (S))) × (ℍ 2 /ρr(π 1 (S))) and ρ- equivariant embeddings σ of S into Anti-de Sitter space, where ρ = (ρl, ρr) is the corresponding representation into PSL 2 ℝ × PSL 2 ℝ. It is known that, if σ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold ΛML of Mρ. We show that, given any ρ-equivariant embedding σ, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to ΛML. Conversely, any Lagrangian submanifold Hamiltonian isotopic to ΛML is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.
Equivariant maps into anti-de sitter space and the symplectic geometry of ℍ 2 × ℍ 2
Seppi A.
2019-01-01
Abstract
Given two Fuchsian representations ρ l and ρ r of the fundamental group of a closed oriented surface S of genus ≥ 2, we study the relation between Lagrangian submanifolds of Mρ = (ℍ 2 /ρ l (π 1 (S))) × (ℍ 2 /ρr(π 1 (S))) and ρ- equivariant embeddings σ of S into Anti-de Sitter space, where ρ = (ρl, ρr) is the corresponding representation into PSL 2 ℝ × PSL 2 ℝ. It is known that, if σ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold ΛML of Mρ. We show that, given any ρ-equivariant embedding σ, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to ΛML. Conversely, any Lagrangian submanifold Hamiltonian isotopic to ΛML is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.
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