Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation ρ:π1(S) → PSL2ℝ× PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h. on S induced by the action of ρ on ℍ2 × H2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h′).
The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry
Seppi A.
2017-01-01
Abstract
Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation ρ:π1(S) → PSL2ℝ× PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h. on S induced by the action of ρ on ℍ2 × H2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h′).
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