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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access December 29, 2017

The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

  • Andrea Seppi EMAIL logo
From the journal Complex Manifolds


Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation p: π1 (S) → PSL2ℝ x 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 p on ℍ2 x 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’).

Keywords MSC 2010: 57M50; 53D05; 30F45


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Received: 2017-10-5
Accepted: 2017-12-7
Published Online: 2017-12-29
Published in Print: 2017-12-20

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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