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Complex Manifolds

Ed. by Fino, Anna Maria

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

Andrea Seppi
  • University of Luxembourg, Mathematics Research Unit, Maison du Nombre, 6 Avenue de la Fonte, Esch-sur-Alzette L-4364 Luxembourg
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Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/coma-2017-0013


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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About the article

Received: 2017-10-05

Accepted: 2017-12-07

Published Online: 2017-12-29

Published in Print: 2017-12-20

Citation Information: Complex Manifolds, Volume 4, Issue 1, Pages 183–199, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2017-0013.

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