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’).
@article{bwmeta1.element.doi-10_1515_coma-2017-0013,
author = {Andrea Seppi},
title = {The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry},
journal = {Complex Manifolds},
volume = {4},
year = {2017},
pages = {183-199},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2017-0013}
}
Andrea Seppi. The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry. Complex Manifolds, Tome 4 (2017) pp. 183-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2017-0013/