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

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’).

The group of symplectomorphisms

This chapter discusses the basic properties of the group of symplectomorphisms of a compact connected symplectic manifold and its subgroup of Hamiltonian symplectomorphisms. It begins by showing that the group of symplectomorphisms is locally path-connected and then moves on to the flux homomorphism. The main result here is a theorem of Banyaga that characterizes the Hamiltonian symplectomorphisms in terms of the flux homomorphism. In the noncompact case there is another interesting homomorphism, called the Calabi homomorphism, that takes values in the reals and may be defined on the universal cover of the group of Hamiltonian symplectomorphisms. The chapter ends with a brief comparison of the topological properties of the group of symplectomorphisms with those of the group of diffeomorphisms.