In this paper we study a broad class of complete Hamiltonian integrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibres. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms ωƒ, parametrized by smooth functions ƒ : [0, + ∞] → [-∞, 0], defined on the deformation space of properly convex ℝℙ²-structures on the torus. Such a symplectic form is part of a family of pseudo-Kähler metrics (gƒ, I, ωƒ) defined on ℬ₀(T²) and introduced by the authors. In the last part of the paper, by choosing ƒ(t) = -kt, k >0 we deduce the expression for an arbitrary isometry of the space.
Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of projective structures in genus one
Nicholas Rungi;Andrea Tamburelli
2024-01-01
Abstract
In this paper we study a broad class of complete Hamiltonian integrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibres. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms ωƒ, parametrized by smooth functions ƒ : [0, + ∞] → [-∞, 0], defined on the deformation space of properly convex ℝℙ²-structures on the torus. Such a symplectic form is part of a family of pseudo-Kähler metrics (gƒ, I, ωƒ) defined on ℬ₀(T²) and introduced by the authors. In the last part of the paper, by choosing ƒ(t) = -kt, k >0 we deduce the expression for an arbitrary isometry of the space.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
