Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one

TL;DR

This paper establishes global Darboux coordinates for complete Lagrangian fibrations in integrable systems and applies this to describe the deformation space of convex real projective structures on a torus, revealing new symplectic and metric properties.

Contribution

The paper introduces a method to obtain global Darboux coordinates for a class of complete Lagrangian fibrations and applies it to the deformation space of convex real projective structures on the torus.

Findings

Explicit global Darboux coordinates for a family of symplectic forms on the deformation space.

Description of a family of pseudo-Kähler metrics on the deformation space.

Derivation of isometries for the constructed metric family.

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 $\boldsymbol{\omega}_f$, parametrized by smooth functions $f:[0,+\infty)\to(-\infty,0]$, defined on the deformation space of properly convex $\mathbb{R}\mathbb{P}^2$-structures on the torus. Such a symplectic form is part of a family of pseudo-K\"ahler metrics $(\mathbf{g}_f,\mathbf{I},\boldsymbol{\omega}_f)$ defined on $\mathcal{B}_0(T^2)$ and introduced by the authors. In the last part of the paper, by choosing $f(t)=-kt, k>0$ we deduce the expression for an arbitrary isometry of the space.