Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
home | mail | papers | authors | news | seminars | events | open positions | login

N. Rungi - A. Tamburelli

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

created by rungi on 02 Dec 2023



Inserted: 2 dec 2023
Last Updated: 2 dec 2023

Year: 2022

ArXiv: 2208.05336 PDF


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.

Credits | Cookie policy | HTML 5 | CSS 2.1