论文标题
全局darboux协调了完整的拉格朗日纤维，并应用于$ \ mathbb {r} \ mathbb {p}^2 $ - 结构属的变形空间
Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one
论文作者
论文摘要
在本文中，我们研究了一系列完整的汉密尔顿整合系统，即与之相关的拉格朗日纤维化且具有非紧凑纤维的系统。通过研究相关的完整拉格朗日纤维化，我们表明，在合适的假设下，运动积分可以作为哈密顿系统的动作坐标。作为一个应用程序，我们找到了全球darboux为一个新的符号形式的家族$ \boldsymbolΩ_f$，由光滑函数参数$ f：[0，+\ infty）\ to（ - \ infty，0] $，定义在适当的变形空间上，在适当的convex $ \ mathbb {r} $ r} $ ntorul torul torul torul torul torul符号形式是伪kähler指标的一部分（\ mathbf {g} _f，\ mathbf {i}，\boldsymbolΩ_f）$在$ \ mathcal {b} _0（b} _0（t^2）$上定义，并由作者介绍。推断出空间任意等轴测图的表达式。
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 $\boldsymbolω_f$, 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-Kähler metrics $(\mathbf{g}_f,\mathbf{I},\boldsymbolω_f)$ 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.
