复地震在Teichmüller空间与模空间紧化下的运用

Teichmüller空间与模空间紧化问题是Teichmüller理论中一个非常重要的研究课题。本项目主要研究复地震在Teichmüller空间Thurston紧化、Gardiner-Masur紧化及模空间Deligne-Mumford紧化意义下的收敛性等问题，特别是唯一遍历的及有理的lamination诱导的复地震分别在Teichmüller 空间Thurston紧化与Gardiner-Masur紧化意义下的收敛性；有理lamination诱导的复地震在模空间里的投影在Deligne-Mumford紧化意义下的收敛性，并通过加权简单闭曲线诱导的复地震在模空间里的投影构造与Deligne-Mumford紧化同胚的分层空间。这些问题的研究有利于将三维双曲空间可测pleated曲面与Teichmüller空间及模空间紧化联系，还有利于对相关热点问题的理解和把握。

The problem on the compactifications of the Teichmüller space and the moduli space is a very important research topic in the Teichmüller theory. The convergence of the complex earthquake in the Thurston compactification and the Gardiner-Masur compactification of the Teichmüller space, and the Deligne-Mumford compactification of the moduli space is studied in this project. Especially, it includes the convergence of the complex earthquake for a uniquely ergodic lamination or a rational lamination in the Thurston compactification or the Gardiner-Masur compactification of the Teichmüller space; the convergence of the projective, in the moduli space, of the complex earthquake for a rational lamination in the Deligne-Mumford compactification and the stratified space, homeomorphic to the Deligne-Mumford compactification of the moduli space, being constructed by the projectives of complex earthquakes for weighted simple closed curves in the moduli space. The study of these problems is helpful to connect the measurable pleated surface in the three dimensional hyperbolic space with the compactifications of the Teichmüller space and the moduli space, and to further understand and grasp the related hot issues.