Teichmüller动力学相关问题研究

The representative work of the applicant is to propose a set of methods to study the Lyapunov exponents of Teichmüller geodesic flows. The applicant and the collaborator prove several conjectures about Lyapunov exponents of Teichmüller geodesic flows. The applicant separately proposes a conjecture about the relationship betwwen stability in algebraic geometry and stability in dynamical system, which is porved by Eskin-Kontsevich-Möller-Zorich. In this project, it is proposed to do research in the following problems: (1)The relationship between the equalities and the thin groups. (2) How to define the Lyapunov exponents in general representations, and find similar conjectures. (3)Does the methords of Teichmüller geodesic flows can be applied to the spaces of Bridgeland stablility condeitions. (4)Establish a relationship between Teichmüller geodesic flows and mapping class groups, and give some applications to low dimension topology.

申请人代表性的工作是提出一整套方法研究Teichmüller测地流上的Lyaounov指数。申请人和合作者在Teichmüller测地流领域证明了几个Kontsevich-Zorich关于Lyapunov指数的猜想; 申请人单独提出比较代数几何稳定性和动力系统稳定性的猜想，此猜想被Eskin-Kontsevich-Möller-Zorich专文证明。在本项目中拟对以下问题展开研究: (1) 猜想等号达到情形与的Thin群的关系。（2)在一般表示中定义Lyapunov指数，以及寻找类似的猜想。（3）Teichmüller测地流这一套方法可否应用在Bridgeland稳定条件空间上。(4)建立Teichmüller测地流与映射类群之间的关系，给出一些可能在低维拓扑中的应用。

我们计算了二次微分的所有超椭圆分支和陈大卫和Moller文章中所有的不变化分支里面的泰希米勒曲线的Hodge丛的不变和反不变部分的Harder-Narasimhan滤链。我们证明了这些不变分支里面除了亏格0分支和不正则分支，这些Hodge丛都分裂成线丛的直和。做为推论，我们得到这些分支上第m个李亚普罗夫指数的渐进结果。这个结果推广了由申请人和Eskin-Kontsevich-Moller-Zorich证明的Kontsevich-Zorich猜想。.我们也在下面的问题中取得了一些进展：1. 二次微分分支的Harder-Narasimhan polygon的上界估计；2.Hodge丛的滤链和Brill-Noether除子之间的紧密联系。.这些工作正与陈大卫合作进行中。