Teichmuller空间的度量性质

本项目将研究Teichmuller空间的度量性质，主要研究Thurston度量。具体如下：（1）对于无穷型曲面的Teichmuller空间，将研究两个无穷型双曲曲面之间的Lipschitz常数是否可以由所有的相应的曲线和弧的长度之比的上确界达到、该空间上是否可以定义Thurston度量、两个无穷型双曲曲面之间是否存在测地线实现相应的Lipschitz常数等。（2）对于有限型曲面的Teichmuller空间，我们将主要研究关于Thurston度量的测地线（Thurston测地线）的性质。包括是否存在模空间中稠密的Thurston测地线、给出Thurston测地线在模空间的分类、给出两条Thurston测地线之间距离有界的充要条件、关于Thurston度量的测地流的刻画、平坦度量空间上的Thurston度量等。这些研究将有助于更好的理解和应用Teichmuller空间。

The research project will study the metric properties of Teichmuller spaces. Mainly we'll study the Thurston metric in Teichmuller spaces. We will study the following: (1) for Teichmuller spaces of infinite type surfaces, we will study whether the Lipschitz constant for two infinite type hyperbolic surfaces can be realized by the supremum of the ratios of the lengths of all the closed curves and arcs; whether we can define Thurston metric in these Teichmuller spaces; whether there exist a geodesic realize the Lipschitz constant for two infinite type hyperbolic surfaces, etc. (2) for Teichmuller spaces of finite type surfaces, we will mainly study the properties of the geodesics for Thurston metric (Thurston geodesic). For example, the characterization of Thurston geodesics in moduli space; the existence of dense Thurston geodesic in moduli space; the characterizations of the geodesic flow for Thurston metric; the sufficient and necessary conditions for the distance between two Thurston geodesics to be bounded; the Thurston metric for the space of flat metrics on Riemann surfaces, etc.This study will push forward the developments and applications of Teichmuller space.