本项目将主要研究 Teichmüller 中双曲长度函数的梯度向量场生成的积分曲线的渐近几何，主要包括：(1)、各类 measured laminations 的双曲长度函数沿着积分曲线的变化规律；(2)、梯度向量场的积分曲线在 Teichmüller 空间、模空间中的分布规律；(3)、考虑在 Teichmüller 的切丛或模空间的切丛的子集上定义负梯度流，并初步研究该动力系统的一些几何性质。

This project will study the asymptotic geometry of the integral curve of the gradient vector field of the hyperbolic length function in Teichmüller space, including: (1) consider the behavior of hyperbolic length function of measured laminations along the integral curve; (2)consider the distribution rule of the integral curve in Teichmüller space and moduls space; (3) find a subset of the tangent bundle of Teichmüller space or module space such that the definition of the negative gradient flow is well-defined, and make some research of the geometric properties of the dynamical system associated negative gradient flow.