Teichmüller 空间的极值长度变分

本项目将集中研究 Teichmüller 空间上极值长度的变分, 及其在Teichmüller 曲率和极小曲面的应用. 我们将深入研究下述问题: (1) 以调和映照为工具, 计算极值长度的二阶变分.(2) 通过估计极值长度沿着 Teichmüller 测地线的二阶变分, 用解析的方法重新证明 Lenzhen-Rafi关于Teichmüller 测地球拟凸性的结果. (3) 估计极值长度及其变分沿着 Teichmüller 测地线的渐近值, 从而刻画 Teichmüller 测地线在Teichmüller空间边界 (Thurston 边界, Gardiner-Masur 边界, Bers边界) 的收敛性. (4) 通过对极值长度二阶变分的计算, 研究极小曲面的唯一性问题. 上述研究将加深对 Teichmüller 几何的理解及其和其它数学分支的联系.

Our program will focus on the variation theory of extremal length on Teichmüller space, and its applications in the study of curvature of Teichmüller metric and minimal surface. The following questions will be throughly studied: (1) Compute the second variational formula of extremal length via harmonic maps. (2) By estimating the second variation of extremal length along Teichmüller geodesics, give an analytic new proof of Lenzhen-Rafi's result on the quasi-convexity of Teichmüller geodesic balls. (3) Estimate the asymptotic values of extremal length and its variations, and then characterize the convergence of Teichmüller geodesics in (Thurston, Gardiner-Masur or Bers) boundary of Teichmüller space. (4) Apply the computations of second variation of extremal length to study the uniqueness problem in minimal surface theory. The study of all the above questions would improve our understanding of Teichmüller geometry and its connections with other branches of mathematics.