万有Teichmuller空间BMO理论的若干问题

本项目主要研究万有Teichmuller空间BMO理论的两个重要课题：Grunsky算子与切空间流的光滑性。这两个课题与实分析、调和分析、偏微分方程等其他数学分支中相关课题密切相关。我们期望通过考虑Grunsky算子在BMOA空间上的作用，给出BMO Teichmuller空间以及VMO Teichmuller空间新的刻画，从而进一步考虑能否刻画弦弧曲线。其次，我们期待通过考虑拉回算子对于拟对称同胚的连续依赖性, 解决BMO Teichmuller空间以及VMO Teichmuller空间切空间流的光滑性问题. 这些问题的解决将给我们研究弦弧曲线空间的连通性问题提供一个新思路。

The project is mainly concerned with two problems on the BMO theory of the universal Teichmuller space, the Grunsky operator and the smoothness of the flow curves of the tangent space, which are closely related to some topics in real analysis, harmonic analysis and partial differential equation. By considering the action of the Grunsky operator on the BMOA space, we hope to give some new characterizations of the BMO and VMO Teichmuller space and the chord-arc curve space as well. We will also discuss the smoothness of the flow curves of the tangent spaces of BMO and VMO Teichmuller space by considering the dependence of the pull-back operators on quasisymmetric homeomorphisms. This will provide a new approach to solving the problem of the connectedness of the chord-arc curve space.