Analysis and Applications of Teichmuller space

这些结果可以分为两类：一类是关于Teichmuller空间的结果，另一类是关于提供Teichmuller空间的对象的结果。其中，无穷维Teichmuller空间的研究是非常重要的，首席研究员Taniguchi阐明了超越整函数的Teichmuller空间的基本结构，包括游荡域的缺失与相应的Teichmuller空间的有限维之间的关系。这些结果是与研究生T.Harada的研究成果的共同产物，非常重要，因为它们为研究超越整函数诱导的复杂动力学（如有理函数和Kleinian群的情况）提供了新的思路。Taniguchi还研究了仅在无限维情况下出现的卷曲性质，当把泛Teichmuller空间中的点看作平面上的分形集时，证明了泛Teichmuller空间上的Bloch收敛等价于Caratheodory收敛.其次，Teichmuller理论的一个关键部分是拟共形映射.研究者Sugawa成功地给出了无尖点的有限Riemann曲面的Teichmuller空间的特征。Sugawa也给出了与一致完备性有关的区域常数的定量估计。因此，我们有有趣的估计的豪斯多夫维数的朱莉娅集的一个复杂的动态，或极限集的克莱因集团。

The results can be devided into two categories ; those on the Teichmuller spaces and those on objects which provide the Teichmuller spaces. Among oter things, the research on the Teichmuller spaces of infinite dimension is very important.The Head investigator Taniguchi has clarified the fundamental structures of the Teichmuller space of a transcendental entire function, including the relationship between the absence of wandering domains and finite dimensionality of the corresponding Teichmuller space. These results are the coproduct with T.Harada, a graduate student, and very important, for they give a new light for the investigations on the complex dynamics induced by transcendental entire functions as in the cases of rational functions and of Kleinian groups.Also Taniguchi has investigated the coiling property, appeared only in the case of infinite dimension, and proved that the Bloch convergence on the universal Teichmuller space is equivalent to Caratheodory convergence, when we consider points of the universal teichmuller space as fractal sets on the plane.Next, a crucial divice for the Teichmuller theory is a quasiconformal map. Investigator Sugawa has suceeded to give a characterization of the Teichmuller spaces of finite Riemann surfaces without cusps. Sugawa has also given quantative estimate on domain constants related to uniform perfectness. As a consequence, we have interesting estimates of the Hausdorff dimensions of the Julia sets of a complex dynamics, or of the limit set of a Kleinian group.