Analysis and Applications of Teichmuller space

结果可分为两类；一个是关于泰希穆勒空间的，另一个是关于提供泰希穆勒空间的物体的。其中，对无限维的Teichmuller空间的研究是非常重要的。首席研究员Taniguchi澄清了超越全函数的Teichmuller空间的基本结构，包括没有流浪域和相应的Teichmuller空间的有限维之间的关系。这些结果是与研究生t.h harada的副积，非常重要，因为它们为研究由超越整体函数引起的复杂动力学（如有理函数和Kleinian群）提供了新的视角。此外，Taniguchi还研究了仅在无限维情况下才出现的卷绕性，并证明了当将普适Teichmuller空间的点视为平面上的分形集时，普适Teichmuller空间上的Bloch收敛等价于Caratheodory收敛。接下来，Teichmuller理论的一个关键的部分是拟共形映射。研究者Sugawa已经成功地给出了有限黎曼曲面无尖的Teichmuller空间的一个表征。Sugawa也给出了与均匀完美性相关的域常数的定量估计。因此，我们对复杂动力学的Julia集或Kleinian群的极限集的Hausdorff维数有了有趣的估计。

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.