Teichmuller空间的星形结构问题

On the Problem of Starlikeness of Teichmuller Spaces

对于任一保持单位圆盘Δ及其外部Δ的Ｆｕｃｈｓ群Γ，利用Ｂｅｒｓ嵌入，Ｔｅｉｃｈｍｕｌｌｅｒ空间Ｔ（Γ）可看成是Δ上Γ的有界全纯二次微分Ｂ（Δ，Γ）中的一个有界区域，本文的目的是讨论Ｔｅｉｃｈｍｕｌｌｅｒ空间Ｔ（Γ）的星形问题。特别地，我们证明了：当Γ是第二类Ｆｕｃｈｓ群时，Ｔ（Γ）不是星形的．

Let Γ be a Fuchsian group acting on the unit disk Δ. The Bers embedding repre-sents the Teichmuller space T(Γ)of Γ as a bounded domain in the space of bounded quadraticdifferentials for Γ. Our main result is: There exists a universal constant d_0>0 such that, for anygiven Fuchaian group Γ,suppose that there exists a hyperbolic disk in Δ of radius whichis precisely invariant under the trivial subgroup{1} of Γ, then T(Γ)is not starlike. Particullary,when Γ is a Fuchsian group of the second kind,T(Γ)is not starlike.