Teichm??ller Space of a Once Punctured Disk: Complex Coordinates on the Space of Abelian Differentials and the Takhtajan-Zograf Metric

Abstract

We construct an embedding of T(Z), the Teichm??ller space of Z--a Fuchsian group with a single parabolic generator--into the space of abelian differentials on the Riemann surface M = Z \ H. This "pre-Bers embedding" is much simpler, yet it still draws many parallels with the Bers embedding, such as the existence of a linear local right inverse. This gives rise to new complex coordinates which are compatible with the Bers coordinates. The differentials of the pre-Bers and the Bers embedding belong to a one parameter family of operators on the space of Beltrami differentials on M, which behaving nicely with the Takhtajan-Zograf metric on T(Z).