The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class II
Katsuhiko Matsuzaki
TL;DR
This work advances the complex-analytic structure of circle-diffeomorphism Teichmüller spaces in the Zygmund smooth class. Building on Part I, it establishes that the pre-Schwarzian and Schwarzian derivatives organize a real-analytic disk-bundle structure over the Teichmüller space $T^Z$, and analyzes the little (asymptotic) subspaces to define quotient spaces $T^Z/T^Z_0$ with a robust Banach-manifold model via $A^Z(\mathbb D)/A^Z_0(\mathbb D)$. It proves the quotient Bers embedding $\hat{\alpha}$ is a local homeomorphism and injective, and constructs a quotient pre-Bers embedding $\hat{\beta}$ that is biholomorphic onto its image in $B^Z(\mathbb D)/B^Z_0(\mathbb D)$; together these yield a coherent complex structure on the quotient space and connect to the asymptotic Teichmüller framework. The results extend to little-subspaces and to related spaces (e.g., $T$, $T^{\gamma}$) and establish a unified, holomorphic, fibered picture linking Bers and pre-Bers embeddings in the Zygmund setting.
Abstract
In our previous paper with the same title, we established the complex Banach manifold structure for the Teichmüller space of circle diffeomorphisms whose derivatives belong to the Zygmund class. This was achieved by demonstrating that the Schwarzian derivative map is a holomorphic split submersion. We also obtained analogous results for the pre-Schwarzian derivative map. In this second part of the study, we investigate the structure of the image of the pre-Schwarzian derivative map, viewing it as a fiber space over the Bers embedding of the Teichmüller space, and prove that it forms a real-analytic disk-bundle. Furthermore, we consider the little Zygmund class and establish corresponding results for the closed Teichmüller subspace consisting of mappings in this class. Finally, we construct the quotient space of this subspace in analogy with the asymptotic Teichmüller space and prove that the quotient Bers embedding and pre-Bers embedding are well-defined and injective, thereby endowing it with a complex structure modeled on a quotient Banach space.