The complex structure of the Teichm\"uller space of circle diffeomorphisms in the Zygmund smooth class II

TL;DR

This paper explores the complex structure of the Teichmüller space of circle diffeomorphisms with Zygmund smoothness, revealing fiber space structures and quotient spaces with complex Banach manifold properties.

Contribution

It extends previous work by analyzing the pre-Schwarzian derivative image as a fiber space and constructing quotient spaces with complex structures.

Findings

Pre-Schwarzian image forms a real-analytic disk-bundle.

Quotient space of the Zygmund class has a well-defined complex structure.

Bers and pre-Bers embeddings are injective on the quotient space.

Abstract

In our previous paper with the same title, we established the complex Banach manifold structure for the Teichm\"uller 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 Teichm\"uller 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 Teichm\"uller subspace consisting of mappings in this class. Finally, we construct the quotient space of this subspace in analogy with the asymptotic Teichm\"uller 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.