The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class

TL;DR

The paper develops a robust complex analytic structure for the Teichmüller space $T_Z$ of circle diffeomorphisms with Zygmund derivatives by proving the Schwarzian map $\Phi:{\rm Bel}_Z({\mathbb D}^*)\to A_Z({\mathbb D})$ is a holomorphic split submersion and that the Bers embedding $\alpha:T_Z\to A_Z({\mathbb D})$ is a homeomorphism onto the open image $\alpha(T_Z)=\Phi({\rm Bel}_Z({\mathbb D}^*))$. This yields $T_Z$ as a complex Banach manifold with a natural structure inherited from $A_Z({\mathbb D})$. The authors also formulate a pre-Schwarzian model using $\Psi:{\rm Bel}_Z({\mathbb D}^*)\to B_Z({\mathbb D})$ and $\Lambda(\psi)=\psi''-\tfrac12(\psi')^2$, proving that both maps are holomorphic split submersions and that $\widetilde{\mathcal T}_Z$ is a disk-bundle over $T_Z$ via $\widetilde{\mathcal T}_Z=\Psi({\rm Bel}_Z({\mathbb D}^*))$ with $\Lambda|_{\widetilde{\mathcal T}_Z}$ mapping onto the Bers image. These results extend the split-submersion framework from universal and Hölder settings to the Zygmund regime, providing parallel Schwarzian and pre-Schwarzian descriptions of $T_Z$.

Abstract

We provide the complex Banach manifold structure for the Teichmüller space of circle diffeomorphisms whose derivatives are in the Zygmund class. This is done by showing that the Schwarzian derivative map is a holomorphic split submersion.

Theorems & Definitions (26)

• Theorem 1: TW1
• Corollary 2
• Theorem 3
• Proposition 4
• proof
• Lemma 5
• proof
• Remark 1
• Lemma 6
• proof