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A fiber bundle over BMO Teichmüller space

Katsuhiko Matsuzaki

TL;DR

The paper studies the fiber space formed by $BMOA$-log-derivatives of conformal maps from the unit disk to bounded quasidisks as a bundle over the $BMO$ Teichmüller space via the Bers embedding. It shows this fiber space is a real-analytic disk bundle over $\alpha(T_B)$, while the corresponding $VMOA$-subbundle is real-analytically trivial. A key contrast is established with Zhuravlev-type phenomena: in the $BMO$ setting multiple components arise from post-composition by Möbius maps, whereas in the $VMO$ setting there is essentially a single component and the intersection with $VMOA$ is empty for the auxiliary families. The paper also provides a global real-analytic trivialization in the $VMO$ case via a global right inverse of the Schwarzian map, and notes the constructions extend to the universal Teichmüller space and other Teichmüller settings. Together, these results give a detailed analytic description of Bers-embedded fibers and clarify the structure of BMO/VMO Teichmüller spaces, with implications for the geometry of conformal maps to quasidisks.

Abstract

We prove that the fiber space consisting of BMOA functions that are the logarithms of derivatives of conformal homeomorphisms of the unit disk onto bounded quasidisks forms a real-analytic disk bundle over the Bers embedding of the BMO Teichmüller space. For the VMO Teichmüller space, we show that the corresponding sub-bundle consisting of VMOA functions is real-analytically trivial.

A fiber bundle over BMO Teichmüller space

TL;DR

The paper studies the fiber space formed by -log-derivatives of conformal maps from the unit disk to bounded quasidisks as a bundle over the Teichmüller space via the Bers embedding. It shows this fiber space is a real-analytic disk bundle over , while the corresponding -subbundle is real-analytically trivial. A key contrast is established with Zhuravlev-type phenomena: in the setting multiple components arise from post-composition by Möbius maps, whereas in the setting there is essentially a single component and the intersection with is empty for the auxiliary families. The paper also provides a global real-analytic trivialization in the case via a global right inverse of the Schwarzian map, and notes the constructions extend to the universal Teichmüller space and other Teichmüller settings. Together, these results give a detailed analytic description of Bers-embedded fibers and clarify the structure of BMO/VMO Teichmüller spaces, with implications for the geometry of conformal maps to quasidisks.

Abstract

We prove that the fiber space consisting of BMOA functions that are the logarithms of derivatives of conformal homeomorphisms of the unit disk onto bounded quasidisks forms a real-analytic disk bundle over the Bers embedding of the BMO Teichmüller space. For the VMO Teichmüller space, we show that the corresponding sub-bundle consisting of VMOA functions is real-analytically trivial.
Paper Structure (4 sections, 12 theorems, 22 equations)

This paper contains 4 sections, 12 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. BMO and VMO Teich-mül-ler spaces
  3. The components in BMOA and VMOA
  4. The structure of the fiber space

Key Result

Proposition 2.1

The map $S:M_B(\mathbb D^*) \to A_B(\mathbb D)$ is holomorphic and admits a local holomorphic right inverse at every point in the image $S(M_B(\mathbb D^*))$. The restriction of $S$ to $M_V(\mathbb D^*)$ enjoys the analogous property as a holomorphic map into $A_V(\mathbb D)$.

Theorems & Definitions (22)

  • Remark 1
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Lemma 4.1
  • proof
  • ...and 12 more