A fiber bundle over BMO Teichm\"uller space

TL;DR

This paper establishes a real-analytic fiber bundle structure over the BMO Teichmüller space using BMOA functions related to conformal homeomorphisms, and shows triviality over the VMO Teichmüller space.

Contribution

It constructs a real-analytic fiber bundle over BMO Teichmüller space and demonstrates triviality over VMO Teichmüller space, advancing understanding of these function spaces.

Findings

Fiber space of BMOA functions forms a real-analytic disk bundle over BMO Teichmüller space.

Sub-bundle over VMO Teichmüller space is real-analytically trivial.

Provides new geometric structure for BMO and VMO Teichmüller spaces.

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 Teichm\"uller space. For the VMO Teichm\"uller space, we show that the corresponding sub-bundle consisting of VMOA functions is real-analytically trivial.