BMO Teichmüller spaces and their quotients with complex and metric structures

Abstract

The paper presents some recent results on the BMO Teichmüller space, its subspaces and quotient spaces. We first consider the chord-arc curve subspace and prove that every element of the BMO Teichmüller space is represented by its finite composition. Moreover, we show that these BMO Teichmüller spaces have affine foliated structures induced by the VMO Teichmüller space. By which, their quotient spaces have natural complex structures modeled on the quotient Banach space. Then, a complete translation-invariant metric is introduced on the BMO Teichmüller space and is shown to be a continuous Finsler metric in a special case.

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

“…Here, the boundedness in SQS is considered regarding a metric structure of SQS ∼ = T b . The invariant metric provided for T b is the Carleson metric (see [38]), and let d c denote the Carleson distance in T b .…”

“…The VMO Teichmüller space T v can be defined as the set of all normalized strongly symmetric homeomorphisms of the unit circle onto itself, and this is a closed subspace of T b (see [34]). The subset T c of T b consisting of all elements corresponding to chord-arc curves contains T v (see [38]). It was shown in [36] that all strongly symmetric homeomorphisms on the unit circle constitute a topological group, and from its proof, we see that T v is contained in char(T b ).…”