Chordal Loewner chains and Teichm\"uller spaces on the half-plane

TL;DR

This paper develops a half-plane analogue of the Ahlfors-Weill formula, constructing quasiconformal extensions of univalent functions with boundary Schwarzian conditions, and characterizes VMO-Teichmüller space elements via Carleson measures.

Contribution

It introduces a new half-plane version of the Loewner chain and quasiconformal extension theory, extending classical disk results to the half-plane setting.

Findings

Existence of chordal Loewner chains under boundary Schwarzian conditions.

Explicit quasiconformal extension with complex dilatation related to Schwarzian derivative.

Characterization of VMO-Teichmüller space elements via vanishing Carleson measures.

Abstract

We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker and also the generalization of the Ahlfors-Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichm\"uller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.