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

Huaying Wei; Katsuhiko Matsuzaki

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arXiv · 2207.02447

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

Huaying Wei·Katsuhiko Matsuzaki

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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.

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Huaying Wei, Katsuhiko Matsuzaki. 2022-07-06. Chordal Loewner chains and Teichm\"uller spaces on the half-plane. https://arxiv.org/abs/2207.02447

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