Loewner chains with quasiconformal extensions: an approximation approach

TL;DR

This paper introduces an approximation method in Loewner Theory to establish quasiconformal extensions of Loewner chains under a new condition on the Herglotz function, unifying radial and chordal cases without extra assumptions.

Contribution

It provides a novel approximation approach to prove quasiconformal extensions for generalized Loewner chains with minimal restrictions on the driving function.

Findings

Established quasiconformal extension criteria for Loewner chains.

Unified treatment of radial and chordal Loewner equations.

Used approximation method based on continuous dependence of evolution families.

Abstract

A new approach in Loewner Theory proposed by Bracci, Contreras, D\'iaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $$ \partial_{t}f_{t}(z) = (z - \tau(t))(1-\overline{\tau(t)}z)\partial_{z}f_{t}(z)p(z,t), $$ where $\tau : [0,\infty) \to \overline{\mathbb{D}}$ is measurable and $p$ is called a Herglotz function. In this paper, we will show that if there exists a $k \in [0,1)$ such that $p$ satisfies $$ |p(z,t) - 1| \leq k |p(z,t) + 1| $$ for all $z \in \mathbb{D}$ and almost all $t \in [0,\infty)$, then $f_{t}$ has a $k$-quasiconformal extension to the whole Riemann sphere for all $t \in [0,\infty)$. The radial case ($\tau =0$) and the chordal case ($\tau=1$) have been proven by Becker [J. Reine Angew. Math. \textbf{255} (1972), 23-43] and Gumenyuk and the author (Math. Z. \textbf{285} (2017), no.3, 1063--1089). In our theorem, no superfluous assumption is imposed on $\tau \in \overline{\mathbb{D}}$. As a key foundation of our proof is an approximation method using the continuous dependence of evolution families.