Loewner chains with quasiconformal extensions: an approximation approach

Abstract

A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-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 $$|p(z,t)-1|\le k|p(z,t)+1|$$ 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.

18 pages

References in corpus (2)

• Scaling limits of anisotropic Hastings-Levitov clusters
• Local duality in Loewner equations