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

$${{\partial {f_t}\left(z \right)} \over {\partial t}} = \left({z - \tau \left(t \right)} \right)\left({1 - \overline {\tau \left(t \right)} z} \right)p\left({z,\;t} \right){{\partial {f_t}\left(z \right)} \over {\partial z}},$$

where τ: [0, ∞) → $\overline{\mathbb{D}}$ is measurable and p is called a Herglotz function. In this paper, we will show that if there exists a k ⩽ [0, 1) such that p satisfies

$$\left| {p\left({z,\;t} \right) - 1} \right| \le k\left| {p\left({z,\;t} \right) + 1} \right|$$

for all z ⩽ $\mathbb{D}$ and almost all t ⩽ [0, ∞), then, for all t ⩽ [0, ∞), f_t has a k-quasiconformal extension to the whole Riemann sphere. The radial case (τ = 0) and the chordal case (τ = 1) have been proven by Becker [J. Reine Angew. Math., vol. 255 (1972), 23–43] and Gumenyuk and the author [Math. Z., vol. 285 (2017), 1063–1089]. In our theorem, no superfluous assumption is imposed on τ ⩽ $\overline{\mathbb{D}}$. As a key foundation of the proof is an approximation method using a continuous dependence of evolution families and Loewner chains.