Loewner Theory in annulus II: Loewner chains

TL;DR

This paper extends Loewner Theory to annuli, establishing a correspondence between Loewner chains and evolution families, and providing a classification and explicit characterization of associated vector fields in doubly connected domains.

Contribution

It introduces a general framework for Loewner Theory in annuli, including a classification and a correspondence with evolution families, expanding the scope beyond simply connected domains.

Findings

Established a 1-to-1 correspondence between Loewner chains and evolution families in annuli.

Provided a conformal classification of Loewner chains via evolution families.

Extended the characterization of semicomplete weak holomorphic vector fields to the annular case.

Abstract

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in [ArXiv 1011.4253]. We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly connected setting similar to that in the Loewner Theory for the unit disk. Futhermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in [ArXiv 1011.4253] to the general case.