Loewner Theory in annulus I: evolution families and differential equations

TL;DR

This paper extends Loewner Theory to annuli, establishing a correspondence between evolution families and vector fields, and providing a constructive characterization similar to classical cases, advancing the understanding of complex dynamics in multiply connected domains.

Contribution

It introduces a new framework for Loewner Theory in annuli, linking evolution families with holomorphic vector fields and generalizing classical results to multiply connected domains.

Findings

Established a one-to-one correspondence between evolution families and semicomplete weak holomorphic vector fields.

Provided a constructive characterization of vector fields analogous to Berkson-Porta representation.

Extended Loewner Theory to the setting of annuli, a multiply connected domain.

Abstract

Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). 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. We construct an analogue of this theory for the annulus. In this paper, the first of two articles, we introduce a general notion of an evolution family over a system of annuli and prove that there is a 1-to-1 correspondence between such families and semicomplete weak holomorphic vector fields. Moreover, in the non-degenerate case, we establish a constructive characterization of these vector fields analogous to the non-autonomous Berkson - Porta representation of Herglotz vector fields in the unit disk [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594].