Loewner Theory in annulus II: Loewner chains

Abstract

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947-962, 2009) and Contreras et al. (Revista Matemática Iberoamericana 26:975-1012, 2010) 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 Contreras et al. (Trans Amer Math Soc to appear, 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 352 M. D. Contreras et.alconnected setting similar to that in the Loewner Theory for the unit disk. Furthermore, 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 Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case.

Local duality in Loewner equations

“…Again, since one can choose T > 0 arbitrarily large, S4 holds for the whole semiaxis 12). The uniqueness of the solution is proved in the same way as in Theorem 3.2.…”

“…To conclude the section, we mention that an analogous approach has been suggested for the Loewner Theory in the annulus [11,12]. However, in this paper we restrict ourselves to the Loewner Theory for simply connected domains.…”

Elementary examples of Loewner chains generated by densities

“…In this note, we are working with the decreasing version of Loewner chains; in many applications, the increasing version, where {f (∆, t)} forms a sequence of growing domains, is more natural. The increasing and decreasing theories are similar, but not completely equivalent (see [7]). In both cases, the classical form of the Loewner equation is recovered by setting µ t = δ λ(t) , where λ is a unimodular driving function.…”

Loewner equations and reductions of dispersionless hierarchies

“…The next example of the Löwner-type equations is an equation for the doubly connected domain found by Komatu ([24], [25] §84). (See also [26,27,28,29,30,31]…”