Loewner theory in annulus I: Evolution families and differential equations

Contreras, Manuel; Díaz-Madrigal, Santiago; Gumenyuk, Pavel

doi:10.1090/S0002-9947-2012-05718-7

Loewner theory in annulus I: Evolution families and differential equations
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by Manuel D. Contreras, Santiago Díaz-Madrigal and Pavel Gumenyuk PDF

Trans. Amer. Math. Soc. 365 (2013), 2505-2543 Request permission

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 the famous Bieberbach conjecture and the well-celebrated Schramm stochastic Loewner evolution (SLE). Recently, Bracci et al. proposed a new approach bringing together all the variants of the (deterministic) Loewner evolution in a simply connected reference domain. We construct an analog 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 one-to-one 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.

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