Loewner theory in annulus I: Evolution families and differential equations

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. © 2012 American Mathematical Society.