Geometry Behind Chordal Loewner Chains

Loewner Theory is a deep technique in ComplexAnalysis affording a basis for many further important developments such as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947-962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math. CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7-32, 1979; Bauer RO in J Math Anal Appl 302: 484-501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209-1217, 1992). © 2010 Birkhäuser / Springer Basel AG.