Geometry behind chordal Loewner chains

Abstract

Loewner Theory is a deep technique in Complex Analysis 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 (SLE). It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the {\it radial} and the {\it 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 [arXiv:0807.1594v1, arXiv:0807.1715v1, arXiv:0902.3116v1] 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 by I.A.Aleksandrov, S.T.Aleksandrov and V.V.Sobolev, 1983; V.V.Goryainov and I.Ba, 1992; R.O.Bauer, 2005 [arXiv:math/0306130v1].