Matching univalent functions and conformal welding

Abstract

Given a conformal mapping $f$ of the unit disk $\mathbb{D}$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk ${\mathbb{D}}^{\ast }$ onto the exterior domain $D^{\ast }$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions.

17 pages