Parametrization of domains in $\mathbf {\hat {C}}$: the logarithmic domains

Abstract:

We prove a generalization of Riemann’s mapping theorem: Every $n$-fold connected domain in $\widehat {\mathbf {C}}$, whose boundary does not contain isolated points, is conformal equivalent to a logarithmic domain. The logarithmic domains are characterized by a Green’s function consisting of a finite sum of logarithms.