A Riemann mapping theorem for two-connected domains in the plane (Q2378586)

The authors prove a new analog of the Riemann mapping theorem for doubly connected planar domains such that neither boundary component is a point. Let \(\Omega\) be such a domain. For \(a\in\Omega\), the Ahlfors map \(f_a\) is the unique holomorphic function of \(\Omega\) onto the unit disk such that \(f_a^\prime(a)\) is real and as large as possible. The authors find a point \(a\in\Omega\), a complex constant \(c\), and a conformal mapping \(\Phi\) of \(\Omega\) onto a model representative doubly connected domain \(A=\{z:|z+\frac{1}{z}|<2|c|\}\) such that \[ \Phi=cf_a+\sqrt{c^2f_a^2-1}. \] The main contribution of the paper is the connection of the mapping \(\Phi\) with the Ahlfors map \(f_a\) of \(\Omega\) at a specially chosen point \(a\). The authors also show how this connection can be used to give formulae for the Bergman kernel of doubly connected domains.