ABSTRACT
Suppose Ω is a bounded simply connected domain. Everyone knows that, among all holomorphic functions h that map Ω into the unit disc, the Riemann mapping function associated to a point a ∈ Ω is the unique function in this class making h′(a) real and as large as possible. Hence, finding the Riemann map is equivalent to solving an extremal problem. In Chapter 8, we showed that the solution to this extremal problem can also be expressed as the quotient of the Szego˝ and the Garabedian kernels, f(z) = S(z, a)/L(z, a). In this chapter, we will consider this quotient when Ω is a multiply connected domain. We will show that it is a mapping of the domain onto the unit disc, that it solves the same extremal problem, and that it has many of the geometric features one would expect of a “Riemann mapping function” of a multiply connected domain. The map is known as the Ahlfors mapping.
