Finding the zeros of ahlfors map using integral equation method on bounded multiply connected regions

Abstract

The Ahlfors map of an n-connected region is a n-to-one map from the region onto the unit disk. The Ahlfors map being n-to-one map has n zeros. Previously, the exact zeros of the Ahlfors map are known only for the annulus region and a particular triply connected region. The zeros of the Ahlfors map for general bounded multiply connected regions has been unknown for many years. The purpose of this research is to find the zeros of the Ahlfors map for general bounded multiply connected regions using integral equation method. This work develops six new boundary integral equations for Ahlfors map of bounded multiply connected regions. The kernels of these integral equations are the generalized Neumann kernel, adjoint Neumann kernel, Neumann-type kernel and Kerzman-Stein type kernel. These integral equations are constructed from a non-homogeneous boundary relationship satisfied by an analytic function on a multiply connected region. The first four integral equations have kernels containing the zeros of the Ahlfors map which are unknown. The fifth integral equation has no zeros of the Ahlfors map in the kernel but involves derivative of the Ahlfors map at the unknown zeros. The sixth integral equation has unknown zeros appearing only at the right-hand side. The sixth integral equation proves to be useful for computing the zeros of the Ahlfors map. This work presents a numerical method for computing the zeros of Ahlfors map of any bounded multiply connected region with smooth boundaries. This work derives two formulas for the derivative of the boundary correspondence function of the Ahlfors map and the derivative of the Szeg¨o kernel. The relation between the Ahlfors map and the Szeg¨o kernel is classical. The Szeg¨o kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. These formulas are then used along with the sixth integral equation to compute all the zeros of the Ahlfors map for any bounded smooth multiply connected regions. Some examples are presented to demonstrate the efficiency of the presented method.