Harmonic shears of regular polygons by hypergeometric functions (Q1961031)

A harmonic mapping \(f\) that maps the unit disk \(\mathbb D\) onto a domain \(\Omega\) in the complex plane has a canonical representation \(f=h+\bar{g}\), where \(h\) and \(g\) are analytic in \(\mathbb D\) and \(g(0)=0\). Its dilatation is \(\omega=g'/h'\). The function \(f\) is called the harmonic shear of the analytic function \(\phi=h-g\). The authors construct harmonic shears (with prescribed dilatations) of functions \(\phi\) that map \(\mathbb D\) onto regular polygons. They also construct minimal surfaces associated to harmonic shears and study the inverse question: what conformal maps have harmonic shears that map \(\mathbb D\) onto a regular polygon. The constructions involve hypergeometric functions.