On an extremal problem for harmonic maps conformal at a point

Abstract

Let \(\mathbb D\) denote the unit disc in \(\mathbb C\). For a domain \(D\subset\mathbb C\) and a point \(p\in D\), let \(M_D(p)\) denote the supremum of \(\|df_0\|\) over all harmonic maps \(f:\mathbb D\to D\) with \(f(0)=p\) whose differential \(df_0\) at \(0\in \mathbb D\) is conformal. If \(f:\mathbb D\to D\) is a conformal diffeomorphism onto \(D\) with \(f(0)=p\), then \(\|df_0\|\le M_D(p)\). In a recent paper, the authors proved that equality holds when \(D=\mathbb D\), and they asked whether equality can hold only when \(D\) is a round disc. We give a negative answer by proving that, among bounded convex pointed domains \(p\in D\subset\mathbb C\) and up to translations, rotations, and reflections, equality holds if and only if, after moving \(p\) to the origin, \(D=F(\mathbb D)\) where \(F:\mathbb D\to\mathbb C\) is a holomorphic map with \(F(0)=0\) and \(F'(z)=\frac{c}{1+az+\lambda z^2}\), where \(c>0\), \(|\lambda|<1\), and \(|a-\bar a\lambda|<1-|\lambda|^2\). This family contains strongly convex examples which are not round discs.