On an extremal problem for harmonic maps conformal at a point

TL;DR

This paper investigates an extremal problem for harmonic maps conformal at a point, providing a characterization of domains where equality holds in a specific supremum inequality, including non-round convex domains.

Contribution

The authors show that equality in the harmonic map supremum condition occurs for a broad family of convex domains, not just round discs, with explicit domain descriptions.

Findings

Equality holds for domains conformally equivalent to specific holomorphic maps.

Includes examples of strongly convex domains that are not round discs.

Provides a complete characterization of domains achieving equality.

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.