A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

TL;DR

The paper constructs a bounded, globally univalent harmonic map with an unbounded analytic part, demonstrating a novel example in harmonic mapping theory.

Contribution

It provides a new explicit construction of harmonic maps where the analytic component is unbounded despite the overall map being bounded and univalent.

Findings

Constructed harmonic maps for all 0<k<1 with bounded, globally univalent properties.

The analytic part h of the map is unbounded, contrasting with the boundedness of the entire map.

Uses a logarithmic spiral and perturbation technique in the construction.

Abstract

We construct, for every \(0<k<1\), a bounded globally univalent harmonic mapping \[ f=h+\overline g \colon \D\to\C \] such that \[ |g'(z)|\le k|h'(z)|,\qquad z\in\D, \] while the analytic part \(h\) is unbounded. The construction is based on a bounded logarithmic spiral map on a far right horizontal strip, together with a smaller logarithmic perturbation.