Harmonic Shears of Slit and Polygonal Mappings

TL;DR

This paper explores harmonic mappings via shear construction, mapping the unit disk onto slit domains and regular polygons, and examines their associated minimal surfaces with visualizations.

Contribution

It introduces new classes of harmonic mappings onto slit and polygonal domains using shear construction and analyzes their minimal surfaces.

Findings

Mappings onto four-slit domains and regular n-gons are constructed.

Associated minimal surfaces are characterized and visualized.

The shear construction effectively generates harmonic mappings with specific geometric properties.

Abstract

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations \omega. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.