Univalent harmonic mappings and lift to the minimal surfaces

TL;DR

This paper constructs specific univalent harmonic mappings with convex horizontal images and varying dilatation, and explores their associated minimal surfaces, solving a recent open problem and illustrating results with computational visualizations.

Contribution

It introduces new univalent harmonic mappings with convex horizontal images and varying dilatation, and links these to minimal surfaces, addressing a recent open problem.

Findings

Construction of sense-preserving univalent harmonic mappings

Association of these mappings with minimal surfaces

Illustrative visualizations using Mathematica

Abstract

We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr. Appl. Anal. (2014)). In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of \texttt{Mathematica} software.