Univalent Harmonic Mappings and Lift to the Minimal Surfaces

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. In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of Mathematica software.