We present a two-parameter family of minimal surfaces constructed by lifting a family of planar harmonic mappings. In the process, we use the Clunie and Sheil-Small shear construction for planar harmonic mappings convex in one direction. This family of minimal surfaces, through a continuous transformation, has connections with three well-known surfaces: Enneper’s surface, the wavy plane, and the helicoid. Moreover, the identification process used to recognize the surfaces provides a connection to surfaces that give tight bounds on curvature estimates first studied in a 1988 work by Hengartner and Schober.
Michael Dorff. Stacey Muir. "A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings." Abstr. Appl. Anal. 2014 (SI07) 1 - 8, 2014. https://doi.org/10.1155/2014/476061