A family of minimal surfaces and univalent planar harmonic mappings (Q1724200)

Summary: 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 \textit{W. Hengartner} and \textit{G. Schober} [in: Complex analysis. Basel etc.: Birkhäuser. 87--100 (1988; Zbl 0664.30012)].