A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

Abstract EN

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.