A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

Michael Dorff and Stacey Muir

Abstract and Applied Analysis, 2014, vol. 2014, 1-8

Abstract:

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.

Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:476061

DOI: 10.1155/2014/476061

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