A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

Michael Dorff and Stacey Muir

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

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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https://doi.org/10.1155/2014/476061

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:476061

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