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A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

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  • Michael Dorff
  • Stacey Muir

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.

Suggested Citation

  • Michael Dorff & Stacey Muir, 2014. "A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, April.
    • Handle: RePEc:hin:jnlaaa:476061
    DOI: 10.1155/2014/476061
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