A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings. (3rd April 2014)

• Record Type: Journal Article
• Authors: Dorff, Michael; Muir, Stacey
• Other Names: Ali Rosihan M. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-04-03
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/476061
• Languages: English
• ISSNs: 1085-3375

Abstract

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.