A family of minimal surfaces and univalent planar harmonic mappings
type: Item
instance of: scholarly article
title: A family of minimal surfaces and univalent planar harmonic mappings
zbMATH Open document ID: 1476.53029
DOI: 10.1155/2014/476061
author: M. Dorff
author: Stacey Muir
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: 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 \textit{W. Hengartner} and \textit{G. Schober} [in: Complex analysis. Basel etc.: Birkhäuser. 87--100 (1988; Zbl 0664.30012)].
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 7022451
zbMATH Keywords: lifting planar harmonic mappings
zbMATH Keywords: curvature estimates
Wikidata QID: Q59038572
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/476061
OpenAlex ID: W2025932615
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description: scientific article; zbMATH DE number 7022451