A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings
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 transfor...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/476061 |
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| _version_ | 1832560598106243072 |
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| author | Michael Dorff Stacey Muir |
| author_facet | Michael Dorff Stacey Muir |
| author_sort | Michael Dorff |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5bd1a84eba764199a52c6180c654aab5 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-5bd1a84eba764199a52c6180c654aab52025-02-03T01:27:11ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/476061476061A Family of Minimal Surfaces and Univalent Planar Harmonic MappingsMichael Dorff0Stacey Muir1Department of Mathematics, Brigham Young University, Provo, UT 84602, USADepartment of Mathematics, University of Scranton, Scranton, PA 18510, USAWe 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.http://dx.doi.org/10.1155/2014/476061 |
| spellingShingle | Michael Dorff Stacey Muir A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings Abstract and Applied Analysis |
| title | A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings |
| title_full | A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings |
| title_fullStr | A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings |
| title_full_unstemmed | A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings |
| title_short | A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings |
| title_sort | family of minimal surfaces and univalent planar harmonic mappings |
| url | http://dx.doi.org/10.1155/2014/476061 |
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