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Embeddedness for
singly periodic Scherk surfaces with higher dihedral
symmetry
Valmir Bucaj, Sarah Cannon, Michael Dorff, Jamal Lawson and Ryan Viertel
Vol. 6 (2013), No. 4, 383–392
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Abstract |
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The singly periodic Scherk surfaces with higher dihedral symmetry have -ends that come together based upon the value of . These surfaces are embedded provided that . Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins–Serrin solution and Krust’s theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces. |
Keywords
minimal surfaces, harmonic mappings, Scherk, univalence
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Mathematical Subject Classification 2010
Primary: 30C45, 49Q05, 53A10
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Milestones
Received: 23 May 2012
Revised: 24 July 2012
Accepted: 25 July 2012
Published: 8 October 2013
Communicated by Frank Morgan
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Authors |
| Valmir Bucaj | |
| Computer Science, Information
Systems and Mathematics Texas Lutheran University Seguin, TX 78155 United States |
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| Sarah Cannon | |
| Department of Mathematics Tufts University Medford, MA 02155 United States |
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| Michael Dorff | |
| Department of Mathematics Brigham Young University Provo, UT 84602 United States |
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| Jamal Lawson | |
| Mathematical Sciences Loyola University New Orleans New Orleans, LA 70118 United States |
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| Ryan Viertel | |
| Department of Mathematics Brigham Young University Provo, UT 84602 United States |
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