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2013 Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry
Valmir Bucaj, Sarah Cannon, Michael Dorff, Jamal Lawson, Ryan Viertel
Involve 6(4): 383-392 (2013). DOI: 10.2140/involve.2013.6.383


The singly periodic Scherk surfaces with higher dihedral symmetry have 2n-ends that come together based upon the value of φ. These surfaces are embedded provided that π2πn<n1nφ<π2. 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.


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Valmir Bucaj. Sarah Cannon. Michael Dorff. Jamal Lawson. Ryan Viertel. "Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry." Involve 6 (4) 383 - 392, 2013.


Received: 23 May 2012; Revised: 24 July 2012; Accepted: 25 July 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1356.31001
MathSciNet: MR3115973
Digital Object Identifier: 10.2140/involve.2013.6.383

Primary: 30C45 , 49Q05 , 53A10

Keywords: harmonic mappings , minimal surfaces , Scherk , univalence

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.6 • No. 4 • 2013
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