Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry. (English) Zbl 1356.31001

Bucaj, Valmir; Cannon, Sarah; Dorff, Michael; Lawson, Jamal; Viertel, Ryan

Involve 6, No. 4, 383-392 (2013).

Summary: The singly periodic Scherk surfaces with higher dihedral symmetry have \(2n\)-ends that come together based upon the value of \(\phi\). These surfaces are embedded provided that \(\frac{\pi}{2}-\frac{\pi}{n}<\frac{n-1}{n}\phi<\frac{\pi}{2}\). 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.

MSC:

31A05	Harmonic, subharmonic, superharmonic functions in two dimensions
49Q05	Minimal surfaces and optimization
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

harmonic mappings; minimal surfaces