Embeddedness for singly periodic Scherk surfaces with higher dihedral symmetry

Abstract

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.