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.