A Teichmüller theoretical construction of high genus singly periodic minimal surfaces invariant under a translation (Q1971684)

The author shows the existence of a single periodic minimal surface being invariant under a translation such that a fundamental piece of this surface has \(2m\) planar ends and genus \(m+n+1\), where \(n\) and \(m\) are arbitrarily chosen integers with \(n\geq m\geq 1\). The proof is based on an application of the Weierstraß representation formula for the quotient surface arising from the division of the surface by its translational group. Furthermore, the author discusses techniques to parametrize these surfaces which are helpful for drawing the surfaces.