Explicit minimal Scherk saddle towers of arbitrary even genera in $\mathbb{R}^3$

Abstract

Starting from works by Scherk (1835) and by Enneper-Weierstrass (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher. In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996. However, Traizet's construction is implicit and excludes towers, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martín and Ramos Batista, all of them with genus one. For genus two, the first such towers were constructed in 2010. Back to 2009, implicit towers of arbitrary genera were found in An end-to-end construction for singly periodic minimal surfaces. In our present work we obtain explicit minimal Scherk saddle towers, for any given genus $2k$, $k\ge3$.