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 (see [9, 10]). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see [23]). 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 Martin and Ramos Batista (see [13]), all of them with genus one. For genus two, the first such towers were constructed in 2010 (see [22]). Back to 2009, implicit towers of arbitrary genera were found in [5]. In our present work we obtain explicit minimal Scherk saddle towers, for any given genus 2k, k 3.