Scherk Saddle Towers of Genus Two in $\R^3$

Abstract

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desingularization of such ``forbidden'' planes into an embedded singly periodic minimal surface. We give explicit examples of genus two and discuss some advances regarding this problem. Moreover, our examples are the first ones containing {\it Gaussian geodesics}, and for the first time we prove embeddedness of the surfaces CSSCFF and CSSCCC from Callahan-Hoffman-Meeks-Wohlgemuth.