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

TL;DR

This paper constructs new genus two singly periodic minimal surfaces called saddle-towers, including the first examples with Gaussian geodesics, and proves embeddedness of specific surfaces, advancing the understanding of minimal surface desingularizations.

Contribution

It introduces explicit genus two saddle-tower minimal surfaces that desingularize intersecting planes with multiple lines, including the first with Gaussian geodesics and embeddedness proofs.

Findings

First examples of genus two saddle-tower minimal surfaces with Gaussian geodesics.

Proof of embeddedness for surfaces CSSCFF and CSSCCC.

Explicit constructions expanding the class of known minimal surfaces.

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.