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

TL;DR

This paper constructs explicit minimal Scherk saddle towers of arbitrary even genus in , extending previous implicit and genus-limited examples to all even genera with a direct explicit approach.

Contribution

It provides the first explicit constructions of minimal Scherk saddle towers for any even genus, broadening the class of known minimal surfaces.

Findings

Explicit minimal Scherk saddle towers for all even genera are constructed.

Previous constructions were implicit or limited to genus one.

The new towers extend the known family of minimal surfaces in .

Abstract

Starting from works by Scherk (1835) and by Enneper-Weierstra\ss \ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}). However, Traizet's construction is implicit and excludes {\it 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 \cite{Martin}), all of them with genus one. For genus two, the first such towers were constructed in 2010 (see \cite{Valerio2}). Back to 2009, implicit towers of arbitrary genera were found in \cite{HMM}. In our present work we obtain {\it explicit} minimal Scherk saddle towers, for any given genus $2k$, $k\ge3$.