Gluing Karcher-Scherk saddle towers II: Singly periodic minimal surfaces

TL;DR

This paper constructs new singly periodic minimal surfaces with Scherk ends by gluing Karcher--Scherk saddle towers, carefully analyzing embeddedness and correcting previous misconceptions about their bending behavior.

Contribution

It introduces a method to construct and analyze embedded singly periodic minimal surfaces with Scherk ends, correcting prior errors regarding their bending and embeddedness.

Findings

Successfully constructed new minimal surfaces with prescribed phase differences.

Identified conditions for embeddedness by analyzing slight bendings.

Corrected previous misconceptions about the bending behavior of Scherk ends.

Abstract

This is the second in a series of papers that construct minimal surfaces by gluing singly periodic Karcher--Scherk saddle towers along their wings. This paper aims to construct singly periodic minimal surfaces with Scherk ends. As in the first paper, we prescribe phase differences between saddle towers, and obtain many new examples without any horizontal reflection plane. This construction is not very different from previous ones, hence we will only provide sketched proofs. We will however study the embeddedness with great care. Previously, embeddedness can not be determined in the presence of "parallel" Scherk ends, as it was not clear if they bend towards or away from each other. In a recent study, the bending was completely ignored and embeddedness was falsely claimed. We correct this mistake by carefully analysing slight bendings, thus identify scenarios where the constructed surfaces are indeed embedded.