Gluing doubly periodic Scherk surfaces into minimal surfaces

TL;DR

This paper develops a method to construct new minimal surfaces by stacking and gluing doubly periodic Scherk surfaces, revealing limitations in extending known constructions beyond specific lattice configurations.

Contribution

It demonstrates that, except for special lattice cases, gluing multiple Scherk surfaces only yields known minimal surfaces like the trivial, KMR, or Meeks' examples.

Findings

Gluing multiple Scherk surfaces generally produces only known minimal surfaces.

The construction is limited to specific lattice configurations, notably the triangular horizontal lattice.

New minimal surfaces beyond known examples are not achievable through this gluing method in most cases.

Abstract

We construct minimal surfaces by stacking doubly periodic Scherk surfaces one above another and gluing them along their ends. It is previously known that the Karcher--Meeks--Rosenberg (KMR) doubly periodic minimal surfaces and Meeks' family of triply periodic minimal surfaces can both be obtained by gluing two Scherk surfaces. There have been hope and failed attempts to glue more Scherk surfaces. But our analysis shows that: Except for the special case where the doubly periodic Scherk surfaces all have triangular horizontal lattice, a glue construction can only produce the trivial Scherk surface itself, the KMR examples, or Meeks' surfaces.