A note on balance equations for doubly periodic minimal surfaces

TL;DR

This paper derives a differential equation equivalent to the balance equations for doubly periodic minimal surfaces, facilitating the generation of more complex surfaces by expanding the solution space.

Contribution

It introduces a differential equation that simplifies solving the balance equations, enabling the construction of a broader class of doubly periodic minimal surfaces.

Findings

Derived a differential equation equivalent to balance equations.

Enabled generation of more solutions and complex surfaces.

Expanded the known family of doubly periodic minimal surfaces.

Abstract

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a corresponding three-parameter family of doubly periodic minimal surfaces. In this note we derive a differential equation that is equivalent to the balance equations for doubly periodic minimal surfaces. This allows for the generation of many more solutions to the balance equations, enabling the construction of increasingly complicated surfaces.