Helicoids and vortices

TL;DR

This paper reveals a novel link between fluid vortex arrangements and minimal surface theory, enabling the discovery of new minimal surfaces and vortex crystals through a comparative approach.

Contribution

It introduces a new connection between vortex crystals and minimal surfaces, leading to the construction of new examples in both fields.

Findings

New minimal surfaces derived from vortex crystal configurations

Identification of new vortex crystal arrangements

Establishment of a correspondence between vortex dynamics and minimal surface geometry

Abstract

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a "vortex crystal", an equilibrium of point vortices in 2D fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.