# A characterization of 3D steady Euler flows using commuting zero-flux homologies

## TL;DR

This paper characterizes 3D steady Euler flows via commuting zero-flux homologies, extending Sullivan's work, and shows such flows cannot be constructed with plugs, with analogous higher-dimensional results.

## Contribution

It introduces a homological characterization of steady Euler flows on 3-manifolds, extending previous work and providing new obstructions to flow constructions.

## Key findings

- Steady Euler flows are characterized by commuting zero-flux homologies.
- Such flows cannot be realized using plug constructions like Wilson's or Kuperberg's.
- Results are extended to higher-dimensional manifolds.

## Abstract

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.