Periodic orbits of analytic Euler fields on 3-manifolds

TL;DR

This paper provides an alternative proof that on most closed Riemannian 3-manifolds, nonvanishing analytic solutions to stationary Euler equations necessarily have periodic trajectories, extending previous results in fluid dynamics and topology.

Contribution

It offers a new proof of the existence of periodic orbits for analytic Euler fields on certain 3-manifolds, broadening the understanding of fluid flows in geometric contexts.

Findings

Nonvanishing analytic solutions have periodic trajectories on most 3-manifolds.

The result applies to all closed Riemannian 3-manifolds except torus bundles.

An alternative proof to existing theorems is provided.

Abstract

On any closed Riemannian 3-manifold which is not a torus bundle, every nonvanishing analytic solution of the stationary Euler equations has a periodic trajectory. This result is originally due to A. Rechtman (arXiv:0904.2719) and K. Cieliebak and E. Volkov (arXiv:1402.6484); here we present an alternative proof of it.