Periodic orbits of Euler vector fields on 3-manifolds

TL;DR

This paper proves that non-singular steady Euler flows on closed 3-manifolds always have a periodic orbit unless the manifold is a torus bundle over the circle, extending previous results to less regular flows.

Contribution

It generalizes earlier work by weakening the regularity requirement from real analytic to $C^2$, establishing the existence of periodic orbits in broader conditions.

Findings

Flow always has a periodic orbit unless the manifold is a torus bundle over the circle.

Extends previous results by reducing regularity assumptions from real analytic to $C^2$.

Provides conditions under which periodic orbits exist in steady Euler flows.

Abstract

In this paper we study the existence of periodic orbits in the flow of non-singular steady Euler fields $X$ on closed 3-manifolds, that is $X$ is a solution of time independent Euler equations. We show, that when $X$ is $C^2$ the flow always posses a periodic orbit unless the manifold is a torus bundle over the circle. This result generalizes preavious result of J. Etnyre and R. Ghrist by weakening the real analytic hypothesis to $C^2$.