The periodic orbit conjecture for steady Euler flows

TL;DR

This paper proves that the periodic orbit conjecture holds for Euler flows and related vector fields on closed manifolds, extending previous results and using recent characterizations of Eulerisable flows.

Contribution

It demonstrates that the periodic orbit conjecture is valid for Eulerisable flows, broadening the class of vector fields for which the conjecture holds.

Findings

The conjecture is true for Eulerisable flows.

Recent characterizations of Eulerisable flows are utilized.

The result extends the validity of the conjecture beyond geodesible flows.

Abstract

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.