Stratified integrals and unknots in invisid flows

TL;DR

This paper proves that steady solutions to real analytic Euler equations on a Riemannian 3-sphere have a periodic orbit bounding an embedded disc, extending topological methods to degenerate integrable systems.

Contribution

It extends Fomenko's topological classification to stratified integrals in degenerate cases, linking it to Euler flows on 3-spheres.

Findings

Existence of a periodic orbit bounding an embedded disc in steady Euler flows

Extension of integrable Hamiltonian system topology to degenerate cases

Connection between contact topology and Euler equations

Abstract

We prove that any steady solution to the real analytic Euler equations on a Riemannian 3-sphere must possess a periodic orbit bounding an embedded disc. One key ingredient is an extension of Fomenko's work on the topology of integrable Hamiltonian systems to a degenerate case involving stratified integrals. The result on the Euler equations follows from this when combined with some contact-topological perspectives and a recent result of Hofer, Wyzsocki, and Zehnder.