Contact Topology and Hydrodynamics

TL;DR

This paper establishes a deep connection between contact topology and hydrodynamics by characterizing Beltrami fields as Reeb fields, leading to significant implications for the existence of closed flowlines in fluid flows on three-manifolds.

Contribution

It provides a metric-independent characterization of Beltrami fields via contact topology and applies recent results to prove the existence of closed flowlines in certain steady fluid flows.

Findings

Reeb fields are equivalent to rotational Beltrami fields on 3-manifolds.

Recent solutions to the Weinstein Conjecture imply closed orbits for Beltrami flows on S^3.

Conditions for closed flowlines in Euler flows on T^3 are derived from homotopy data.

Abstract

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields on three-manifolds. Thus, we characterise Beltrami fields in a metric-independant manner. This correspondence yields a hydrodynamical reformulation of the Weinstein Conjecture, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all $C^\infty$ rotational Beltrami flows on $S^3$. This is the key step for a positive solution to the hydrodynamical Seifert Conjecture: all $C^\omega$ steady state flows of a perfect incompressible fluid on $S^3$ possess closed flowlines. In the case of Euler flows on $T^3$, we give general conditions for closed flowlines derived from the homotopy data of the normal bundle to the flow.