Existence and structure of symmetric Beltrami flows on compact $3$-manifolds

TL;DR

This paper proves the existence of symmetric Beltrami eigenvector fields on compact 3-manifolds with boundary, and extends Arnold's structure theorem to these symmetric eigenfields under certain conditions.

Contribution

It establishes the existence of curl eigenfields invariant under manifold symmetries and extends Arnold's structure theorem to symmetric, real analytic Beltrami fields.

Findings

Existence of symmetric curl eigenfields on compact 3-manifolds.

Extension of Arnold's structure theorem to symmetric Beltrami fields.

Characterization of flows of real analytic Killing fields.

Abstract

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $\phi_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field $X$, tangent to the boundary, which is an eigenfield of curl and satisfies $(\phi_t)_{*}X=X$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\bar{M},g)$ has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable $3$-manifolds with and without boundary.