No. 2020.13
Existence and structure of symmetric Beltrami flows on compact 3-manifolds
W. Gerner
Subject: Beltrami fields, helicity, (magneto-)hydrodynamics, field line dynamics, killing fields, isometries

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 $\bm{X}$, tangent to the boundary, which is an eigenfield of curl and satisfies $(\phi_t)_{*}\bm{X}=\bm{X}$, 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.

Reference

Differential Geometry and its Applications, Volume 78, 2021, 101801,ISSN 0926-2245

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arXiv:2006.14507