Asymmetry of curl eigenfields solving Woltjer's variational problem

TL;DR

This paper demonstrates that the lowest curl eigenfields in certain symmetric domains can be asymmetric, challenging the common assumption that eigenfields inherit domain symmetry, with implications for Woltjer's variational principle.

Contribution

It constructs specific domains where the first curl eigenfield is either symmetric or asymmetric, showing eigenfield symmetry is not guaranteed by domain symmetry.

Findings

Some domains have symmetric first curl eigenfields.

Other domains have asymmetric first curl eigenfields.

Eigenfield symmetry does not necessarily follow domain symmetry.

Abstract

We construct families of rotationally symmetric toroidal domains in $\mathbb R^3$ for which the eigenfields associated to the first (positive) Amp\`erian curl eigenvalue are symmetric, and others for which no first eigenfield is symmetric. This implies, in particular, that minimizers of the celebrated Woltjer's variational principle do not need to inherit the rotational symmetry of the domain. This disproves the folk wisdom that the eigenfields corresponding to the lowest curl eigenvalue must be symmetric if the domain is.