Asymmetry of Curl Eigenfields Solving Woltjer’s Variational Problem

Abstract We construct families of rotationally symmetric toroidal domains in $${\mathbb {R}}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> for which the eigenfields associated to the first (positive) Ampèrian 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.