  We say that a bounded domain $\Omega$ is optimal for the first positive curl
eigenvalue $\mu_1(\Omega)$ if $\mu_1(\Omega)\leq \mu_1(\Omega')$ for any domain
$\Omega'$ with the same volume. In spite of the fact that $\mu_1(\Omega)$ is
uniformly lower bounded in terms of the volume, in this paper we prove that
there are no axisymmetric optimal (and even locally minimizing) domains with
$C^{2,\alpha}$ boundary that satisfies a mild technical assumption. As a
particular case, this rules out the existence of $C^{2,\alpha}$ optimal
axisymmetric domains with a convex section. An analogous result holds in the
case of the first negative curl eigenvalue.
