Minimizing and non-minimizing degree one $W^{s,1\mathrm{/}s}$-harmonic maps between spheres

We show that $id:{\mathbb{S}}^1\to {\mathbb{S}}^1$ is \emph{not} a minimizing $W^{s,\frac{1}{s}}$-harmonic map for $s\in (0,\frac{1}{8}$). On the other hand, for $s\in (\frac{1}{3},1)$ it is a local minimizing map, and for $s\in [\frac{1}{2}-\epsilon ,\frac{1}{2}+\epsilon ]$ it is a global minimizer. The usual extension or Fourier techniques being unavailable, our argument relies instead of stability analysis in $s$.