s-stability for W^{s,n/s}-harmonic maps in homotopy groups

We study $s$-dependence for minimizing $W^{s,n\mathrm{/}s}$-harmonic maps $u\text{\hspace{0.17em}⁣}:{\mathbb{S}}^n\to {\mathbb{S}}^{\mathrm{\ell }}$ in homotopy classes. Sacks–Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of ${\pi }_n({\mathbb{S}}^{\mathrm{\ell }})$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies the minimal $W^{s,n\mathrm{/}s}$-energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis–Mironescu.