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

Abstract

We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps $u:{\mathbb{S}}^n\to {\mathbb{S}}^{\ell }$ in homotopy classes. Sacks–Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of ${\pi }_n({\mathbb{S}}^{\ell })$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies, the minimal $W^{s,n/s}$-energy in each homotopy class changes continuously. In particular, we provide progress on a question raised by Mironescu [in: Perspectives in nonlinear partial differential equations (2007), 413–436] and Brezis–Mironescu [Sobolev maps to the circle (2021)].