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

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We study s -dependence for minimizing W^{s,n/s} -harmonic maps u\colon \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)].