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 $u\colon \mathbb{S}^n \to \mathbb{S}^\ell$ 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 to a question raised by Mironescu and Brezis--Mironescu.