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

TL;DR

This paper investigates how minimal $W^{s,n/s}$-harmonic maps between spheres depend on the parameter s, showing local constancy of homotopy generators and continuous energy variation, advancing understanding in geometric analysis.

Contribution

It demonstrates that the generating subset of homotopy classes can be chosen locally constant in s and that minimal energies vary continuously with s, addressing open questions.

Findings

Generating subset of homotopy classes is locally constant in s.

Minimal $W^{s,n/s}$-energy varies continuously with s.

Progress on open questions by Mironescu and Brezis--Mironescu.

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.