Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes

TL;DR

This paper extends classical harmonic map results to fractional Sobolev spaces, proving existence and regularity of minimal fractional harmonic maps in various homotopy classes, and developing new analytical tools for the fractional setting.

Contribution

It introduces new methods for fractional harmonic maps, including removability of singularities and energy estimates, and establishes existence and regularity results in the fractional framework.

Findings

Existence of minimizing fractional harmonic maps in trivial homotopy classes.

Existence of non-trivial fractional harmonic maps in non-trivial homotopy classes.

Development of new analytical tools for fractional harmonic map theory.

Abstract

Let $\Sigma$ a closed $n$-dimensional manifold, $\mathcal{N} \subset \mathbb{R}^M$ be a closed manifold, and $u \in W^{s,\frac ns}(\Sigma,\mathcal{N})$ for $s\in(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if $\pi_n(\mathcal{N})=\{0\}$ then there exists a minimizing $W^{s,\frac ns}$-harmonic map homotopic to $u$. If $\pi_n(\mathcal{N})\neq \{0\}$, then we prove that there exists a $W^{s,\frac{n}{s}}$-harmonic map from $\mathbb{S}^n$ to $\mathcal{N}$ in a generating set of $\pi_{n}(\mathcal{N})$. Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when $\frac{n}{s} \neq 2$ one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing $W^{s,\frac{n}{s}}$-maps into manifolds.