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

Let a closed -dimensional manifold, be a closed manifold, and let for . We extend the monumental work of Sacks and Uhlenbeck by proving that if , then there exists a minimizing -harmonic map homotopic to . If , then we prove that there exists a -harmonic map from to in a generating set of . Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular, when , 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 nonscaling invariant energies. Moreover, we prove the regularity theory for minimizing -maps into manifolds.