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

Abstract

Let Σ a closed 𝑛-dimensional manifold,  ⊂ ℝ 𝑀 be a closed manifold, and let 𝑢 ∈ 𝑊 𝑠, 𝑛 𝑠 (Σ,  ) for 𝑠 ∈ (0, 1). We extend the monumental work of Sacks and Uhlenbeck by proving that if 𝜋 𝑛 ( ) = {0}, then there exists a minimizing 𝑊 𝑠, 𝑛 𝑠 -harmonic map homotopic to 𝑢. If 𝜋 𝑛 ( ) ≠ {0}, then we prove that there exists a 𝑊 𝑠, 𝑛 𝑠harmonic map from 𝕊 𝑛 to  in a generating set of 𝜋 𝑛 ( ). Since several techniques, especially Pohozaevtype arguments, are unknown in the fractional framework (in particular, when 𝑛 𝑠 ≠ 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 nonscaling invariant energies. Moreover, we prove the regularity theory for minimizing 𝑊 𝑠, 𝑛 𝑠 -maps into manifolds.