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

Abstract

Let Σ a closed n-dimensional manifold, N ⊂ R M be a closed manifold, and u ∈ W s, n s (Σ, N ) for s ∈ (0, 1). We extend the monumental work of Sacks and Uhlenbeck by proving that if π n (N ) = {0} then there exists a minimizing W s, n s -harmonic map homotopic to u. If π n (N ) = {0}, then we prove that there exists a W s, n s -harmonic map from S n to N in a generating set of π n (N ).Since several techniques, especially Pohozaev-type 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 non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing W s, n s -maps into manifolds. ContentsW s, n s -HARMONIC MAPS IN HOMOTOPY CLASSES 5• K.M.

A fractional version of Rivière's GL(N)-gauge

“…. , K. Then by (4.32) we get In order to conclude we will need a removability of singularities lemma, compare with [18,Proposition 4.7].…”