Limits of $\alpha$-harmonic maps

Abstract

Critical points of approximations of the Dirichlet energy à la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are the only critical points of $E_{\alpha }$ for maps from $S^2$ to $S^2$ whose $\alpha$-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $\alpha$-harmonic maps.