Limits of alpha-harmonic maps

Critical points of approximations of the Dirichlet energy `a 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 criti- cal points of such perturbed energies. Indeed, we prove that constant maps and the rotations of S^2 are the only critical points of E_α for maps from S^2 to S^2 whose α-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of α-harmonic maps.