  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 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.
