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 maps of the form $u^R (x) = Rx, R \in O(3)$, 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.