Variational Methods in Geometry Seminar

In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the Dirichlet energy, the so-called $\alpha$-energy $E_\alpha$, $\alpha > 1$, to construct non-trivial harmonic maps of the two-sphere in manifolds with a non-contractible universal cover. The Dirichlet energy corresponds to $\alpha = 1$ and, as $\alpha$ decreases to 1, critical points of $E_\alpha$ are known to converge to harmonic maps in a suitable sense. However, in a joint work with Andrea Malchiodi and Mario Micallef, 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, which is independent of $\alpha$ (sufficiently close to 1). In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $\alpha$-harmonic maps. We also show the optimality of our threshold assumption.