Tobias Lamm, Tristan Rivière
Limits of α-harmonic maps
In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the Dirichlet energy, the so-called α-energy Eα, α>1, to construct non-trivial harmonic maps of the two-sphere in manifolds with a non-contractible universal cover. The Dirichlet energy corresponds to α=1 and, as α decreases to 1, critical points of Eα 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 S2 are the only critical points of Eα for maps from S2 to S2 whose α-energy lies below some threshold, which is independent of α (sufficiently close to 1). In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of α-harmonic maps. We also show the optimality of our threshold assumption.
Minmax hierarchies for minimal surfaces and harmonic maps