Topics in Geometric Analysis

Tobias Lamm

(KIT Karlsruhe)

Limits of α-harmonic maps

In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the Dirichlet energy, the so-called α-energy E_α, 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 E_α 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 E_α 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.

Tristan Rivière

(ETH Zürich)

Minmax hierarchies for minimal surfaces and harmonic maps