Ben Sharp

Low energy $\alpha$-harmonic maps into the round sphere

In the celebrated works of Sacks-Uhlenbeck, a sub-critical functional $E_\alpha$ is utilised in order to prove existence of non-trivial harmonic maps from a surface to a Riemannian manifold. For $\alpha >1$ this functional satisfies the Palais-Smale condition, meaning that it is easy to find critical points ($\alpha$-harmonic maps) e.g. via direct minimisation or min-max methods. Moreover all $\alpha$-harmonic maps are smooth up to some uniform regularity scale which depends on $\alpha$. As $\alpha \to 1$, $E_\alpha$ becomes the Dirichlet energy and the regularity scale may degenerate leading to the formation of harmonic spheres and potentially long geodesic necks. Nevertheless, one recovers harmonic maps as a result of sending $\alpha$ to 1. We will classify low-energy $\alpha$-harmonic maps from non-spherical surfaces to the round two-sphere via their bubble scales and centres (e.g. where a bubble is blown and at what rate). When a bubble is blown, this information is determined entirely by the behaviour of holomorphic one-forms on the domain.