Low-energy α-harmonic maps into the round sphere

We classify low-energy α α -harmonic maps from a closed non-spherical Riemannian surface Σ Σ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as 1<α ↓ 1 1 < α ↓ 1 and assuming E_α E α is close to |Σ |+4π | Σ | + 4 π then degree-one α α -harmonic maps blow a bubble based at a critical point ac a c of a an explicit function J J and at scale √ |J(ac)|⁻¹(α -1). | J ( a c ) | - 1 ( α - 1 ) . Up to a constant, J J is the sum of the squares of any L² L 2 -orthonormal basis of holomorphic one-forms on the domain.