Low-energy α -harmonic maps into the round sphere

We classify low-energy $\alpha$ -harmonic maps from a closed non-spherical Riemannian surface $\Sigma$ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as $1<\alpha \downarrow 1$ and assuming $E_{\alpha }$ is close to $\left|\Sigma \right|+4\pi$ then degree-one $\alpha$ -harmonic maps blow a bubble based at a critical point $a_c$ of a an explicit function $J$ and at scale $\sqrt{|J\left(a_c\right){|}^{-1}(\alpha -1)}.$ Up to a constant, $J$ is the sum of the squares of any $L^2$ -orthonormal basis of holomorphic one-forms on the domain.