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

We classify low-energy $\alpha$-harmonic maps from a closed non-spherical Riemannian surface $\mathrm{\Sigma }$ 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 $\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }+4\pi$ then degree-one $\alpha$-harmonic maps blow a bubble based at a critical point $a_c$ of a an explicit function $\mathcal{J}$ and at scale $\sqrt{\mathrm{\mid }\mathcal{J}(a_c){\mathrm{\mid }}^{-1}(\alpha -1)}$. Up to a constant, $\mathcal{J}$ is the sum of the squares of any $L^2$-orthonormal basis of holomorphic one-forms on the domain.