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

Abstract

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