The qualitative behavior for 𝛼-harmonic maps from a surface with boundary into a sphere

Abstract

Let u α u_\alpha be a sequence of smooth α \alpha -harmonic maps from a compact Riemann surface M M with boundary ∂ M \partial M to a compact Riemannian manifold N N with free boundary u α ( ∂ M ) u_\alpha (\partial M) on a supporting submanifold Γ \Gamma of N N and with uniformly bounded α \alpha -energy. If the target manifold N N is a sphere S K − 1 S^{K-1} , we show that there is no energy loss for such a sequence of maps during the blow-up process as α ↘ 1 \alpha \searrow 1 . Moreover, the image of the weak limit map and bubbles is a connect set. Also, the case of Dirichlet boundary is considered.