The qualitative behavior for $\alpha$-harmonic maps from a surface with boundary into a sphere

Abstract:

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