THE QUALITATIVE BEHAVIOR FOR α-HARMONIC MAPS FROM A SURFACE WITH BOUNDARY INTO A SPHERE

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