MiS Preprint Repository

We investigate the possible blow-up behavior of sequences of Sacks-Uhlenbeck $\alpha$-harmonic maps from a compact Riemann surface with boundary to a compact Riemannian manifold $N$ with a free boundary on a closed submanifold $K\subset N$. We discover and explore a new phenomenon, that the connection between bubbles, instead of being a geodesic joining them, can be a more general curve that involves the geometry of both $N$ and $K$. In technical terms, by comparing the blow-up radius with the distance between the blow-up position and the boundary, we define a new quantity, based on which we show a generalized energy identity for the blow-up sequence and give new length formulas for the necks in the case that there is only one bubble occurring at a boundary blow-up point.