On the blow-up of Yang-Mills fields in dimension four

In this paper, we study the blow-up of a sequence of Yang-Mills connection with bounded energy on a four manifold. We prove a set of equations relating the geometry of the bubble connection at the infinity with the geometry of the limit connection at the energy concentration point. These equations exclude certain scenarios from happening, for example, there is no sequence of Yang-Mills $SU(2)$ connections on $S^4$ converging to an ASD one-instanton while developing a SD one-instanton as a bubble. The proof involves the expansion of connection forms with respect to some Coulomb gauge on long cylinders.