A Compactness Theorem for Invariant Connections

It is well-known that the Palais-Smale Condition C does not hold for the YangMills functional on a principal bundle over a compact four-manifold. According to the Uhlenbeck weak compactness theorem, a Palais-Smale sequence will in general only have a subsequence that converges on the complement of a finite set of points. Moreover, even on the complement of these points, the convergence is not as good as one would desire. One only gets convergence with one derivative in L. It is inconvenient to work with connections with one derivative in L as there is no slice theorem for such connections.