Compactness Theorems for Invariant Connections

The Palais-Smale Condition C holds for the Yang-Mills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for Yang-Mills over compact manifolds of dimension ≥ 4. The example of SO(3)-invariant SU(2)-connections over S, see [2], [14], and [16], suggested that Condition C holds for Yang-Mills over compact manifolds of any dimension when restricted to connections that are invariant under a group action on the manifold with orbits of codimension ≤ 3. Such a result, essentially Theorem 3 below, was established by T. Parker [14]. In this paper we generalize his result.