Relaxed Yang-Mills functional over 4-manifolds (Q1922766)

Let \(M\) be a \(4\)-dimensional compact orientable Riemannian manifold and \({\mathcal P}_k\) the space of all principal \(SU(2)\)-bundles over \(M\) with second Chern number \(-k\). For a \(SU(2)\)-bundle \(P\) in \({\mathcal P}_k\), the second Chern class can be considered as a Radon measure on \(M\). Denote by \({\mathcal R}(M)\) the space of Radon measures on \(M\). Then, given a sequence of connections \(A_i\) with bounded Yang-Mills energy, the weak limit of the measure \(\text{ tr} (F_{A_i}\wedge F_{A_i})/8\pi^2\), if it exists, comprises a subspace in \({\mathcal R}(M)\), which is denoted by \({\mathcal P}^2_k\) in the author's notation. The author proves that any element in this space is written as \(\text{ tr} (F_{A_i}\wedge F_{A_i})/8\pi^2 + \sum d_j\delta_{a_j}\), where \(d_j\in {\mathbb{Z}}\) and \(\delta_{a_j}\) is the Dirac measure supported at \(a_j\), that the Yang-Mills functional can be extended over the space \({\mathcal P}^2_k\), and that any minimizing sequence relative to this extended functional has limit inside \({\mathcal P}^2_k\). A regularity theorem of the limit is given in certain cases.