Energy identity for stationary Yang Mills

Given a principal bundle \(P\rightarrow M\) over a Riemannian manifold with compact structure group G, let us consider a stationary Yang–Mills connection A with energy \(\int _M |F_A|^2\le \Lambda \). If we consider a sequence of such connections \(A_i\), then it is understood by Tian (Ann Math 151(1):193–268, 2000) that up to subsequence we can converge \(A_i\rightarrow A\) to a singular limit connection such that the energy measures converge \(|F_{A_i}|^2 dv_g\rightarrow |F_A|^2dv_g +\nu \), where \(\nu =e(x)d\lambda ^{n-4}\) is the \(n-4\) rectifiable defect measure. Our main result is to show, without additional assumptions, that for \(n-4\) a.e. point the energy density e(x) may be computed explicitly as the sum of the bubble energies arising from blow ups at x. Each of these bubbles may be realized as a Yang Mills connection over \(S^4\) itself. This energy quantization was proved in Riviere (Commun Anal Geom 10(4):683–708, 2002) assuming a uniform \(L^1\) hessian bound on the curvatures in the sequence. In fact, our second main theorem is to show this hessian bound holds automatically. Precisely, given a connection A as above we have the a-priori estimate \(\int _M |\nabla ^2 F_A| < C(\Lambda ,\dim G,M)\) for the curvature. It is important to note this result is proved in tandem with the energy quantization, and not before it. Indeed, we will in fact prove an effective version of the energy identity, and it is this effective version which will lead to both the \(L^1\) hessian bound and the classical energy quantization results. In the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections. To produce strongest estimates in the paper we introduce an \(\epsilon \)-gauge condition, which generalizes the usual Coulomb gauge and which will exist, with effective control, even over singular regions. On these \(\epsilon \)-gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher-dimensional annular regions.