LAUSR

Given a principal bundle $$P\rightarrow M$$P→M over a Riemannian manifold with compact structure group G, let us consider a stationary Yang–Mills connection A with energy $$\int _M |F_A|^2\le \Lambda $$∫M|FA|2≤Λ. If we consider a sequence of such connections $$A_i$$Ai, then it is understood by Tian (Ann Math 151(1):193–268, 2000) that up to subsequence we can converge $$A_i\rightarrow A$$Ai→A to a singular limit connection such that the energy measures converge $$|F_{A_i}|^2 dv_g\rightarrow |F_A|^2dv_g +\nu $$|FAi|2dvg→|FA|2dvg+ν, where $$\nu =e(x)d\lambda ^{n-4}$$ν=e(x)dλn-4 is the $$n-4$$n-4 rectifiable defect measure. Our main result is to show, without additional assumptions, that for $$n-4$$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 $$S^4$$S4 itself. This energy quantization was proved in Rivière (Commun Anal Geom 10(4):683–708, 2002) assuming a uniform $$L^1$$L1 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 $$\int _M |\nabla ^2 F_A| < C(\Lambda ,\dim G,M)$$∫M|∇2FA|