Coulomb Gauges and Regularity for Stationary Weak Yang$-$Mills Connections in Supercritical Dimension

We prove that stationary Yang$-$Mills fields in dimensions 5 belonging to the variational class of weak connections are smooth away from a closed singular set $S$ of vanishing 1-dimensional Hausdorff measure. Our proof is based on an $\epsilon$-regularity theorem, which generalizes to this class of weak connections the existing previous $\epsilon$-regularity results by G. Tian for smooth connections, by Y. Meyer and the second author for Sobolev and approximable connections, and by T. Tao and G. Tian for admissible connections (which are weak limits of smooth Yang$-$Mills fields). On the path towards establishing $\epsilon$-regularity, a pivotal step is the construction of controlled Coulomb gauges for general weak connections under small Morrey norm assumptions.