Morrey Saces and Regularity for Yang-Mills Higgs Equations

We start with background on regularity theory for the equations of gauge theory. Morrey spaces arise naturally from monotinicity theorems in dimensions greater than 4. Our main technical result is that functions in a Morrey space which satisfy an elliptic inequality off a singular set of Hausdorf codimension 4 can be bounded in a much better Morrey space in the interior. This can be used to show that when the curvature is small in the relevant Morrey space, solutions of a Yang-Mills-Higgs equation off a singular set of codimensions 4 are in interior domains gauge equation to a connection which is smooth. This simplifies a classic paper of Tao-Tian.