A singularity removal theorem for Yang-Mills fields in higher dimensions

Abstract:

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price’s monotonicity formula, which allows us to assert that the curvature is small not only in the $L^2$ norm, but also in the Morrey norm $M_2^{n/2}$. The second ingredient is a new inductive (averaged radial) gauge construction and truncation argument which allows us to approximate a singular gauge as a weak limit of smooth gauges with curvature small in the Morrey norm. The second ingredient is a variant of Uhlenbeck’s lemma, allowing one to place a smooth connection into the Coulomb gauge whenever the Morrey norm of the curvature is small; This variant was also proved independently by Meyer and Rivière. It follows easily from this variant that a $W^{1,2}$-connection can be placed in the Coulomb gauge if it can be approximated by smooth connections whose curvatures have small Morrey norm.