Yang-Mills Replacement

We develop an analog of harmonic replacement in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron. The technique, as introduced by Jost and further developed by Colding and Minicozzi, involves taking a map $v\colon\Sigma\to M$ defined on a surface $\Sigma$ and replacing its values on a small ball $B^2\subset\Sigma$ with a harmonic map $u$ that has the same values as $v$ on the boundary $\partial B^2$. The resulting map on $\Sigma$ has lower energy, and repeating this process on balls covering $\Sigma$, one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection $B$ on a bundle over a four-manifold $X$, and replace it on a small ball $B^4\subset X$ with a Yang--Mills connection $A$ that has the same restriction to the boundary $\partial B^4$ as $B$. As in the harmonic replacement results of Colding and Minicozzi, we have bounds on the difference $\lVert B-A\rVert_{L^2_1(X)}^2$ in terms of the drop in energy, and we only require that the connection $B$ have small energy on the ball, rather than small $C^0$ oscillation. Throughout, we work with connections of the lowest possible regularity $L^2_1(X)$, the natural choice for this context, and so our gauge transformations are in $L^2_2(X)$ and therefore almost but not quite continuous, leading to more delicate arguments than in higher regularity.