Yang-Mills replacement

Abstract

We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function v: ... defined on a surface ... and replacing its values on a small ball B2 ... with a harmonic function u that has the same values as v on the boundary &B2 . The resulting function on ... has lower energy, and repeating this process on balls covering ..., 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 ... with a Yang-Mills connection A that has the same restriction to the boundary [alpha]B4 as B, and we obtain bounds on the difference ... in terms of the drop in energy. Throughout, we work with connections of the lowest possible regularity ... (X), the natural choice for this context, and so our gauge transformations are in ... (X) and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.