Geometry & Topology Seminar: "Yang-Mills Replacement"

Abstract: We develop an analog of Jost's harmonic replacement technique in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique 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 function 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 $L^2_1$ norm of the difference $B-A$ 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 as in Jost's work. 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.

Given a map, we can change its values on a small ball in the domain: On the small ball, we replace the original map with a harmonic map with the same boundary values. I'll describe Yang-Mills connections by way of analogy with harmonic maps. Then, I'll exploit the similarities and attack the differences between these two contexts in order to perform the analogous replacement technique in this new setting.

Host: Xiang Tang