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Title:
Yang-Mills Replacement
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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. As a result, 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, including extensions of Uhlenbeck's gauge fixing results.
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