The Yang-Mills equation near instanton-anti-instanton configurations

We study the question of whether a sequence of non-instanton Yang-Mills connections can limit to a bubbling configuration composed only of instantons. In the case that the Uhlenbeck limit and the bubbles are of opposite charge, we determine an obstruction coming from deformations of the Uhlenbeck limit. As an application, we prove that instantons are the only solutions of the $\mathrm{SU}(2)$ Yang-Mills equation on $\mathbb{R}^4$ with energy less than $4\pi^2 \left( |\kappa| + 2 \right) + \varepsilon_\kappa,$ where $\kappa$ is the charge. We also prove discreteness of the energy spectrum on the trivial $\mathrm{SU}(2)$-bundle in the range $\left[ 0, 16 \pi^2 \right).$