Instantons and Singularities in the Yang-mills Flow

Several results on existence and convergence of the Yang-Mills flow in dimension four are given. We show that a singularity modeled on an instanton cannot form within finite time. Given low initial self-dual energy, we then study convergence of the flow at infinite time. If an Uhlenbeck limit is anti-self-dual and has vanishing self-dual second cohomology, then no bubbling occurs and the flow converges exponentially. We also recover Taubes’s existence theorem, and prove asymptotic stability in the appropriate sense.