Self-duality and singularities in the Yang-Mills flow

Description

We investigate the long-time behavior and smooth convergence properties of the Yang-Mills flow in dimension four. Two chapters are devoted to equivariant solutions and their precise blowup asymptotics at infinite time. The last chapter contains general results. We show that a singularity of pure + or - charge cannot form within finite time, in contrast to the analogous situation of harmonic maps between Riemann surfaces. This implies long-time existence given low initial self-dual energy. In this case we study convergence of the flow at infinite time: if a global weak Uhlenbeck limit is anti-self-dual and has vanishing self-dual second cohomology, then the limit exists smoothly and exponential convergence holds. We also recover the classical grafting theorem, and derive asymptotic stability of this class of instantons in the appropriate sense.

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