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Theses Doctoral

Self-duality and singularities in the Yang-Mills flow

Waldron, Alex

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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More About This Work

Academic Units
Mathematics
Thesis Advisors
Daskalopoulos, Panagiota
Degree
Ph.D., Columbia University
Published Here
July 7, 2014
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