Instantons and singularities in the Yang-Mills flow

TL;DR

This paper investigates the behavior of the Yang-Mills flow in four dimensions, demonstrating conditions under which singularities do not form and establishing convergence results, including exponential convergence and stability, for certain initial conditions.

Contribution

The paper provides new results on the non-formation of instanton singularities in finite time and characterizes the flow's convergence and stability under specific energy and topological conditions.

Findings

Instantons cannot form singularities in finite time.

Flow converges exponentially if the Uhlenbeck limit is anti-self-dual with vanishing self-dual second cohomology.

Recovery of Taubes's existence theorem and proof of asymptotic stability.

Abstract

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.