Parabolic gap theorems for the Yang-Mills energy

Abstract

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $\hat{I}º$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4\ddot{I}{€}^2\left(|\hat{I}º|+2\right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-Kähler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive.

51 pages