Parabolic Gap Theorems for the Yang-Mills Energy

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an SU (r) SU ( r ) -bundle of charge κ κ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than 4 π ² ( |κ | + 2 ) 4 π 2 | κ | + 2 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 sp(1) s p ( 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.