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 \({\text {SU} }(r)\)-bundle of charge \(\kappa \) over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than \(4 \pi ^2 \left( |\kappa | + 2 \right) \) 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 \(\mathfrak {s}\mathfrak {p}(1)\) 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.