Parabolic gap theorems for the Yang-Mills energy

TL;DR

This paper establishes parabolic gap theorems for Yang-Mills energy, showing how Yang-Mills flow simplifies the structure of connection spaces on various manifolds.

Contribution

It introduces parabolic versions of known gap theorems, providing new deformation-retraction results under Yang-Mills flow on different geometric settings.

Findings

Connections with energy below a certain threshold deformation-retract onto instantons.

Yang-Mills flow simplifies the topology of connection spaces on 4-sphere and quaternion-Kähler manifolds.

Positive lower bounds for Morrey norm of curvature on nontrivial bundles.

Abstract

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-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-K\"ahler 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.