Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds

TL;DR

This paper generalizes an energy gap result for Yang-Mills connections on four-dimensional closed manifolds, extending it from positive to good Riemannian metrics, with specific conditions on the topology and Lie groups involved.

Contribution

It extends the $L^2$ energy gap result for Yang-Mills connections to a broader class of Riemannian metrics on four-dimensional manifolds, including generic metrics.

Findings

Established energy gap for Yang-Mills connections under good Riemannian metrics.

Included cases where the topology of the bundle and manifold meet certain mild conditions.

Applicable to principal bundles with structure groups SU(2) or SO(3).

Abstract

We extend an $L^2$ energy gap result due independently to Min-Oo and Parker (1982) for Yang-Mills connections on principal $G$-bundles, $P$, over closed, connected, four-dimensional, oriented, smooth manifolds, $X$, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics that are generic and where the topologies of $P$ and $X$ obey certain mild conditions and the compact Lie group, $G$, is $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$.