The stability of Yang-Mills connections on $\delta$-pinched manifolds

TL;DR

This paper identifies specific curvature conditions on compact manifolds that guarantee all weakly stable Yang-Mills connections are flat, thereby ruling out non-flat solutions under certain pinching conditions.

Contribution

It establishes new pinching criteria ensuring the flatness of weakly stable Yang-Mills connections on compact manifolds, including a dimension-dependent constant.

Findings

No non-flat weakly stable Yang-Mills connections on $oldsymbol{ ext{delta(n)}}$-pinched simply-connected manifolds.

Derived a dimension-dependent constant $oldsymbol{ ext{delta(n)}}$ for flatness conditions.

Proved flatness results under specific curvature pinching conditions.

Abstract

In this article, we establish pinching conditions under which all weakly stable Yang-Mills connections on compact manifolds are flat. As a corollary, we provide a dimension-dependent constant $\delta(n)$ and prove that there exist no non-flat weakly stable Yang-Mills connections on $\delta(n)$-pinched compact simply-connected Riemannian manifolds.