Nonexistence of weakly stable Yang-Mills fields

TL;DR

The paper establishes rigidity results for Yang-Mills connections under conformal and warped-product geometries. By deploying gradient-conformal vector-field variations and Bochner–Weitzenböck calculus, it derives sign conditions on the second variation that force nonexistence of nontrivial weakly stable (and sometimes stable) Yang-Mills connections in high dimensions on conformal spheres and their products. Specifically, on $(S^n,\tilde{g}=e^{2\varphi}g)$ with $n\ge5$, a positivity condition on $\varphi$ eliminates nontrivial weakly stable Yang-Mills fields, and a stronger bound eliminates stability as well; analogous statements hold for products of spheres and for warped-product noncompact manifolds under curvature and growth assumptions. These results extend classical stability analyses and reveal rigidity of Yang-Mills fields under conformal deformations and warped geometries, highlighting the delicate interplay between dimension, curvature, and gauge-theoretic energy.

Abstract

In this paper we prove that there is a neighborhood in the $C^2$ topology of the usual metric on the Euclidean sphere $S^n (n\geq 5)$ such that there is no nontrivial weakly stable Yang-Mills connections for any metric $\tilde{g}$ in this neighborhood. We also study the stability of Yang-Mills connections on the warped product manifolds.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.2
• Theorem 2.1
• Remark 2.1
• Definition 2.1
• Proposition 3.1