Nonexistence of weakly stable Yang-Mills fields

In this paper we prove that there is a neighborhood in the $C^2$ topology of the usual metric on the Euclidean sphere $S^n(n\ge 5)$ 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.