Nonclosure of stationary Yang-Mills connections under weak convergence

Abstract

We construct a sequence of stationary weak Yang-Mills connections on the trivial vector bundle $B^5\times\mathbb{R}^{15}$ with structure group $\operatorname{SO}(15)$ whose weak limit is a nonstationary weak Yang-Mills connection. Motivated by the stationarity question arising in Tian's work on Yang-Mills compactness, our result establishes the failure of weak closure for stationary weak Yang-Mills connections with an isolated singularity. It also provides a Yang-Mills analogue of the phenomenon exhibited by Ding, Li and Li for harmonic maps.