The Space of Weak Connections in High Dimensions

The space of Sobolev connections, as it has been introduced for studying the variation of Yang-Mills Lagrangian in the critical dimension $4$, happens not to be weakly sequentially complete in dimension larger than $4$. This is a major obstruction for studying the variations of this important Lagrangian in high dimensions. The present paper generalizes the previous result of the authors from 'The resolution of the Yang-Mills Plateau problem in super-critical dimensions', Adv. Math. 316 (arxiv:1306.2010), valid in $5$ dimensions to arbitrary dimension and introduces a space of so called 'weak connections' for which we prove the weak sequential closure under Yang-Mills energy control. We also establish a strong approximation property of any weak connection by smooth connections away from codimension $5$ polyhedral sets. This last property is used in a subsequent work in preparation for establishing the partial regularity property for general stationary Yang-Mills weak connections.