2018/12/11 by Mircea Petrache, Petrache, Mircea, Tristan Rivière +1
Mathematics · #49Q15 #49Q20 #53C07 #53C65 #57R57 #58E15 #81T13 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1812.04432
openalex publication_date 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.