Abstract

The small energy regularity for Yang-Mills solutions in dimension 4 was the key to the concentration-compactness result for Yang-Mills fields by Karen Uhlenbeck. This result, proven about 3 decades ago, was relevant for the construction of 4-manifold invariants by Simon Donaldson. The study is based on the theory of Sobolev connections over smooth bundles. I will describe in what sense dimension 4 is critical for Uhlenbeck’s approach.

I will then introduce a notion of weak curvatures on possibly very singular bundles. Such class allows to apply a variational approach to the Yang-Mills Lagrangian in superctitical dimensions, where we don’t have a priori control on the topology of the bundle. This measure-theoretic notion of bundles allowing wild topological singularities bears analogies to integral currents. The integer multiplicity condition of Herbert Federer and Wendell Fleming is replaced in our setting by the integrality of relevant Chern classes.

I will present the solution to the Yang-Mills-Plateau problem in the abelian case. In particular I will describe a closure theorem for abelian weak curvatures on singular bundles and the successive regularity result for the Yang-Mills-Plateau minimizers. These results are part of an ongoing project in collaboration with my advisor Tristan Rivière.