Abstract: I will present approximation, existence and regularity results for Yang-Mills minimizers in supercritical dimensions, based on a joint project with Tristan Rivière.
The starting point are Uhlenbeck’s results which provided the analytic foundations for Donaldson’s study of Yang-Mills connections on bundles over 4-manifolds. The object of study in that case was the class of Sobolev connections on smooth bundles.
In dimensions 5 and higher the space of Sobolev connections over smooth bundles does not allow to apply the direct methods of the Calculus of Variations to obtain Yang-Mills minimizers. The substitute is a space of weak connections over singular bundles, in which a weak closure result allows constructing Yang-Mills connections by direct minimization. This space is a real measure-theoretic counterpart to singular objects of more algebraic flavour, e. g. coherent reflexive sheaves.
The main tool for the optimal partial regularity result for Yang-Mills minimizers in 5 dimensions is an approximation of weak connections by connections with finitely many topological defects. Such approximation allows to apply a Morrey space analogue of Uhlenbeck’s result, relaxing the approximability hypothesis from previous singularity removal results by Tao-Tian and Meyer-Rivière. We will contrast this new approximation result with approximation results for nonlinear Sobolev maps and for integral currents.