The Yang-Mills Plateau Problem in Supercritical Dimensions
Speaker: Mircea Petrache, ETH Zurich
Location: Warren Weaver Hall 1302
Date: Thursday, November 6, 2014, 11 a.m.
The celebrated results of K. K. Uhlenbeck furnished the bases for a complete variational study of the Yang-Mills functional in dimension 4, leading to the definition of new differential invariants of 4-manifolds by S. K. Donaldson. In this setting the objects of study were Sobolev connections over smooth bundles. In the first part of the talk I will explain how this setting works, and show why it is not sufficient for studying the Yang-Mills functional in dimensions higher than 4.
I will then define a space of weak connections introduced in collaboration with Tristan Riviere, and present our weak closure and approximability results in that setting. These results lead to the an existence-and-regularity theory for Yang-Mills minimizers in dimensions greater than 4. This bears a strong analogy to the work of Federer-Fleming on the Plateau problem. The existence of minimizers requires the definition of new weak connection spaces and we have a proof that they correspond to smooth connections over classical bundles, up to a codimension 5 singular set. This gives a first step for connecting the variational theory to the algebraic geometry framework which motivated some conjectures of Gang Tian. The optimality of the codimension 5 is shown by an explicit example of a minimizer, and the proof of its minimality uses a new combinatorial technique together with a decomposition theorem for vector fields by S. Smirnov.