Mathematics Colloquium

Energy Identity for Stationary Yang Mills

Speaker: Aaron Naber, Northwestern University

Location: Warren Weaver Hall 1302

Date: Monday, November 14, 2016, 3:45 p.m.


Yang Mills connections over a principle bundle are critical points of the energy functional \(\int |F|^2\), the \(L^2\) norm of the curvature, and thus may be viewed as a solution to a nonlinear pde. In many problems, e.g. compactifications of moduli spaces, one considers sequences \(A_i\) of such connections which converge to a potentially singular limit connection \(A_i \rightarrow A\). The convergence may not be smooth, and we can understand the blow up region by converging the energy measures \(|F_i|^2 dv_g \rightarrow |F|^2dv_g +\nu\), where \(\nu=e(x)d\lambda^{n-4}\) is the \(n-4\) rectifiable defect measure (e.g. think of \(\nu\) as being supported on an \(n-4\) submanifold). It is this defect measure which explains the behavior of the blow up, and thus it is a classical problem to understand it. The main open problem on this front is to compute \(e(x)\) explicitly as the sum of the bubble energies which arise from blow ups at \(x\), a formula known as the energy identity. This talk will primarily be spent explaining in detail the concepts above, with the last part focused on the recent proof of the energy quantization, which is joint with Daniele Valtorta. The techniques may also be used to give the first apriori higher derivative estimates on Yang Mills connections, and we will discuss these results as well.