Energy Identity for Stationary Yang Mills

Aaron Charles Naber, Northwestern University
Fine Hall 314

The first part of this talk will introduce and discuss the basics of Yang Mills connections, which are connections over a principle bundle which are critical points of the energy functional \int |F|^2, the L^2 norm of the curvature of A, and thus A 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-> 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 -> |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 conjectural picture is that e(x) may be computed 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 identity, which is joint with Daniele Valtorta. The techniques may also be used to prove a conjectured L^1 estimate on the hessian of the curvature, and we will discuss these results as well.