Renormalized Yang-Mills Energy on Poincar\'e-Einstein Manifolds

Abstract

We prove that the renormalized Yang-Mills energy on six dimensional Poincar\'e-Einstein spaces can be expressed as the bulk integral of a local, pointwise conformally invariant integrand. We show that the latter agrees with the corresponding anomaly boundary integrand in the seven dimensional renormalized Yang-Mills energy. Our methods rely on a generalization of the Chang-Qing-Yang method for computing renormalized volumes of Poincar\'e-Einstein manifolds, as well as known scattering theory results for Schr\"odinger operators with short range potentials.