Conformally compact and higher conformal Yang-Mills equations

For the Yang-Mills equations on the interior of a conformally compact manifold we determine the appropriate boundary conditions, formal asymptotics, and describe Dirichlet-to-Neumann type maps. Given the "magnetic" Dirichlet data of a boundary connection, by constructing a recursion for asymptotic solutions, we find that smooth solutions are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion. This provides higher derivative conformally invariant analogs of Yang-Mills equations and energies. The invariant energy is the anomaly for the interior renormalized Yang-Mills energy. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they (i) give obstructions to solving the Yang-Mills boundary problem, (ii) determine the asymptotics of formal solutions, and (iii) yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincar\'e-Einstein structures.