Bubble tree construction for harmonic maps using Deligne-Mumford moduli space

We formulate and prove a general compactness theorem for harmonic maps.Convergence is defined using Deligne-Mumford space and families of curves.Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of "non-regular'' nodes.This convergence result extends existing bubble-tree construction to the case of varying domains.