Weak compactness of wave maps and harmonic maps

We show that a weak limit of a sequence of wave maps in (1 + 2) dimensions with uniformly bounded energy is again a wave map. Essential ingredients in the proof are Hodge structures related to harmonic maps, ¹ estimates for Jacobians, ¹-BMO duality, a “monotonicity” formula in the hyperbolic context and the concentration compactness method. Application of similar ideas in the elliptic context yields a drastically shortened proof of recent results by Bethuel on Palais-Smale sequences for the harmonic map functional on two dimensional domains and on limits of almost H-surfaces.