On the global regularity of wave maps in the critical Sobolev norm

We extend the recent result of T. Tao [6] to wave maps defined from the Minkowski space Rⁿ+1, n ≥ 5, to a target manifold N which possesses a “bounded parallelizable” structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces H^N. General compact Riemannian manifolds can be imbedded as totally geodesic submanifolds in bounded parallelizable manifolds, see [1], and therefore are also covered, in principle, by our result. Compactness of the target manifold, which seemed to play an important role in [6], turns out however to play no role in our discussion. Our proof follows closely that of [6] and is based, in particular, on its remarkable microlocal gauge renormalization idea.