On the Global Regularity of Wave Maps in the Critical Sobolev Norm

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We extend the recent result of T.Tao to wave maps defined from the Minkowski space of dimension >4 to a target Riemannian manifold which possesses a ``bounded parallelizable'' structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces. General compact Riemannian manifolds can be imbedded as totally geodesic submanifolds in bounded parallelizable manifolds, and therefore are also covered, in principle, by our result. Compactness of the target manifold, which seemed to play an important role in Tao's result, turns out however to play no role in our discussion. Our proof follows closely that of Tao's recent paper and is based, in particular, on its remarkable microlocal gauge renormalization idea.