The Cauchy Problem for Wave Maps on a Curved Background

We consider the Cauchy problem for wave maps u: \R times M \to N for Riemannian manifolds, (M, g) and (N, h). We prove global existence and uniqueness for initial data that is small in the critical Sobolev norm in the case (M, g) = (\R^4, g), where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe for proving global existence and uniqueness of small data wave maps u : \R \times \R^d \to N in the critical norm, for d at least 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru.