  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.

Analysis of PDEs