Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R3,g), where the metric g is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|2)−ρ/2 with ρ>1. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix A.