  We consider the problem of small data global existence for quasilinear wave
equations with null condition on a class of Lorentzian manifolds
$(\mathbb{R}^{3+1}, g)$ with \textbf{time dependent} inhomogeneous metric. We
show that sufficiently small data give rise to a unique global solution for
metric which is merely $C^1$ close to the Minkowski metric inside some large
cylinder $\{\left.(t, x)\right||x|\leq R\}$ and approaches the Minkowski metric
weakly as $|x|\rightarrow \infty$. Based on this result, we give weak but
sufficient conditions on a given large solution of quasilinear wave equations
such that the solution is globally stable under perturbations of initial data.
