  In this article we prove a local energy estimate for the linear wave equation
on metrics with slow decay to a Kerr metric with small angular momentum. As an
application, we study the quasilinear wave equation $\Box_{g(u, t, x)} u = 0$
  where the metric $g(u, t, x)$ is close (and asymptotically equal)to a Kerr
metric with small angular momentum $g(0,t,x)$. Under suitable assumptions on
the metric
  coefficients, and assuming that the initial data for $u$ is small enough, we
prove global existence and decay of the solution $u$.
