A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr

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 □ _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.