Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes

This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e |a| 1, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide complete proofs for Theorems M1 and M2 as well the curvature estimates of Theorem M8, which were stated without proof in Sections 3.7.1 and 9.4.7 of [56]. Our procedure is based on a new general interest formalism (detailed in Part I of this work), which extends the one used in the stability of Minkowski space. Together with [56] and the GCM papers [54], [55], [69], this work completes proof of the Main Theorem stated in Section 3.4 of [56].