This is our main paper in which we prove the full, unconditional, nonlinear stability of the Kerr family Kerr(a,m) for small angular momentum, i.e. |a|/m« 1, in the context of asymptotically flat solutions of the Einstein vacuum equations (EVE). We rely on our GCM papers  and , the recently released , as well as on the general formalism contained in part I of , see also the older version in . The recently released  also contains, in parts II, III, all hyperbolic type estimates needed in our work. Our work extends the strategy developed in , in which only axial polarized perturbations of Schwarzschild were treated, by developing new geometric and analytic ideas on how to deal with with general perturbations of Kerr. We note that the restriction to small angular momentum is needed only in , mainly for the Morawetz type estimates derived in that paper.
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