  Let an initial data metric $\overline{g}$ be, outside a ball $B_{R_0}$
centered in the origin, the induced metric on $\Sigma_0$ of a Kerr spacetime
(with a mass $M$ and angular momentum $J$ whose ratio, $J/M$, depends on the
size of $R_0$) plus small corrections which decay at spacelike infinity faster
than $r^{-3}$; let, in the same region, a symmetric tensor $\overline{k}$ be
the second fundamental form of the Kerr spacetime plus small corrections which
decay at spacelike infinity faster than $r^{-4}$, let $\overline{g}$ and
$\overline{k}$ satisfy the constraint equations. Then, using the previous
results of \cite{Ch-Kl:book} and \cite{Kl-Ni:book}, the global existence of the
external region of a global spacetime, outside the region of influence of
$B_{R_0}$ follows. In this external region the various components of the
Riemann tensor decay along the outgoing null directions in agreement with the
"Peeling Theorem".
