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After describing the inhomogeneous equation suitable to describe W{\nu}{\mu}{\rho}{\sigma}, a solution of the linearized version of the full quasilinear equation for the conformal part of the Riemann tensor connected to the perturbations of the Kerr spacetime far from the origin, we find the right decays we have to impose to the source term to obtain the peeling decays for this linearized solution. We basically follow the ideas of the Christodoulou Klainerman approach, [Ch-Kl] and [Kl-Ni1]. This result requires some new detailed estimates which could be considered a useful result by themselves., Comment: 61 pages, no figures

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the apriori estimates for the Weyl equations, associated to the "Bel-Robinson norms". In particular if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a "geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper., Comment: 68 pages

We prove, outside the influence region of a ball of radius $R_0$ centered in the origin of the initial data hypersurface, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr; This external region is the "far" part of the outer region of the perturbed Kerr spacetime. Moreover if we assume that the corrections to the Kerr metric decay sufficiently fast, $o(r^{-3})$, we prove that the various null components of the Riemann tensor decay in agreement with the "Peeling theorem"., Comment: 118 pages

This is the second part of our result on a class of global characteristic problems for the Einstein vacuum equations with small initial data. In the previous work denoted by (I), our attention was focused on prescribing the initial data satisfying the costraints imposed by the characteristic problem. Here we show how the global existence result can be achieved. This part is heavily based on the global results of D.Christodoulou, S.Klainerman and S.Klainerman, F.Nicolo`., Comment: 32 pages, 7 figures

We show how to prescribe the initial data of a characteristic problem satisfying the costraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non characteristic problem in [Kl-Ni], will be the content of a subsequent paper., Comment: 77 pages, no figures; corrected misprints and minor errors, to appear on Journal of Hyperbolic Differential Equations (JHDE)