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We introduce , the inhomogeneous equation suitable to describe the solution of the linearized version of the conformal part of the Riemann tensor connected to the perturbations of the Kerr spacetime far from the origin. Then we find the right decays we have to impose to the source term to obtain the peeling decays for this linearized solution. We explain how these decays are compatible with the ones we need to attack the full nonlinear problem following the Christodoulou-Klainerman approach, see [7, 11]. This result is expressed explicitly in Theorem 1.1 and requires some new detailed estimates for the connection coefficients related to the null cone foliation in Kerr, see [10], which could be considered as a useful result by itself. [ABSTRACT FROM AUTHOR]

We study a class of global characteristic problems for the Einstein vacuum equations with small initial data. In a previous work [Caciotta and Nicolò, 'Global characteristic problem for Einstein vacuum equations with small initial data: (I) The initial data constraints,' Journal of Hyperbolic Differential Equations (JHDE) 2, 201 (2005)], denoted by Paper I, our attention was focused on prescribing the initial data satisfying the constraints imposed by the characteristic problem. In this paper we show how the global existence result can be achieved. This result is heavily based on the global results of Christodoulou and Klainerman [The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series Vol. 41 (Princeton University Press, Princeton, NJ, 1993)] and Klainerman and Nicolò [The Evolution Problem in General Relativity, Progress in Mathematical Physics Vol. 25 (Birkhauser, Boston, MA, 2002)]. [ABSTRACT FROM AUTHOR]

Abstract: We prove, outside the influence region of a ball of radius centered at the origin of the initial data hypersurface, , the existence of global solutions near to the 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, , we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling conjecture”. [Copyright &y& Elsevier]

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, 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 [19], will be the content of a subsequent paper. [ABSTRACT FROM AUTHOR]