We provide a classification of Λ > 0-vacuum spacetimes which admit a Killing vector field with respect to which the associated "Mars-Simon tensor" (MST) vanishes and having a conformally flat I − (or I + ). To that end, we also give a complete classification of conformal Killing vector fields on the 3-sphere S 3 up to Möbius transformations shedding light on the two fundamental constants that characterize the family of Kerr-de Sitterlike spacetimes, which turn out to be well-defined geometrical invariants. The topology of I is determined in every case, and a characterization result at I of the Kerr-de Sitter family presented.

We investigate solutions (M , g) to Einstein's vacuum field equations with positive cosmological constant Λ which admit a smooth past null infinity I −à la Penrose and a Killing vector field whose associated Mars-Simon tensor (MST) vanishes. The main purpose of this work is to provide a characterization of these spacetimes in terms of their Cauchy data on I − .Along the way, we also study spacetimes for which the MST does not vanish. In that case there is an ambiguity in its definition which is captured by a scalar function Q. We analyze properties of the MST for different choices of Q. In doing so, we are led to a definition of "asymptotically Kerr-de Sitter-like spacetimes", which we also characterize in terms of their asymptotic data on I − .PACS numbers: 02.30.Jr, 04.20.Ex, 04.20.Ha, on a 4-dimensional spacetime (M , g), where g is smooth and Λ is a ("cosmological") constant. We focus on the case Λ > 0 but compare occasionally with Λ = 0. Spacetime indices are greek, while coordinates in 1 + 3 splits are denoted by {x α } = {t, x i } (rather than {x 0 , x i }), with corresponding tensorial indices. Our conventions for the signature, the curvature tensor R µνσ κ , the Weyl tensor C µνσ κ , the Ricci tensor R µσ and the scalar curvature R follow e.g. [36]. The Levi-Civita connection of g is denoted by ∇.The setting of our work is an asymptotic structureà la Penrose [10,28]. By that we mean that an appropriate conformal rescaling of (M , g)( 1.2)

Abstract. We review various aspects of the characteristic initial value problem for the Einstein equations, presenting new approaches to some of the issues arising.

We analyze the Cauchy problem for the vacuum Einstein equations with data on a complete light-cone in an asymptotically Minkowskian space-time. We provide conditions on the free initial data which guarantee existence of global solutions of the characteristic constraint equations. We present necessary-and-sufficient conditions on characteristic initial data in 3 + 1 dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.

We derive, in 3 + 1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich's conformal field equations. We analyse their equivalence to Einstein's vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.