We review the asymptotic initial value problem in all dimensions for the Λ>0-vacuum Einstein metrics in the Fefferman-Graham picture.
We give a geometric definition of the initial data in the conformally flat $\mathscr{I}$ case and a Killing initial data equation for the analytic data case.
We use these results to characterize the Kerr-de Sitter family of metrics in all dimensions in terms of asymptotic initial data, which are given by a conformally flat boundary metric γ and a TT tensor canonically constructed from a particular conformal Killing vector (CKV) ξ_{K dS} of γ. The data naturally generalize for an arbitrary CKV ξ, extending the definition of the Kerr-de Sitter-like class (with conformally flat $\mathscr{I}$) to arbitrary dimensions by means of asymptotic data.
We prove that each metric in the class corresponds to a (conformal) class of CKVs, i.e. the set generated from ξ by local conformal transformations of γ. This fact endows the space of metrics with a structure of limits, which we analyze. This allows us to make explicit all the metrics in the class and identify them with the set of Λ>0-vacuum and conformally extendable Kerr-Schild metrics with an additional decay condition at $\mathscr{I}$.