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We study the free data in the Fefferman-Graham expansion of asymptoticallyEinstein metrics with non-zero cosmological constant. We prove that if$\mathscr{I}$ is conformally flat, the rescaled Weyl tensor at $\mathscr{I}$agrees up to a constant with the free data at $\mathscr{I}$ , namely thetraceless part of the $n$-th order coefficient of the expansion. In thenon-conformally flat case, the rescaled Weyl tensor is generically divergent at$\mathscr{I}$ but one can still extract the free data in terms of thedifference of the Weyl tensors of suitably constructed metrics, in fullgenerality when the spacetime dimension $D$ is even and provided the so-calledobstruction tensor at $\mathscr{I}$ is identically zero when $D$ is odd. Theseresults provide a geometric definition of the data, particularly relevant forthe asymptotic Cauchy problem of even dimensional Einstein metrics withpositive $\Lambda$ and also for the odd dimensional analytic caseirrespectively of the sign of $\Lambda$. We establish a Killing initial dataequation at spacelike $\mathscr{I}$ in all dimension for analytic data. Theseresults are used to find a geometric characterization of the Kerr-de Sittermetrics in all dimensions in terms of its geometric data at null infinity.

Free data at spacelike $\mathscr{I}$ and characterization of Kerr-de Sitter in all dimensions