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

Abstract

Abstract We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $${\mathscr {I}}$$ I , namely a certain traceless and transverse part of the n-th order coefficient of the expansion $$\mathring{g}_{(n)}$$ g ˚ ( n ) . In the case $$\Lambda 0$$ Λ > 0 case where the Cauchy problem at $${\mathscr {I}}$$ I of the Einstein vacuum field equations is known to be well-posed when the data at $${\mathscr {I}}$$ I are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at $${\mathscr {I}}$$ I to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.