Scattering-data conjecture for Kerr-de Sitter spacetimes

Fix Kerr-de Sitter parameters $\mathsf{b}$. Let $R^\alpha H_{\mathrm{b}}^\infty$ and $R^\alpha H_{\mathrm{b}}^d$ denote the indicated weighted b-Sobolev spaces, and suppose that for $m=0,3$ we are given tensors $h_{(m),ij}=h_{(m),ij}(R,\omega)$ in $R^\alpha H_{\mathrm{b}}^\infty$ for $R\leq R_0$, with sufficiently small $R^\alpha H_{\mathrm{b}}^d$ norms for some sufficiently large $d$. Set

$$g_{(0)}=\mathrm{d}x^2+h_{(0)},\qquad g_{(3)}=g_{\mathsf{b},(3)}+h_{(3)}.$$

Assume that $g_{(3)}$ is a weighted transverse-traceless tensor with respect to $g_{(0)}$. Scattering-data conjecture. There exists a solution $g$ of the Einstein vacuum equations in a neighborhood of $\{R\leq R_0\}\subset\mathcal I^+$ with scattering data $g_{(0)},g_{(3)}$, and $g$ is asymptotic to the Kerr-de Sitter metric with parameters $\mathsf{b}$ at $\mathcal K$. This conjecture proposes nonlinear local realization of sufficiently small admissible perturbations of Kerr-de Sitter scattering data; the preceding discussion explains that the Fefferman--Graham expansion is determined by this data, while the existence assertion remains unresolved in the supplied text.