Dafermos–Holzegel–Rodnianski conjecture on scattering solutions of the Einstein vacuum equations

Let $\mathcal{H}$ and $\mathcal{I}$ denote the horizon and null infinity, respectively. Consider smooth scattering data settling down to a possibly extremal Kerr solution on $\mathcal{H}$ and $\mathcal{I}$ at a sufficiently fast inverse-polynomial decay rate. A vacuum spacetime $(M,g)$ is said to be bounded by $\mathcal{H}$ and $\mathcal{I}$ if it attains these data on the corresponding boundaries. Dafermos–Holzegel–Rodnianski conjecture. For the Einstein vacuum equations, there exists a vacuum spacetime $(M,g)$ bounded by $\mathcal{H}$ and $\mathcal{I}$, attaining the data and regular away from $\mathcal{H}$. However, for generic such data converging to a subextremal Kerr solution, the Christoffel symbols of the resulting metric fail to be locally square integrable near the horizon. This conjecture expresses the expected obstruction to smooth evolution from generic polynomially decaying scattering data on subextremal black holes; the source presents it as an expectation motivated by known scalar-wave results, and its resolution is not supplied.