Kerr-de Sitter rigidity conjecture for stationary vacuum black holes

Let $(\mathcal{M},\mathbf{g})$ be a regular, stationary (appropriately defined), four-dimensional vacuum black hole solution of the Einstein vacuum equations with positive cosmological constant $\Lambda$, and let its domain of exterior communication be the region outside the black hole. Kerr-de Sitter rigidity conjecture. The domain of exterior communication of $(\mathcal{M},\mathbf{g})$ is isometrically diffeomorphic to the domain of exterior communication of a Kerr-de Sitter black hole. This is the positive-cosmological-constant analogue of Kerr rigidity and of the Carter–Robinson uniqueness theorem. Unlike the asymptotically flat case, the source notes that no complete analogue of the Carter–Robinson theorem is known for nonzero cosmological constant; the conjecture remains unresolved.