Near-extremal Kerr–de Sitter failure of $L^2_{\rm loc}$ strong cosmic censorship

Let $(\mathcal{M},g)$ be the maximal Cauchy development of sufficiently small perturbations of Kerr–de Sitter data with $n$ black-hole regions, initial topology $\mathbb{S}^1\times\mathbb{S}^2$, and parameters sufficiently close to extremality. Let $(\widetilde{\mathcal{M}},\widetilde{g})$ be extensions as in the preceding conjecture. Near-extremal Kerr–de Sitter conjecture. One can choose the extensions so that $\widetilde{g}$ has locally square-integrable Christoffel symbols and the extensions satisfy the Einstein equations in a generalized sense. In particular, the formulation of strong cosmic censorship requiring inextendibility as a metric with $L^2_{\rm loc}$ Christoffel symbols does not hold for positive cosmological constant. This predicts a low-regularity extension beyond the Cauchy horizon in the near-extremal vacuum case and remains open.