Near-extremal Reissner–Nordström–de Sitter regularity conjecture

Let the initial data be sufficiently close to a Reissner–Nordström–de Sitter solution whose parameters are sufficiently close to extremal but still subextremal, and let $(\widetilde{\mathcal{M}},\widetilde{g})$ be extensions supplied by the preceding stability theorem. Near-extremal regularity conjecture. The extensions can be chosen so that $\widetilde{g}$ is Lipschitz and the scalar field extends to $\widetilde{\mathcal{M}}$ as an $H^1_{\rm loc}$ function. For generic such initial data, for all subextremal values of the Reissner–Nordström–de Sitter reference solution, $\widetilde{g}$ cannot be $C^2$ and $\phi$ cannot extend $C^1$. The conjecture predicts improved metric and scalar-field regularity near extremality while retaining a lower-order singularity; it is motivated by linearized analysis and is not established in the source.