On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. this article, we study Cauchy problem and give rigorous proof phenomenon sufficiently thereby providing first constructive stable blowup (without symmetry assumptions) hypersurface as effect pure gravity. Our applies subset regular initial data satisfying assumptions Hawking's celebrated singularity theorem, which shows solution is geodesically incomplete but does not reveal nature incompleteness. Specifically, our main result dynamic stability Kasner whose metrics exhibit only moderately (as opposed severely) spatially anisotropic behavior. Of independent interest method proof, more robust than earlier approaches i) it rely on approximate monotonicity identities ii) accommodates possibility develops very singular high-order spatial derivatives, rates allowed be, within scope bootstrap argument, much worse those base-level quantities driving fundamental blowup. For these reasons, approach could be used obtain similar results various Einstein-matter systems any number dimensions corresponding data, thus beyond nearly isotropic regime treated works.