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

Understand

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface.

• In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity.
• Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated "singularity" theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness.
• Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior.