Stable Big Bang Formation in General Relativity: The Sub-Critical Regime

The celebrated theorems of Hawking and Penrose show that under appropriate assumptions on the matter model, a large, open set of initial data for Einstein's equations lead to geodesically incomplete solutions. However, these theorems are "soft" in that they do not yield any information about the nature of the incompleteness. Despite the uncertainty, in the mathematical physics literature, there are heuristic results, going back 50 years, suggesting that whenever a certain "sub-criticality" condition holds, the Hawking-Penrose incompleteness is caused by the formation of a cosmological Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with G. Fournodavlos and I. Rodnianski, we have given a rigorous proof of this phenomenon, i.e., a proof of stable monotonic curvature blowup along a spacelike hypersurface in the complete regime where the heuristics suggest it might occur. More precisely, our results apply to Sobolev-class perturbations - without symmetry - of generalized Kasner solutions whose exponents satisfy the sub-criticality condition. Our main theorem shows that - like the generalized Kasner solutions - the perturbed solutions develop Big Bang singularities. In this talk, I will provide an overview of our result and explain how it is tied to some of the main themes of investigation by the mathematical general relativity community, including the remarkable works of Dafermos-Luk on the stability of Kerr Cauchy horizons and of Van de Moortel on the structure of singularities inside black holes. I will also discuss the new gauge that we used in our work, as well as intriguing connections to other problems concerning stable singularity formation.