Stability of Singularities in Geometric Evolutionary PDE

We study questions of stability of two types of singularities encountered in geometric evolutionary PDE, one in Ricci flow and the other in the context of the Einstein field equations in vacuum. In the first part of the thesis we introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow ``pushes away'' from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. We study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the ``opening up'' of the singularity persists for the perturbations as well. The second problem we study concerns the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface $\Sigma_0$ in the interior of the black hole region, tangent to the singular hypersurface $\{r=0\}$ at a single sphere, we study the problem of perturbing the Schwarzschild data on $\Sigma_0$ and solving the Einstein vacuum equations backwards in time. We obtain a local backwards well-posedness result for small perturbations lying in certain weighted Sobolev spaces. The perturbed spacetimes all have a singularity at a ``collapsed'' sphere on $\Sigma_0$, where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order. As in the Schwarzschild backward evolution, the pinched initial hypersurface $\Sigma_0$ `opens up' instantly, becoming a regular spacelike (cylindrical) hypersurface. This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from regular initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.Ph.D