Geometric Flow Singularities with Bounded Curvature Quantities and the Uniqueness Question for Weak Solutions

Geometric flows refer to processes where a manifold moves in some way related to its geometry. For even the simplest geometric flows relevant to physical applications, the evolving manifold often becomes singular in finite time. Ideally, one would like to have a simple way to detect singularities and a well-posed notion of weak solution that allows one to continue the flow through the singularity in some generalized sense. For the mean curvature flow and Ricci flow, however, we’ll discuss constructions that highlight the difficulty of detecting singularities and establishing the uniqueness of weak solutions in general. These solutions exist in high dimensions and see the evolving manifold converge to a cone near the singularity.