Mathematics Colloquium

In this talk I will survey recent results regarding uniqueness and continuous dependence of two closely related geometric flows: the Ricci flow and the Mean Curvature flow.

In the first part of my talk, I will focus on Ricci flow in dimension 3. I will briefly revisit Perelman's groundbreaking construction of Ricci flow with surgery, which famously resolved the Poincaré and Geometrization Conjectures. Building on this, I will discuss work of Lott, Kleiner and myself, on an enhanced version of this flow, called "singular Ricci flow". The advantage of this flow is that it is uniquely determined by its initial data and depends continuously on it. This fact has allowed us to resolve two topological conjectures: the Generalized Smale Conjecture on diffeomorphism groups of 3-manifolds and a conjecture on the contractibility of the space of positive scalar curvature metrics on 3-manifolds.

In the second part of the talk, I will focus on the Mean Curvature flow. I will present joint work with Bruce Kleiner in which we resolve the Multiplicity One Conjecture. Combined with work from other people, our result allows us to establish uniqueness and continuous dependence results similar to the Ricci flows case, though these properties hold only under certain additional conditions, highlighting key differences between the two flows.