Quebec Mathematical Sciences Colloquium

January 22, 2021 from 15:00 to 16:00 (Montreal/EST time)

Mean curvature flow through neck-singularities

Colloquium presented by Robert Haslhofer (University of Toronto, Canada)

2020 André Aisenstadt Prize in Mathematics Recipient

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces and has been extensively studied over the last 40 years.

In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken’s classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. Specifically, singularities can be either of neck-type or conical-type. We will discuss examples from the 90s, which show, both experimentally and theoretically, that flow through conical singularities is utterly non-unique.

In the last part of the talk, I will report on recent work with Kyeongsu Choi, Or Hershkovits and Brian White, where we proved that mean curvature flow through neck-singularities is unique. The key for this is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms the mean-convex neighborhood conjecture. Assuming Ilmanen’s multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed.