Ancient flows and the mean-convex neighborhood conjecture
In this talk, I will explain our recent proof of the mean-convex neighborhood conjecture for the mean curvature flow of surfaces. Namely, if the flow develops a neck-singularity, then it is mean-convex in a space-time neighborhood. The major difficulty is to promote the infinitesimal information captured by the tangent flow to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient mean curvature flows that arise as potential limit flows near a neck-singularity. As an application, we prove the uniqueness conjecture for mean curvature flow through cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits. Finally, I will briefly mention the higher-dimensional case, which is joint work with Kyeongsu Choi, Or Hershkovits and Brian White.