Speaker: Robert Haslhofer (University of Toronto)

Title: Recent progress on mean curvature flow

Abstract: 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 is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or 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.