Lectures on Mean Curvature Flow and Stability (MAT 1063 HS)

The mean curvature flow (MCF) arises material science and condensed matter physics and has been recently successfully applied to topological classification of surfaces and submanifolds. It is closely related to the Ricci and inverse mean curvature flow. The most interesting aspect of the mean curvature flow is formation of singularities, which is the main theme of these lectures. In dealing with this, we emphasize the key notion of stability which isolates ‘physically’ interesting dynamics. We also consider briefly the closely related volume preserving mean curvature flow (VPF), which is important on its own right. In the last section we ketch some features of the Ricci flow and emphasize the parallels as well as the distinctions between two flows. Background on geometry of surfaces and some technical statements are given in appendices. These are rough, unedited notes, based on the course given a few years back and will changed and further developed in the course of the present course. My thanks go to Greg Fournodavlos, Dan Ginsberg and Wenbin Kong for the help with the material of the notes.