Lecture 2: Mean curvature flow with surgery

The lecture studies hypersurfaces moving in direction of their mean curvature vector, a flow governed by a system of quasi-linear parabolic equations of second order. This non-linear deformation exhibits on the one hand the characteristic smoothing properties of the heat equation while exhibiting certain singularities due to the reaction-diffusion type properties of the flow. We show how the structure of singularities can be controlled with suitable a priori estimates obtained with techniques from PDE theory, allowing the circumvention of singularities by surgery in certain cases. It will become apparent that some of this behavior in mean curvature flow is similar to the behavior of Ricci flow discovered in the work of Hamilton and Perelman.