Surveys in Differential Geometry XII Singularities of Mean Curvature Flow and Flow with Surgeries

We collect in this paper several results on the formation of singularities in the mean curvature flow of hypersurfaces in euclidean space, under various kinds of convexity assumptions. We include some recent estimates for the flow of 2-convex surfaces, i.e. the surfaces where the sum of the two smallest principal curvatures is positive everywhere. Such results enable the construction of a flow with surgeries for these surfaces similar to the one introduced by Hamilton and Perelman for the Ricci flow. The topological applications of the construction are also described.