Blow-ups of Two-Convex, Type-I Mean Curvature Flow

Much of geometric analysis can be described as the study of (hyper)surfaces changing shape subject to certain equations. Here we study one such equation, mean curvature ow, which decreases the area of a surface as fast as possible. However, solutions to this equation develop singularities. I present a detailed analysis of this development under suitable restrictions on curvature.Assuming mean-convexity and type-I growth of curvature in time, there are three main parts to the results:1) I collect well-known results to describe the shape of (rescalings) blow-ups near singularities in a specific way with high precision.2) Colding and Minicozzi showed uniqueness of blow-ups and Andrews showed a restriction on the surface collapsing. I combine these to ensure the formation of a certain neck shape to emulate the neck-pinching argument of Angenent to control when the singularity occurs. In turn I use this to show the conditions at the singular time depend in a nice way on the initial conditions.3) Since blow-ups are often stationary, their study leads to the study of ancient solutions, which exist for all negative time. I classify all possible ancient solutions arising from blow-ups under certain conditions.