A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector.
In the last 15 years, White developed a far-reaching structure theory for weak solutions with positive mean curvature, and Huisken-Sinestrari constructed a flow with surgery for hypersurfaces where the sum of the smallest two principal curvatures is positive.
In this course, I will present joint work with Kleiner, where we give a streamlined and unified treatment of the theory of White and Huisken-Sinestrari.
After reviewing the necessary background, I'll discuss our new a priori estimates for mean convex mean curvature flow and mean curvature flow with surgery. Our estimates are local and universal. They are based on the beautiful noncollapsing result of Andrews, which says that the condition of admitting interior and exterior balls of radius $c/H$ is preserved under the flow.
In the last two lectures, I'll sketch how we use our estimates to obtain the main structural results for mean convex level set flow, and to prove the existence of mean curvature flow with surgery.