LECTURES ON MEAN CURVATURE FLOW

Abstract. A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curva-ture vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively stud-ied ever since the pioneering work of Brakke [Bra78] and Huisken [Hui84]. In the last 15 years, White developed a far-reaching reg-ularity and structure theory for mean convex mean curvature flow [Whi00, Whi03, Whi11], and Huisken-Sinestrari constructed a flow with surgery for two-convex hypersurfaces [HS99a, HS99b, HS09]. In this course, I first give a general introduction to the mean curva-ture flow of hypersurfaces and then present joint work with Bruce Kleiner [HK13a, HK14], where we give a streamlined and unified treatment of the theory of White and Huisken-Sinestrari. These notes are from summer schools at KIAS Seoul and SNS Pisa. Content