The mean curvature flow of submanifolds of high codimension

Abstract

The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study of the mean curvature flow of submanifolds of higher codimension has only recently started to receive attention. The mean curvature flow of submanifolds is the main object of investigation in this thesis, and indeed, the central results we obtain can be considered as high codimension analogues of some early hypersurface theorems. The result of Huisken's 1984 paper roughly says that convex hypersurfaces evolve under the mean curvature flow to round points in finite time. Here we obtain the result that if the ratio of the length of the second fundamental form to the length of the mean curvature vector is bounded (by some explicit constant depending on dimension but not codimension), then the submanifold will evolve under the mean curvature flow to a round point in finite time. We investigate evolutions in flat and curved backgrounds, and explore the singular behaviour of the flows as the first singular time is approached.