Compact Mean Curvature Flows from Non-Compact Initial Data

Mean curvature flow describes the motion of hypersurfaces which seek to most efficiently decrease their surface area. If the initial hypersurface is compact, then this flow is unique, and the long-time behavior up to curvature blow-up is well understood. However, if the initial hypersurface is non-compact, the situation is more subtle. Work of Ecker-Huisken demonstrates the existence of a non-compact mean curvature flow in the case that the initial hypersurface can be written as a locally Lipschitz entire graph, and if we further assume rotational symmetry, work of Daskalopoulos-Saez upgrades this picture to imply uniqueness of this flow. Outside of these rigid regimes however, uniqueness, in general, fails. Indeed, Angenent-Ilmanen-Chopp directly compute an example of non-uniqueness of mean curvature flow with non-compact initial data. More recently, Bourni-Reiris showed the remarkable phenomenon in the one-dimensional case, the curve shortening flow, where if the initial data encloses a finite area, then there exists a compact curve shortening flow flowing out of this initial data. In this talk, we discuss the generalization of these construction to higher dimensions and discuss the non-uniqueness implications of our construction. All work is with Theodora Bourni and Marcus Flook.