Uniqueness problems in mean curvature flow

Abstract

We investigate uniqueness phenomena in mean curvature flow, focusing on two central problems: the behavior of the flow near singularities and the extension of the flow beyond singular times. These questions have significant applications in geometry, topology, and analysis. For the first problem, with Jingze Zhu, we formulate a canonical way to study the limit model near a singularity of a generic closed mean curvature flow of surfaces. Using this framework, we establish a uniqueness result for singularity models. As a consequence, we resolve a uniqueness problem for gradient flow lines in ordinary differential equation theory, related to a question posed by Thom and Arnold, and revisited by Colding–Minicozzi. For the second problem, with Alec Payne, we examine the level set flow as a weak formulation that ensures long-time existence and uniqueness of mean curvature flow past singularities. This approach, however, can lead to fattening, a phenomenon reflecting genuine non-uniqueness of the extended flow. While genuine uniqueness cannot always be expected, we address this challenge by establishing an intersection principle for comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness. Finally, with Larry Guth, we explore a problem concerning homotopy classes of maps between spheres. Recent progress on this problem relies on delicate analysis of high codimensional graphical mean curvature flow. We use a direct method to refine a homotopy criterion for maps between low-dimensional spheres.