Abstract:

Mean curvature flow can be continued through singularities using a weak solution to the flow. Two major weak solutions are Brakke flow and level set flow. Brakke flow is defined with an inequality which makes it tantamount to a subsolution to mean curvature flow, whereas level set flow may attain positive measure and is properly thought of as a supersolution. In this talk, we will discuss these weak solutions, and we will relate uniqueness problems to multiplicity problems for mean curvature flow. In particular, we will discuss how Brakke flows with only generic singularities achieve equality in their defining inequality. This uses an analysis of worldlines in Brakke flows, analogous to the theory of singular Ricci flows. This talk will be aimed at those without background in mean curvature flow.