Abstract

This talk is concerned with the Brakke flow, which is a weak measure theoretic solution to the mean curvature flow. I will start by introducing the Brakke flow, and talk about some important properties, as well as some key difficulties of the flow. I shall then move onto my own recent joint work with Liu, in which we have proven the existence of a space-time-Grassmann measure for the flow, and given a new characterisation of the flow with respect to the space-time weight of this measure. This leads to a new definition of the Brakke flow as that of a space-time measure which satisfies the Brakke inequality in an appropriate distributional sense. The aim of this talk is to motivate this study of the Brakke flow as a space-time measure. This will include proving that fundamental geometric properties along the flow (the mean curvature vector, the tangent map, and the density), are all measurable with respect to this space-time weight measure.