The space-time-Grassmann measure of the Brakke flow

Abstract

For a $k$-dimensional Brakke flow on an open subset $U \subset \mathbf{R}^{n}$, over an open time interval $J$, we prove the existence of a canonical space-time-Grassmann measure $\lambda$, over $J \times \mathbf{G}_{k} (U)$, and give a characterisation of the flow with respect to the space-time weight of this measure. This results in a new definition of the Brakke flow, as that of a space-time measure which satisfies the Brakke inequality in a distributional sense. Each such space-time measure corresponds to a class of equivalent (classical) Brakke flows, thus yielding an equivalence between the classical definitions of the Brakke flow, and this new definition. Moreover, we prove that the mean curvature vector, density, and tangent map along the flow, are all measurable with respect to this space-time weight measure.