The space-time-Grassmann measure of the Brakke flow

TL;DR

This paper introduces a new measure-theoretic framework for Brakke flows, establishing a canonical space-time-Grassmann measure that characterizes the flow and links classical and measure-based definitions.

Contribution

It develops a novel space-time measure approach for Brakke flows, providing a new distributional definition and demonstrating equivalence with classical formulations.

Findings

Existence of a canonical space-time-Grassmann measure for Brakke flows

Characterization of flows via a space-time weight measure

Measurability of mean curvature, density, and tangent map with respect to this measure

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.