The space-time-Grassmann measure of the Brakke flow

TL;DR

The paper develops a canonical space-time-Grassmann measure $\lambda$ associated to a $k$-dimensional Brakke flow, enabling a space-time formulation of Brakke flow as a measure that satisfies the Brakke inequality in a distributional sense. By projecting the flow into $J \times \mathbf{G}_k(U)$ and defining the space-time weight $\|\lambda\|$, it shows that each Brakke flow corresponds to an equivalence class of classical flows and that key geometric quantities (mean curvature, density, tangent map) are measurable with respect to $\|\lambda\|$. The authors prove the existence of $\lambda$, establish an exact characterization of the flow in terms of $\|\lambda\|$, and demonstrate that the space-time formulation is equivalent to the classical definitions (Ilmanen, Lahiri) of Brakke flow. This framework also yields left and right continuous representatives that capture jump behavior, mirroring BV theory, and lays groundwork for extending integral varifold theory to parabolic settings. The results connect weak measure-theoretic solutions with a robust space-time measure-theoretic structure, enriching the analytic toolkit for mean curvature flow in geometric measure theory.

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 $λ$, 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.

Theorems & Definitions (40)

• Theorem A: see \ref{['thm:Brakke flow:space time measure']}
• Theorem B
• Theorem C: see \ref{['thm:measurable function:tangent,density,mean curvature vector']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 3.1
• Remark 3.2