The space-time-Grassmann measure of the Brakke flow

Technical Summary: The Space-Time-Grassmann Measure of the Brakke Flow

Problem Statement
The paper addresses the structural characterization of the $k$-dimensional Brakke flow, a weak, measure-theoretic solution to the mean curvature flow defined on an open subset $U \subset \mathbb{R}^n$ over a time interval $J$. While the classical definition (Brakke [4]) describes the flow as a family of Radon measures $\{\mu(t)\}_{t \in J}$ satisfying a specific differential inequality, the authors seek to reformulate this concept using a canonical space-time measure. The primary challenge lies in rigorously defining a measure on the product space $J \times G_k(U)$ (time $\times$ Grassmannian) that captures the flow's evolution, handles jump discontinuities inherent in the flow, and provides a unified framework equivalent to existing classical definitions.

Methodology
The authors employ techniques from geometric measure theory, specifically the theory of varifolds and distributional derivatives. Their approach proceeds through the following logical steps:

1. Measurability of the Varifold Map: The authors first establish that for a Brakke flow $\{\mu(t)\}$, the associated map $t \mapsto V(t)$, where $V(t)$ is the unique integral varifold such that $\|V(t)\| = \mu(t)$ (which exists for $L^1$-almost every $t$), is $L^1$-measurable. This relies on the upper semicontinuity of the Brakke variation functional $B(\mu, \phi)$.
2. Construction of the Space-Time Measure: Using the measurability of $t \mapsto V(t)$, they define a canonical Radon measure $\lambda$ on $J \times G_k(U)$ via integration:
   $$\lambda(f) = \int_J \int_{G_k(U)} f(t, x, S) \, dV(t)(x, S) \, dL^1_t$$
   for test functions $f$. The associated space-time weight measure is denoted $\|\lambda\|$.
3. Characterization via Distributional Inequality: The authors define a "space-time-Grassmann Brakke flow" as a Radon measure $V$ on $J \times G_k(U)$ satisfying a distributional Brakke inequality involving the time derivative and the functional $B$.
4. Analysis of Discontinuities: Recognizing that Brakke flows behave like one-dimensional functions of bounded variation (BV), the authors analyze the left and right limits of the flow. They define measures $\langle V^-, t \rangle$ and $\langle V^+, t \rangle$ representing the left-continuous and right-continuous representatives of the flow at time $t$, derived from the space-time measure.
5. Measurability of Geometric Quantities: The paper utilizes Borel regularity arguments (referencing Federer [6]) to prove that the tangent map, mean curvature vector, and density are measurable with respect to the space-time weight measure $\|\lambda\|$.

Key Contributions and Results

• Theorem A (Existence of Canonical Measure): Proves that for any Brakke flow, there exists a unique Radon measure $\lambda$ on $J \times G_k(U)$ such that the flow can be recovered by integrating $\lambda$ over time. This measure is defined by the integral of test functions against the family of varifolds $V(t)$.
• Theorem B (Characterization and Equivalence): Establishes that a Radon measure $V$ on $J \times G_k(U)$ satisfying the distributional Brakke inequality corresponds to a class of equivalent classical Brakke flows. Specifically:
  • The measures $\|\langle V^-, \cdot \rangle\|$ and $\|\langle V^+, \cdot \rangle\|$ are themselves Brakke flows.
  • Any classical Brakke flow $\nu(t)$ associated with $V$ must satisfy $\|\langle V^-, t \rangle\| \geq \nu(t) \geq \|\langle V^+, t \rangle\|$ for all $t$.
  • This yields a new definition of the Brakke flow as a space-time measure, which is shown to be equivalent to the classical definition (Definition 3.4) and the alternate definition found in Lahiri [9].
• Theorem C (Measurability of Geometric Data): Proves that the tangent plane map $Tan(V(t), x)$, the generalized mean curvature vector $h(V(t), x)$, and the density $\Theta_k(\|V(t)\|, x)$ are all measurable with respect to the space-time weight measure $\|\lambda\|$. This is achieved by showing these maps are Borel functions on the space of varifolds and composing them with the measurable map $(t, x) \mapsto (V(t), x)$.
• Resolution of Approximation Issues: In Section 3.6, the authors provide an alternative proof for the upper semicontinuity of the Brakke variation $B(\|V_i\|, \phi)$. They explicitly address a gap in previous literature (Ilmanen [8]) where an interior approximation of the test function $\phi$ was assumed to exist for $C^2$ functions. The authors demonstrate via Example 3.8 that such approximations do not generally exist for $C^2$ functions and provide a proof that avoids this assumption.

Significance and Claims
The paper claims to provide a "new definition" of the Brakke flow that unifies the classical measure-theoretic evolution with a static space-time measure framework. The significance of this formulation is threefold:

1. Equivalence: It rigorously establishes the equivalence between the classical definition (a family of measures evolving in time) and a definition based on a single space-time measure satisfying a distributional inequality.
2. Handling Discontinuities: The framework naturally accommodates the jump discontinuities inherent in Brakke flows by distinguishing between left-continuous and right-continuous representatives, characterizing the flow's behavior at singular times.
3. Foundational Groundwork: The authors state that the formulation of these measures and the proven properties (specifically the measurability of geometric quantities) lay the groundwork for adapting results concerning integral varifolds with locally bounded first variation to the parabolic setting of mean curvature flow.

The paper does not propose new experimental applications or future conjectures beyond the immediate mathematical implications of the equivalence and measurability results. It positions itself as a technical refinement and unification of existing definitions, citing the work of Hensel and Laux [7] and Buet et al. [5] as related explorations of space-time-Grassmann measures in weak solution concepts.