Parabolic rectifiability of the Brakke flow

Abstract: We prove that the support of the canonical space-time measure for a Brakke flow is a parabolic $(k+2)$-rectifiable set. As a consequence, we obtain that at almost all points along the flow, with respect to this canonical space-time measure, there exists a unique, static, planar tangent flow, and that various notions of density for the flow agree at these points. Moreover, following on from our previous work The space-time-Grassmann measure of the Brakke flow', we continue to develop the approach to the Brakke flow as a space-time-Grassmann measure. We prove that the standard notion of convergence for Brakke flows, coming from the compactness theorem of Ilmanen (7.1 ofElliptic regularization and partial regularity for motion by mean curvature'), is equivalent to the convergence of these space-time-Grassmann Radon measures. This gives an alternate notion of varifold convergence to the one exhibited in 7.1(ii) of `Elliptic regularization and partial regularity for motion by mean curvature'.

• The paper proves that the support of every Brakke flow’s canonical space-time measure is vertically parabolically (k+2)-rectifiable using tangent-measure analysis, density bounds, and cone-avoidance arguments.
• The paper shows that at almost every space-time point, the flow has a unique static planar tangent, and Gaussian, spatial, and parabolic density values coincide.
• The paper establishes that weak convergence of space-time-Grassmann measures, weak convergence of their weight measures, and Ilmanen’s time-slice convergence are equivalent under the stated hypotheses.

Overview

The paper under discussion, "Parabolic rectifiability of the Brakke flow" by Yu Tong Liu and Myles Workman, establishes a fundamental regularity property of the canonical space-time measure associated with a $k$-dimensional Brakke flow. The central result states that the support of the weight measure $\|V\|$ of any Brakke flow over $J \times U \subset \mathbf{R} \times \mathbf{R}^n$ is a (vertically) parabolic $(k+2)$-rectifiable set, in the sense developed by Mattila. Two significant consequences follow: at $\|V\|$ almost every point the flow admits a unique static planar tangent flow, and the various density notions for Brakke flows — Gaussian density, spatial $k$-density of time slices, and parabolic $(k+2)$-densities with respect to both standard parabolic metrics — coincide at those points. A secondary contribution is an equivalence theorem showing that Ilmanen's standard notion of convergence of Brakke flows coincides with weak convergence of the associated space-time-Grassmann Radon measures.

The work continues the authors' program, initiated in their earlier paper on the space-time-Grassmann measure of the Brakke flow, of treating a Brakke flow as a single Radon measure on $J \times \mathbf{G}_k(U)$ rather than as a one-parameter family of measures on $U$. This viewpoint is not merely cosmetic: it permits direct application of classical geometric measure theory tools — differentiation theory for Radon measures, approximate continuity, and tangent measure analysis in the parabolic metric — to objects that are otherwise studied through time-slice-dependent quantities.

Background and framework

A Brakke flow is a family $\{\mu(t)\}_{t \in J}$ of Radon measures evolving as a weak solution of mean curvature flow, defined via Brakke's inequality against non-negative class 2 test functions $\|V\|$0, using the functional $\|V\|$1 built from generalized mean curvature when the relevant integrability conditions hold. For such a flow there exists a canonical Radon measure $\|V\|$2 on $\|V\|$3, obtained by integrating the integral varifolds $\|V\|$4 (which exist uniquely for $\|V\|$5 almost all $\|V\|$6) against Lebesgue measure in time; its weight is

$\|V\|$7

The authors adopt the equivalent definition of a space-time-Grassmann Brakke flow: a Radon measure $\|V\|$8 on $\|V\|$9 whose time slices are integral varifolds almost everywhere and which satisfies the distributional form of Brakke's inequality. They record several structural facts in this language: the first variation $J \times U \subset \mathbf{R} \times \mathbf{R}^n$0 is itself a Radon measure on $J \times U \subset \mathbf{R} \times \mathbf{R}^n$1 (with an $J \times U \subset \mathbf{R} \times \mathbf{R}^n$2 bound on mean curvature on compact sets), and the extended Brakke inequality holds for test functions depending on both time and space.

Two parabolic metrics are used throughout:

$J \times U \subset \mathbf{R} \times \mathbf{R}^n$3

which are bi-Lipschitz equivalent. Rectifiability is formulated via vertical parabolic Lipschitz graphs over $J \times U \subset \mathbf{R} \times \mathbf{R}^n$4 for $J \times U \subset \mathbf{R} \times \mathbf{R}^n$5, following Mattila's characterization: a set is parabolic $J \times U \subset \mathbf{R} \times \mathbf{R}^n$6-rectifiable if and only if, at almost every point, its tangent measures are precisely the multiples of $J \times U \subset \mathbf{R} \times \mathbf{R}^n$7 for some plane $J \times U \subset \mathbf{R} \times \mathbf{R}^n$8. Crucially, the authors invoke Itoh's Besicovitch covering theorem for parabolic balls, which makes the covering relation by $J \times U \subset \mathbf{R} \times \mathbf{R}^n$9-balls a Vitali relation and hence licenses Federer's differentiation theory and the notion of approximate continuity in space-time.

Density bounds play a key technical role. From the mean value inequality (equivalently, Brakke's clearing out lemma), the authors derive a uniform lower bound $(k+2)$0 on $(k+2)$1, implying $(k+2)$2. Under a total mass bound they also obtain a local upper density estimate via a monotonicity-formula argument with a carefully constructed cutoff, giving mutual absolute continuity of $(k+2)$3 and $(k+2)$4. This absolute continuity is what allows rectifiability information about the Hausdorff-restricted support to transfer to the weight measure itself.

Convergence of Brakke flows

The convergence theorem asserts the equivalence of three statements for space-time-Grassmann Brakke flows $(k+2)$5 and $(k+2)$6: weak convergence of the Grassmann measures $(k+2)$7; weak convergence of the weights $(k+2)$8; and convergence of representatives $(k+2)$9 for all $\|V\|$0 outside a countable exceptional set $\|V\|$1.

The proof proceeds in two directions. That weight convergence implies Grassmann convergence uses the uniform $\|V\|$2 control on mean curvature: any subsequential limit $\|V\|$3 has $\|V\|$4 a Radon measure, and a uniqueness argument (two Grassmann measures with equal weights and Radon first variations must coincide, since their time slices agree as integral varifolds almost everywhere) forces $\|V\|$5. The converse direction is more delicate: writing $\|V\|$6, the functions $\|V\|$7 are non-increasing for suitable $\|V\|$8, uniformly bounded in BV by the mass bound derived from the clearing-out lemma, and converge weakly to $\|V\|$9; a contradiction argument using Ambrosio–Fusco–Pallara then upgrades this to $k$0 convergence, and a lemma on monotone functions yields pointwise convergence at continuity points of $k$1. The exceptional set can be taken to be the jump set where $k$2.

This result is notable because condition (3), after passing to a subsequence converging along a countable dense set of times, is exactly Ilmanen's standard notion of Brakke flow convergence. The equivalence therefore provides an alternative varifold-level convergence notion to the one exhibited in Ilmanen's compactness theorem, and it identifies the space-time-Grassmann measure as the natural object carrying the full convergence structure.

Parabolic rectifiability of the support

The main theorem states that $k$3 is a (vertical) parabolic $k$4-rectifiable set. The proof strategy combines three ingredients:

Approximate continuity of tangent planes. By the authors' earlier work and the Vitali relation machinery, the map $k$5 is approximately continuous with respect to $k$6 at almost every point, and $k$7 holds $k$8 almost everywhere. Removing the null set where either fails leaves $k$9.

Flat blow-ups. At points of $(k+2)$0, any blow-up limit $(k+2)$1 is stationary and supported entirely on planes parallel to $(k+2)$2. A classification lemma for such flat flows shows that each time slice of $(k+2)$3 is an at most countable sum of parallel planes with integer multiplicities given by non-increasing functions $(k+2)$4, constant and equal to the Gaussian density $(k+2)$5 for negative times, with $(k+2)$6.

Cone avoidance and graph covering. A companion lemma shows that points off $(k+2)$7 stay a definite distance away from the rescaled supports, using the clearing-out lemma to rule out mass appearing near the complement of the plane. Consequently, for each point of $(k+2)$8 and each sufficiently small scale, the support avoids a parabolic cone about $(k+2)$9 for some approximating plane $J \times \mathbf{G}_k(U)$0 from a fixed countable dense family. Mattila's Lipschitz graph criterion then decomposes $J \times \mathbf{G}_k(U)$1 into countably many pieces, each covered by vertical parabolic Lipschitz graphs of arbitrarily small slope, establishing rectifiability.

An important consequence follows immediately from Mattila's characterization: since $J \times \mathbf{G}_k(U)$2 is absolutely continuous with respect to $J \times \mathbf{G}_k(U)$3, tangent measures of $J \times \mathbf{G}_k(U)$4 at almost every point have the form $J \times \mathbf{G}_k(U)$5 for a unique plane $J \times \mathbf{G}_k(U)$6 and finite positive constant $J \times \mathbf{G}_k(U)$7.

Unique tangent flows and equality of densities

Combining the flat-flow classification with the rectifiability-induced rigidity of tangent measures yields the second main theorem: for $J \times \mathbf{G}_k(U)$8 almost every $J \times \mathbf{G}_k(U)$9, the set of varifold tangents $U$0 consists of the single element

$U$1

That is, the tangent flow is unique, static, and planar. The mechanism is worth noting: the self-shrinking equation forces the multiplicity function of a flat tangent flow to be constant for negative times, but in principle the multiplicity could drop at non-negative times. Rectifiability forbids this drop almost everywhere, because tangent measures to $U$2 must be scalar multiples of a single static planar measure.

At these points all density notions agree:

$U$3

The authors emphasize that uniqueness of the static planar tangent flow was previously inaccessible: White's stratification guarantees existence of some static planar tangent flow at $U$4 almost every point, but stratification alone cannot rule out multiple distinct planar tangents, and hence cannot yield equality of densities. The final corollary expresses the weight measure as a variable-density restriction of parabolic Hausdorff measure,

$U$5

obtained by combining the rectifiability theorem, Mattila's density theorem for parabolic rectifiable sets, and the Vitali differentiation theorem.

Limitations and open questions

Several qualifications attach to these results. First, the conclusions are almost-everywhere statements with respect to the canonical space-time measure $U$6; nothing is asserted about the behavior at individual singular points, and the exceptional set where the static planar tangent fails or densities disagree may be large in other senses. Second, the rectifiability obtained is vertical parabolic rectifiability — graphs are taken over $U$7 with the time direction included in the base — which is the notion suited to the heat-flow scaling; whether stronger structural conclusions hold at specific strata of the singular set is not addressed. Third, the equality of densities identifies the Gaussian density with the spatial density of time slices only at $U$8 almost every point; the relationship at times belonging to the jump set $U$9, where $\{\mu(t)\}_{t \in J}$0 is possible, remains outside the scope of the theorem. Finally, the convergence theorem characterizes convergence of flows whose limit is already known to be a Brakke flow; the compactness statement still requires the separate hypothesis of locally bounded mass, inherited from Ilmanen's work.

Conclusion

This paper establishes that the canonical space-time measure of any Brakke flow lives on a parabolically rectifiable set, and leverages this to obtain uniqueness of static planar tangent flows and agreement of Gaussian, spatial, and parabolic densities at almost every space-time point — results previously unattainable through stratification alone. The equivalence between Ilmanen's convergence notion and weak convergence of space-time-Grassmann measures validates the authors' program of treating Brakke flows as single space-time-Grassmann objects, to which the classical differentiation and tangent-measure theory of Federer and Mattila applies directly. The results provide a measure-theoretic foundation on which finer questions about the size and structure of the singular set of general Brakke flows can subsequently be posed.