Brakke Flow Research Articles

Generic Level Sets in Mean Curvature Flow are BV Solutions

We show that a generic level set of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate. This extends the fundamental work of Evans and Spruck (J Geom Anal 5(1):79–116, 1995), which draws a similar connection between the viscosity solution and Brakke flows.

Time-global existence of generalized BV flow via the Allen–Cahn equation

We show that a mean curvature flow obtained as the limit of the Allen–Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa (2024).

End-time regularity theorem for Brakke flows

AbstractFor a general k-dimensional Brakke flow in $${\mathbb {R}}^n$$ R n locally close to a k-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to 1, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White’s local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.

A Short Proof of Allard’s and Brakke’s Regularity Theorems

Abstract We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.

Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

Given a double cone $${\mathcal {C}}$$ with entropy at most two that is symmetric across some hyperplane, we show that any integral Brakke flow coming out of the cone must inherit the reflection symmetry for all time, provided the flow is smooth for a short time. As a corollary we prove that any such flow coming out of a rotationally symmetric double cone must stay rotationally symmetric for all time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity. Additionally, we show the existence of self-expanders with triple junctions, which are exceptions to our main theorem.

Structural descriptions of limits of the parabolic Ginzburg-Landau equation on closed manifolds

In the setting of a compact Riemannian manifold of dimension $N\ge3$, we provide a structural description of the limiting behavior of the energy measures of solutions to the parabolic Ginzburg-Landau equation. In particular, we provide a decomposition of the limiting energy measure into a diffuse part, which is absolutely continuous with respect to the volume measure, and a concentrated part supported on a codimension $2$ rectifiable subset. We also demonstrate that the time evolution of the diffuse part is determined by the heat equation while the concentrated part evolves according to a Brakke flow. This paper extends the work of Bethuel, Orlandi, and Smets from [8].

On the existence of canonical multi-phase Brakke flows

Abstract This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established in [J. Fischer, S. Hensel, T. Laux and T. M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, preprint 2020, https://arxiv.org/abs/2003.05478], under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $${\mathbb {R}}^{n+1}$$ for all $$n\ge 3$$ : we show that if a mean curvature flow $$\{M_t\}$$ in $${\mathbb {R}}^{n+1}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists an $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such that $$M_t\cap B(x_0,\varepsilon )$$ is mean-convex for all $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$$ . As in the case $$n=2$$ , which was resolved by the first three authors in Choi et al. (Acta Math, 2018), the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors (Hershkovits and White in Commun Pure Appl Math 73(3):558–580, 2020), we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $$n\ge 3$$ , smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call “moving plane method without assuming smoothness”—where smoothness and symmetry are established in tandem.

Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2\pi/e}+\delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.

An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably n-rectifiable set in a strictly convex (n+1)-dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As t uparrow infty , the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and some uniform boundedness property of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

Existence and regularity theorems of one-dimensional Brakke flows

Given a closed countably 1-rectifiable set in \mathbb R^2 with locally finite 1 -dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.

Regularity of minimal submanifolds and mean curvature flows with a common free boundary

Let N be a smooth (n + l)-dimensional Riemannian manifold. We show that if V is an area-stationary union of three or more n-dimensional submanifolds-with-boundary with a common boundary Γ, then Γ is smooth and each Mk is smooth up to Γ (real-analytic in the case N is real-analytic). This extends a previous result of the author for codimension l = 1. We additionally show that if is a Brakke flow such that each time-slice Vt is a union of three or more n-dimensional submanifolds-with-boundary with a common boundary Γt and with parabolic regularity in time-space, then and are smooth (second Gevrey with real-analytic time-slices in the case N is real-analytic).

Consequences of Strong Stability of Minimal Submanifolds

Abstract In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang [12] directly extends to the enhanced Brakke flows of Ilmanen [5]. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow, which are proved to occur by Neves [7].

The free-boundary Brakke flow

Abstract We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must “pop” upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen’s elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

A local regularity theorem for mean curvature flow with triple edges

Abstract Mean curvature flow of clusters of n-dimensional surfaces in ℝ n + k {\mathbb{R}^{n+k}} that meet in triples at equal angles along smooth edges and higher order junctions on lower-dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

The Blow Up Method for Brakke Flows: Networks Near Triple Junctions

We introduce a parabolic blow-up method to study the asymptotic behavior of an integral Brakke flow of planar networks (i.e. a 1-dimensional integral Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction $J$. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 degree angles, staying smoothly close to $J$ and moving smoothly. Using this result and White's stratification theorem, we deduce that whenever an integral Brakke flow of networks in a space-time region ${\mathcal R}$ has no static tangent flow with density $\geq2$, there exists a closed subset $\Sigma \subset {\mathcal R}$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in ${\mathcal R} \setminus \Sigma$, i.e. near every point in ${\mathcal R} \setminus \Sigma$, the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 degree angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $\frac{1}{2}$.

Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let \({\mathcal{M}}\) be a Brakke flow of n-dimensional surfaces in \({\mathbb{R}^N}\). The singular set \({\mathcal{S} \subset \mathcal{M}}\) has a stratification \({\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}\), where \({X \in \mathcal{S}^j}\) if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata \({\mathcal{S}^j_{\eta, r}}\) satisfying \({\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}\). Sharpening the known parabolic Hausdorff dimension bound \({{\rm dim} \mathcal{S}^j \leq j}\), we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of \({\mathcal{S}^j_{\eta, r}}\) satisfies \({{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}\). Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by \({\mathcal{B}_r \subset \mathcal{M}}\) the set of points with regularity scale less than r, we prove that \({{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}\). This gives L p-estimates for the second fundamental form for any p < n + 1 − k. In fact, the estimates are much stronger and give L p-estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).

Hamiltonian stationary cones and self-similar solutions in higher dimension

In an upcoming paper by Lee and Wang, we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimensions. We construct new higher-dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are constructed as well. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show that the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author.