arXiv research

New measure defined for Brakke flow, linking classical and new definitions.

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 v…

Foundations for free boundary Brakke flows established.

Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.

Survey discusses recent advances on Brakke flows and their properties.

Proves regularity for Brakke flow near stationary half-plane.

Brakke flow support is parabolically rectifiable

The paper proves smoothness of Brakke flows up to the end-time.

Proves existence of multi-phase flows from arbitrary initial data.

The study examines mass drop and multiplicity in mean curvature flow.

Proves strong solutions for graphical Brakke flows with $L^2$ normal velocity.

The abstract proves the existence and regularity of Brakke flows starting from a given set.

The paper studies the consistency of mean curvature flow via volumetric varifolds.

The paper proves a regularity theorem for Brakke flows near triple junctions.

Study shows non-uniqueness of Brakke flow near flat singular points.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

Consider an integral Brakke flow $(μ_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $μ_{0}$ has small height, its measure does not deviate too much from that of a plane and if $μ_{T}$ is non-empty, then Brakke's local regularity theorem yields that $(μ_t)$ is actually smooth and graphical inside a smaller b…

Constructs approximate mean curvature flows for general varifolds.

In 1978 Brakke introduced the mean curvature flow in the setting of geometric measure theory. There exist multiple variants of the original definition. Here we prove that most of them are indeed equal. One central point is to correct the proof of Brakke's §3.5, where he develops an estimate for the evolution of the mea…

Study proves existence of a specific type of flow in geometry.

New, shorter proofs for varifolds and flows with improved decay of flatness.

Study proves a new flow method for mean curvature with volume change analysis.

A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a conse…

The paper analyzes flows related to Higgs energies on manifolds.

New non-canonical flows found via parabolic Allen-Cahn equations.

In [LW], 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 form…

Flow of curves with curvature and forcing vector field exists.

In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable c…

Study resolves flow through cylindrical singularities, proving nonfattening.

Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.

The paper proves smoothness of transition layers in the Allen-Cahn equation.

In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

Paper introduces a modified Allen-Cahn equation for better energy equipartition.

New varifold solutions for mean curvature flow converge and are unique.

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.

Generic level sets in mean curvature flow are BV solutions.

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 directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stab…

Study shows flows from double cones remain symmetric, finds non-symmetric example.

Study proves existence of weak mean curvature flow with contact angle.

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.

A theorem proves a surface evolution graph satisfies a PDE under specific conditions.

For decades, the sphere eversion has been a classic subject for mathematical visualization. The 1998 video "The Optiverse" shows geometrically optimal eversions created by minimizing elastic bending energy. We contrast these minimax eversions with earlier ones, including those by Morin, Phillips, Max, and Thurston. The…

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 me…

Mean curvature flow of clusters of n-dimensional surfaces in 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 smoot…

Let $\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} …

Suppose that $Γ_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus Γ_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $Γ_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we…

A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively studied ever since the pioneering work of Brakke and Huisken. In the last 15 years, Whi…