The paper studies the consistency of mean curvature flow via volumetric varifolds.
problem Consistency of mean curvature flow.
method Discretization using volumetric varifolds and derivation of Brakke approximate equality.
result Derivation of a Brakke approximate equality involving varifold masses and approximate mean curvatures.
The paper proves smoothness of Brakke flows up to the end-time.
problem Smoothness of Brakke flows up to the end-time.
method Local regularity theorem for Brakke flows, extending White's theorem.
result Smooth extension of Brakke flows up to the end-time.
The paper analyzes flows related to Higgs energies on manifolds.
problem Analyzing flows related to Higgs energies on manifolds.
method Developing asymptotic analysis for gradient flow of self-dual U(1)-Higgs energies. result Solutions converge to codimension-two mean curvature flows.
Consider an integral Brakke flow (μt), t∈[0,T], inside some ball in Euclidean space. If μ0 has small height, its measure does not deviate too much from that of a plane and if μT is non-empty, then Brakke's local regularity theorem yields that (μt) is actually smooth and graphical inside a smaller b…
The paper proves a regularity theorem for Brakke flows near triple junctions.
problem Understanding the structure of triple junctions in Brakke flows.
method Establishes the ε-regularity theorem for k-dimensional Brakke flows near static, multiplicity-one triple junctions.
result The regular structure of triple junctions persists under weak mean curvature flow.
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
New measure defined for Brakke flow, linking classical and new definitions.
problem Defining and characterizing the Brakke flow.
method Introduced a space-time-Grassmann measure to characterize the flow.
result Equivalence between classical and new definitions of the Brakke flow.
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.
problem Analyzing free boundary flows through singularities
method Introducing unit-regular and cyclic free boundary flows, proving avoidance principle, and introducing inner and outer flows.
result General tools for analyzing free boundary flows through singularities.
Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.
problem Understanding Brakke flow's non-triviality for smooth boundaries in codimension 1.
method Analyzing spacetime Brakke flow constructed by Buet et al. for initial varifolds.
result Support of mass measure of spacetime Brakke flow coincides with classical mean curvature flow's support.
Survey discusses recent advances on Brakke flows and their properties.
problem Existence and regularity of Brakke flows.
method Analyzes recent progress including new theorems and proofs.
result Proof that branching singularities trigger dynamical instability.
Proves regularity for Brakke flow near stationary half-plane.
problem Regularity of Brakke flow near stationary half-planes.
method Viscosity techniques from Savin applied to Brakke's flow.
result Proves C1,α regularity for flows close to stationary half-planes. Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
Proves existence of multi-phase flows from arbitrary initial data.
problem Non-uniqueness issue in Brakke flows.
method Global existence proof for multi-phase mean curvature flow.
result Validates explicit identity for evolving grain volumes.
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
Proves strong solutions for graphical Brakke flows with L2 normal velocity.
problem Proving strong solutions for graphical Brakke flows with specific velocity conditions.
method Combining L2 normal velocity with parabolic regularity theory. result Graphical Brakke flows with forcing term in Lp,q and C0,α are strong and classical solutions. The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result 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 0, 60, or 120 degrees.
Develops weak formulation for spacelike flows in pseudo-Euclidean space.
problem Weak formulation of spacelike mean curvature flow in pseudo-Euclidean space.
method Based on spacelike integer rectifiable varifolds and pseudo-Euclidean first variation.
result Existence and compactness of spacelike Brakke flows with fixed boundary.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
Constructs approximate mean curvature flows for general varifolds.
problem Mean curvature flow for general initial data.
method Approximation of mean curvature flows using varifolds and iterated push-forwards.
result Approximate mean curvature flow converges to a spacetime Brakke flow under certain conditions.
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.
problem Existence of canonical multi-phase free boundary Brakke flows.
method Global-in-time existence established using Brakke flow and uniform density ratio assumption.
result Existence of the flow with no positive mass on the free boundary for some short time.
Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.
problem Understanding eternal solutions to the Allen-Cahn equation on the 3-sphere.
method Realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, using classifications and rigidity results.
result Construction of eternal integral Brakke flows connecting Clifford tori to equatorial spheres.
Study proves a new flow method for mean curvature with volume change analysis.
problem Existence of BV flow via mean curvature flow.
method Elliptic regularization to prove existence of generalized BV flow.
result Proves existence of generalized BV flow with volume change expression.
Study shows flows from double cones remain symmetric, finds non-symmetric example.
problem Understanding flows from double cones under mean curvature flow.
method Analyzes Brakke flows, proves symmetry, constructs non-symmetric examples.
result Non-self-similar flows exist for entropy at most two.
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…
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…
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.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
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…
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…
The study examines singularities in flows with curvature bounds and identifies unique tangent flows.
problem Analyzing singularities in mean curvature flows with curvature bounds.
method Examines tangent flows and uses stationary and area-minimizing cones to identify unique flows.
result For flows with H∈L∞Llocp, the tangent flow is unique when p=∞ and C is a regular cone. A theorem proves a surface evolution graph satisfies a PDE under specific conditions.
problem Prove a surface evolution graph satisfies a PDE under specific conditions.
method Use Brakke's formulation of velocity and analyze the distributional time derivative of the graph.
result The graph satisfies the PDE pointwise under the given conditions.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
The paper proves smoothness of transition layers in the Allen-Cahn equation.
problem Proving uniform C2,α regularity for transition layers. method Utilizes Allen-Cahn monotonicity formula, Lipschitz approximation, and blowups.
result Shows uniform C2,α regularity for transition layers converging to smooth mean curvature flows. 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.
problem Improving energy equipartition in mean curvature flow.
method Developed a modified Allen-Cahn equation (MAC) by time-dependent parameter and Z-transform.
result Improved estimate on discrepancy energy for better equipartition.
In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in Rn+1 for all n≥3: we show that if a mean curvature flow {Mt} in Rn+1 has an Sn−1×R singularity at (x0,t0), then there exists an $\varepsilon…
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
New varifold solutions for mean curvature flow converge and are unique.
problem Mean curvature flow and Allen-Cahn equation convergence and uniqueness.
method Evolving varifolds coupled to phase volumes, weak-strong uniqueness principle.
result Limits of Allen-Cahn solutions are varifold solutions, and classical flows 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.
Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface…
In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in R3. Namely, if the flow has a spherical or cylindrical singularity at a space-time point X=(x,t), then there exists a positive ε=ε(X)>0 such that the flow is mean convex in a …
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
In this note we show that the recent dynamical stability result for small C1-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 proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. 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.