Neckpinch — Phase fields & multiphase flows

Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators

Wed, 14 Oct 2026 00:00:00 GMT

Jingwen Chen — In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions n≥2. We construct eternal Allen–Cahn flows on Sⁿ⁺¹ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.

Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

Wed, 30 Sep 2026 00:00:00 GMT

Andrea Poiatti — The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.

Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

Thu, 27 Aug 2026 00:00:00 GMT

Tim Laux, Keisuke Takasao — Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.

Generalised stochastic curvature flow in d ≥ 2, and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion

Wed, 12 Aug 2026 00:00:00 GMT

Weijun Xu, Shuhan Zhou — We construct the local-in-time solution of the generalised anisotropic direction-dependent curvature flow in dimension d ≥ 2 forced by a white-in-time and smooth-in-space Gaussian noise. This seems to be the first construction with a white-in-time noise which also allows spatial dependence, even in the simpler case of isotropic stochastic mean curvature flow. The main difficulty is that the stochastic PDE describing the flow has a multiplicative noise depending nonlinearly on both the solution and its gradient. The key technique is a transform developed in based on rough characteristics that removes this rough multiplicative term. We also illustrate the relationship of this transform with previously known special situations. As an application of the construction, we show that in a short time interval, the sharp interface limit of the stochastic Allen-Cahn equation with nonlinear diffusion and the same noise (slightly smoothened in time) is given by the above direction-dependent stochastic curvature flow.

A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

Fri, 31 Jul 2026 00:00:00 GMT

Yuchuan Yang, Selim Esedoglu — We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

Parabolic free boundary phase transition and mean curvature flow

Tue, 16 Dec 2025 00:00:00 GMT

Jingeon An-Lacroix, Kiichi Tashiro — It is known that there is a strong relation between the parabolic Allen–Cahn equation and the mean curvature flow, in the sense that the parabolic Allen–Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow v=−H-∂_νlog |∇u|+f(u)/|∇u|, satisfied by level surfaces of any solution to the nonlinear parabolic equation ∂ₜu=Δu-f(u). Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen–Cahn equation \{beginalignedat2 ∂ₜu&=Δu &&in \|u|<1\ |∇u|&=1/ε &&on ∂\|u|<1\, endalignedat and confirm that under reasonable assumptions, the C^α norm of the forcing term ∂_νlog|∇u| converges to zero at an algebraic rate as ε→ 0, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in ε and in time) in the C^(2,α) sense.

Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctions

Tue, 02 Dec 2025 00:00:00 GMT

Wei-Hung Liao — We consider a multiphase surface 𝓒₀ in ℝ³ consisting of a finite number of surfaces passing through the origin , where all 1-dimensional junctions are regular triple junctions in which three planes meet at the same angle and each surface scales down homothetically to a limit curve of finite length. We prove the existence of self-similar expanding solutions of the mean curvature flow on the multiphase surface initially given by 𝓒₀. For this initial condition, there are multiple solutions that are combinations of the regular triple junctions and regular quadruple points, where four regular triple junctions meet at an angle of approximately 109.5^(∘).

Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

Wed, 08 Oct 2025 00:00:00 GMT

J. M. Daniels-Holgate — When mean curvature flow evolves non-uniquely, the flow is said to fatten. The work of Ilmanen shows that any weak MCF is supported inside the fattening, and work of Hershkovits–White identified canonical weak flows supported on the boundary of the fattening, known as the outermost flows. It is natural to ask, when the flow fattens, are there weak mean curvature flows supported strictly inside the fattening? Outside of some special cases (e.g. flow from cones), this question was entirely open. We show these interior flows exist, providing a general construction for non-outermost flows as limits of solutions to the parabolic ε-Allen–Cahn. This gives the first examples of closed, non-trivial, non-canonical, integral Brakke motions. As part of this construction, we study the ε-Allen–Cahn flow from low regularity initial data, and our results demonstrate the existence of integral Brakke motions from fractal sets. This includes the existence portion of Hershkovits's work on mean curvature flow from Reifenberg sets.

Curvature Flow of Networks with Triple Junction Drag

Sun, 07 Sep 2025 00:00:00 GMT

Yuchuan Yang, Selim Esedoglu — We consider a PDE system that describes curvature motion of networks with a dynamic boundary condition known as triple junction drag. This model arises in the study of grain boundary evolution in polycrystalline materials. In this system, the surface tension coefficients depend on the crystallographic orientations of the grains, which are allowed to rotate. We prove existence and uniqueness of solutions to this system in the parabolic Hölder class C^(2+α,1+α/2). Moreover, we extend our existence result to accommodate a wider class of initial networks by relaxing the compatibility conditions on the angles and curvatures at the triple junction. As an important by-product of our result, we demonstrate the possibility of a new type of topological change during the evolution of the network. We also revisit the question of stability of stationary networks and how it is affected by the choice of surface tensions.

Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

Tue, 12 Aug 2025 00:00:00 GMT

Fabius Krämer — The famous thresholding scheme by Merriman, Bence, and Osher (Motion of multiple junctions: A level set approach. Journal of Computational Physics 112.2 (1994): 334-363.) proved itself as a very efficient time discretization of mean curvature flow. The present paper studies a multiphase, volume constrained version on a weighted manifold that naturally arises in the context of data science. The main result of this work proves convergence to multiphase, weighted, volume-preserving mean curvature flow on a smooth, closed manifold. The proof is only conditional in the sense that convergence of the approximating energy to the weighted perimeter is assumed. These type of assumptions are natural and common in the literature. The proof shows convergence in the energy dissipation inequality, induced by the underlying gradient flow structure, similar to the work of Laux and Otto (The thresholding scheme for mean curvature flow and De Giorgi's ideas for minimizing movements. The Role of Metrics in the Theory of Partial Differential Equations 85 (2020): 63-94.). This leads to a new De Giorgi solution concept that describes weakly the limiting flow. Estimates regarding the derivatives of the heat kernel are developed in order to pass to the limit in the variations of the energy and metric.

Short-time existence of Lagrangian mean curvature flow

Mon, 14 Jul 2025 00:00:00 GMT

Spandan Ghosh — In his paper `Conjectures on Bridgeland Stability', Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez.

Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

Fri, 11 Jul 2025 00:00:00 GMT

Andrea Poiatti — In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

Thu, 29 May 2025 00:00:00 GMT

Andrea Chiesa, Keisuke Takasao — In this paper, we propose a new notion of Brakke inequality for volume preserving mean curvature flow. We show the existence of integral varifolds solving the flow globally-in-time in the corresponding Brakke sense using the phase field method. Moreover, such varifolds are solutions to volume preserving mean curvature flow in the L²-flow sense as well. We thus extend a previous result by one of the authors [25].

Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle

Wed, 26 Mar 2025 00:00:00 GMT

Andrea Chiesa, Karel Svadlenka — We extend the analysis by Esedoglu and Otto (2015) of thresholding energies for the celebrated multiphase Bence-Merriman-Osher algorithm for computing mean curvature flow of interfacial networks, to the case of differing space-dependent anisotropies. In particular, we address the special setting of an obstacle problem, where anisotropic particles move on an inhomogeneous substrate. By suitable modification of the surface energies we construct an approximate energy that uses a single convolution kernel and is monotone with respect to the convolution width. This allows us to prove that the approximate energies Γ-converge to their sharp interface counterpart.

Diffusion-aggregation equations and volume-preserving mean curvature flows

Tue, 25 Mar 2025 00:00:00 GMT

Jiwoong Jang, Antoine Mellet — The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature driven free boundary problems. In this paper, we focus on the Elliptic-Parabolic PKS model and we obtain the first unconditional convergence result in dimension 2 and 3 towards the volume preserving mean-curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on Brakke type formulation of the mean-curvature flow and a reinterpretation of the problem as an Allen-Cahn equation with a nonlocal forcing term.

Existence of weak solutions to volume-preserving mean curvature flow with obstacles

Tue, 07 Jan 2025 00:00:00 GMT

Jiwoong Jang — We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.

Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

Tue, 03 Dec 2024 00:00:00 GMT

Likhit Ganedi, Alice Marveggio, Kerrek Stinson — We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of the diffuse interface vanishes. This convergence result allows us to deduce a Gibbs-Thomson relation for heterogeneous surface tensions. Proceeding from this information, we prove that (weak) solutions of the Allen-Cahn equation with space dependent potential converge to a BV solution of weighted mean curvature flow, under an energy convergence hypothesis. Additionally, relying on the relative energy technique, we establish a weak-strong uniqueness principle for solutions of weighted mean curvature flow.

Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

Fri, 11 Oct 2024 00:00:00 GMT

Helmut Abels, Christoph Hurm, Maximilian Moser — We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a W^(1,1)-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an L²-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform H³-estimates for the nonlocal solutions.

Median filter method for mean curvature flow using a random Jacobi algorithm

Thu, 10 Oct 2024 00:00:00 GMT

Anton Ullrich, Tim Laux — We present an efficient scheme for level set mean curvature flow using a domain discretization and median filters. For this scheme, we show convergence in L^∞-norm under mild assumptions on the number of points in the discretization. In addition, we strengthen the weak convergence result for the MBO thresholding scheme applied to data clustering of Lelmi and one of the authors. This is done through a strong convergence of the discretized heat flow in the optimal regime. Different boundary conditions are also discussed.

Deep Learning as Ricci Flow

Mon, 22 Apr 2024 00:00:00 GMT

Anthony Baptista, Alessandro Barp, Tapabrata Chakraborti, Chris Harbron, Ben D. MacArthur, Christopher R. S. Banerji — Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow - a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of `global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than 1,500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Stability of multiphase mean curvature flow beyond circular topology changes

Wed, 03 Apr 2024 00:00:00 GMT

Julian Fischer, Sebastian Hensel, Alice Marveggio, Maximilian Moser — We prove a weak-strong uniqueness principle for varifold-BV solutions to planar multiphase mean curvature flow beyond a circular topology change: Assuming that there exists a classical solution with an interface that becomes increasingly circular and shrinks to a point, any varifold-BV solution with the same initial interface must coincide with it, and any varifold-BV solution with similar initial data must undergo the same type of topology change. Our result illustrates the robustness of the relative energy method for establishing weak-strong uniqueness principles for interface evolution equations, showing that it may also be applied beyond certain topological changes.

Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

Sat, 27 Jan 2024 00:00:00 GMT

Giovanni Bellettini, Shokhrukh Kholmatov, Firdavsjon Almuratov — We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.

New advances on the existence and regularity of Brakke's mean curvature flows

Thu, 09 Nov 2023 00:00:00 GMT

Salvatore Stuvard — In this survey paper, I discuss some recent progress on the existence and regularity of Brakke flows. These include: an "end-time version" of Brakke's local regularity theorem, which allows to extend the validity of the celebrated regularity theorem by White from limits of smooth mean curvature flows to arbitrary Brakke flows; a global-in-time existence theorem for multi-phase Brakke flows of grain boundaries satisfying suitable rm BV regularity in time, in both the unconstrained and the fixed boundary settings, with applications to Plateau's problem for the latter; and the proof that branching singularities of minimal surfaces are a trigger for dynamical instability, in the sense that they may be "perturbed away" by a non-trivial canonical Brakke flow. This note is an extended version of a talk given by the author at the MATRIX Research Institute on the occasion of the workshop entitled "Minimal surfaces and geometric flows: interaction between the local and the nonlocal worlds".

Stability analysis for the anisotropic curve shortening flow of planar networks

Mon, 09 Oct 2023 00:00:00 GMT

Michael Gößwein, Matteo Novaga, Paola Pozzi — In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are C^(2,α)-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.

Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

Thu, 21 Sep 2023 00:00:00 GMT

Milan Kroemer, Tim Laux — We prove a quantitative convergence result of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak-strong uniqueness results for mean curvature flow and convergence of the Allen–Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.

Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

Fri, 03 Feb 2023 00:00:00 GMT

Xinfu Chen, Bendong Lou, Xiaoliu Wang, Lixia Yuan — We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen–Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a fixed profile and finite speed. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient |Du| (away from the center) and the instantaneous speed uₜ diverge exponentially as t→∞. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time C⁰, C¹, or C² estimates are available, our analysis relies on a new approach based on the zero number argument.

Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

Sat, 28 Jan 2023 00:00:00 GMT

Pascal Steinke — This paper presents a conditional convergence result of solutions to the Allen–Cahn equation with arbitrary potentials to a De Giorgi type BV-solution to multiphase mean curvature flow. Moreover we show that De Giorgi type BV-solutions are De Giorgi type varifold solutions, and thus our solution is unique in a weak-strong sense.

Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

Thu, 22 Dec 2022 00:00:00 GMT

Tim Laux, Kerrek Stinson, Clemens Ullrich — The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Type-0 singularities in the network flow – Evolution of trees

Fri, 08 Jul 2022 00:00:00 GMT

Carlo Mantegazza, Matteo Novaga, Alessandra Pluda — The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a Type-0 singularity, in contrast to the well known Type-I and Type-II ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree-like networks till the first singular time, under the assumption that all the tangents flows have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results by Ilmanen-Neves-Schulze/Lira-Mazzeo-Pluda-Saez to restart the flow after the singularity.

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Thu, 31 Mar 2022 00:00:00 GMT

Julian Fischer, Alice Marveggio — Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε↘ 0. We even establish the rate of convergence O(ε^(1/2)). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.