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Mean curvature flow

Mean curvature flow is a geometric evolution equation in which a family of hypersurfaces in Euclidean space evolves over time such that the normal velocity at each point is equal to the mean curvature of the surface at that point.^[1] Formally, if F: M \times [0, T) \to \mathbb{R}^{n+1} parametrizes the evolving hypersurface M_t = F(M, t), the flow satisfies \frac{\partial F}{\partial t} = \vec{H}, where \vec{H} is the mean curvature vector.^[2] This flow, first formalized in a weak sense by Kenneth Brakke in 1978 using varifolds to handle singularities, models the motion of interfaces driven by surface tension, such as in grain boundary evolution during metal annealing.^[3] For smooth initial hypersurfaces, Gerhard Huisken established short-time existence and uniqueness in 1984, showing that the flow acts as a gradient descent for the area functional, monotonically decreasing the surface area while smoothing irregularities.^[4] Key properties include the maximum principle, which preserves geometric inclusions like convexity, and monotonicity formulas that control the evolution.^[2] Compact convex hypersurfaces contract to round points in finite time, as proven by Huisken.^[4] However, the flow can develop singularities, such as neckpinch phenomena, necessitating weak formulations like Brakke flows or level-set methods for continuation beyond singularities.^[1] In higher dimensions and codimensions, the flow reveals rich singularity structures, with generic singularities often of lower dimension, and has applications in proving theorems like the Riemannian Penrose inequality via rescaled limits.^[2] Ongoing research explores non-generic behaviors, stability, and extensions to Riemannian manifolds, including recent advances in singularity analysis and uniqueness theorems as of 2025.^[5]^[6]^[7]

Fundamentals

Definition

Mean curvature flow is a geometric evolution equation in which a hypersurface in Euclidean space evolves over time such that each point on the hypersurface moves in the direction of the unit normal with speed equal to the mean curvature of the hypersurface at that point.^[4] This process can be viewed as the L^2 gradient flow of the area functional, which decreases the total area of the hypersurface while preserving its topology under suitable conditions.^[8] The mean curvature H at a point on the hypersurface is defined as the trace of the second fundamental form, or equivalently, the sum (or average, up to scaling) of the principal curvatures \kappa_1, \dots, \kappa_n of the hypersurface, H = \kappa_1 + \dots + \kappa_n.^[4] For a family of hypersurfaces M_t in \mathbb{R}^{n+1} parameterized by time t \geq 0, with a smooth immersion X: M \times [0, T) \to \mathbb{R}^{n+1} describing the position, the flow is governed by the equation

\frac{\partial X}{\partial t} = \vec{H}, $&#36; where $\vec{H}$ is the mean curvature vector.[](https://web.stanford.edu/~ochodosh/MCFnotes.pdf) This setup applies to both open and closed hypersurfaces, though much of the classical theory focuses on compact, embedded cases. ### Historical Development The origins of mean curvature flow can be traced to 19th-century investigations into minimal surfaces, where Belgian physicist Joseph Plateau's experiments with soap films revealed that these films span wire frames to form surfaces minimizing area, characterized by zero [mean curvature](/page/Mean_curvature). Plateau's 1873 treatise formalized these observations, linking physical phenomena to the mathematical problem of finding area-minimizing surfaces bounded by given curves, laying foundational groundwork for later geometric evolution equations. In the mid-20th century, the flow emerged in [materials science](/page/Materials_science) through the work of W. W. Mullins, who in 1957 modeled the evolution of grain boundaries and thermal grooves in metals as motion proportional to [curvature](/page/Curvature), deriving equations that align with the modern mean curvature flow for interfaces driven by [surface diffusion](/page/Surface_diffusion). This physical motivation provided early applications in understanding annealing processes and boundary dynamics in crystalline materials. Mullins' 1959 collaboration with P. G. Shewmon further refined these kinetics for [copper](/page/Copper) bicrystals, verifying the predicted rates experimentally.[](https://doi.org/10.1016/0001-6160(59)90069-0) The rigorous mathematical framework for [mean curvature](/page/Mean_curvature) flow was established in 1978 by K. A. Brakke, who introduced weak solutions using the theory of varifolds to handle [singularities](/page/Singularity) and topological changes that smooth flows cannot accommodate, enabling the study of evolving surfaces beyond classical partial differential equations. Building on this, Gerhard Huisken in 1984 analyzed smooth solutions for convex hypersurfaces, proving convergence to spheres and deriving a monotonicity formula for the Gaussian area functional, which quantifies entropy-like decrease and aids in [singularity](/page/Singularity) analysis. Concurrently, Claus Gerhardt's work on [quasilinear](/page/Quasilinear) parabolic systems provided [existence](/page/Existence) and regularity results for [mean curvature](/page/Mean_curvature) flows in general settings, including [Lorentzian](/page/Lorentzian) manifolds.[](https://link.springer.com/article/10.1007/BF01420219) The 1980s and 1990s saw expansions, with Steven Altschuler, Sigurd Angenent, and Yoshikazu Giga's 1992 collaboration demonstrating flow through singularities for rotationally symmetric surfaces using viscosity solutions. Meanwhile, connections to higher-dimensional geometry grew through [Richard Hamilton](/page/Richard)'s 1982 introduction of [Ricci flow](/page/Ricci_flow) on Riemannian manifolds, an analogue of mean curvature flow for metrics, which evolved to address topological questions. This culminated in Grigori Perelman's 2002–2003 proofs resolving the [Poincaré conjecture](/page/Poincaré_conjecture) via [Ricci flow](/page/Ricci_flow) with surgery, highlighting structural parallels to mean curvature flow singularities and weak continuations. ## Mathematical Formulation ### Evolution Equation Mean curvature flow (MCF) arises as the gradient flow of the area functional for a time-dependent family of hypersurfaces $M_t$ in Euclidean space, defined by $A(M_t) = \int_{M_t} \, d\mu_t$, where $d\mu_t$ denotes the area element induced by the metric $g_t$ on $M_t$. The first variation of the area under a normal variation $\phi \nu$, with $\phi$ a scalar function and $\nu$ the unit normal, yields $\frac{d}{dt} A(M_t) \big|_{t=0} = -\int_{M} \phi H \, d\mu$, where $H = \mathrm{tr}_g h$ is the scalar mean curvature, with $h$ the second fundamental form. The $L^2$ gradient of the area functional is thus the mean curvature vector $\mathbf{H} = H \nu$, and the flow that decreases area most rapidly is given by the evolution equation

\frac{\partial}{\partial t} X = -\mathbf{H} = -H \nu

for the position vector $X$ of points on the hypersurface, where $\nu$ is chosen outward for closed hypersurfaces. Under this flow, the area evolves as $\frac{d}{dt} A(M_t) = -\int_{M_t} H^2 \, d\mu_t \leq 0$. In local coordinates on the hypersurface, the evolution equation takes the normal form $\frac{\partial}{\partial t} X^\alpha = -H \nu^\alpha$, where Greek indices denote ambient Euclidean coordinates. The induced metric $g_{ij}$ on $M_t$ evolves according to

\frac{\partial}{\partial t} g_{ij} = -2 H h_{ij},

derived by differentiating $g_{ij} = \langle \partial_i X, \partial_j X \rangle$ and projecting onto the tangent space, using the compatibility of the second fundamental form and the flow's normal direction. The full parabolic system under MCF includes evolution equations for the geometric quantities. The second fundamental form $h_{ij}$ satisfies

\frac{\partial}{\partial t} h_{ij} = \Delta h_{ij} + |A|^2 h_{ij} - 2 H h_{ik} h^k_j,

[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) where $\Delta$ denotes the Laplace-Beltrami operator on $M_t$, $A$ is the shape operator with $|A|^2 = g^{ik} g^{jl} h_{ij} h_{kl}$, and the Laplacian term arises from commuting time and spatial derivatives via Gauss-Weingarten relations. The [Christoffel symbols](/page/Christoffel_symbols) $\Gamma^k_{ij}$, determined by the [metric](/page/Metric) via $\partial_k g_{ij} = \Gamma^l_{ki} g_{lj} + \Gamma^l_{kj} g_{il}$, evolve through differentiation of the [metric](/page/Metric) equation, yielding

\frac{\partial}{\partial t} \Gamma^k_{ij} = g^{kl} \left( \nabla_i (H h_{lj}) + \nabla_j (H h_{li}) - \nabla_l (H h_{ij}) \right),

ensuring compatibility with the evolving geometry. This system is [quasilinear](/page/Quasilinear) parabolic, as the principal symbol corresponds to the Laplace-Beltrami operator on tensor fields. The scalar mean curvature $H$ itself satisfies the reaction-diffusion equation

\frac{\partial}{\partial t} H = \Delta H + |A|^2 H,

obtained by tracing the evolution of $h_{ij}$ and using the contracted Bianchi identity $\nabla_i H = \nabla_j h^i_j$. This equation highlights the diffusive smoothing effect of MCF, modulated by the nonlinear term involving the squared norm of the second fundamental form. ### Parametric and Graphical Representations In the parametric representation of mean curvature flow, a hypersurface is described as an evolving immersion $ F: M \times [0, T) \to \mathbb{R}^{n+1} $, where $ M $ is a smooth manifold without boundary, and the evolution satisfies $ \frac{\partial F}{\partial t} \perp T F(M_t) $ at each time $ t $, ensuring the velocity is normal to the tangent space of the instantaneous hypersurface $ M_t = F(\cdot, t) $.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) This formulation preserves the intrinsic geometry of the manifold and is particularly suited for analyzing smooth flows of compact hypersurfaces, as introduced in the study of convex surfaces contracting to spheres.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) For graphical representations, the mean curvature flow is expressed when the hypersurface can be realized as the graph of a function $ u: \Omega \times [0, T) \to \mathbb{R} $ over a domain $ \Omega \subset \mathbb{R}^n $, leading to the quasilinear parabolic equation

\frac{\partial u}{\partial t} = \sqrt{1 + |D u|^2} \operatorname{div} \left( \frac{D u}{\sqrt{1 + |D u|^2}} \right),

where $ D u $ denotes the spatial gradient.[](https://www.jstor.org/stable/1971452) This equation arises as the projection of the mean curvature vector onto the vertical direction and enables the treatment of entire graphs in Euclidean space, with long-time existence under suitable tilt conditions on the initial data.[](https://www.jstor.org/stable/1971452) The [level set method](/page/Level-set_method) represents the evolving [hypersurface](/page/Hypersurface) implicitly as the zero [level set](/page/Level_set) $ \{ x \in \mathbb{R}^{n+1} \mid \phi(x, t) = 0 \} $ of a scalar [function](/page/Function) $ \phi: \mathbb{R}^{n+1} \times [0, T) \to \mathbb{R} $, satisfying the Hamilton-Jacobi-type equation

\frac{\partial \phi}{\partial t} = |\nabla \phi| \operatorname{div} \left( \frac{\nabla \phi}{|\nabla \phi|} \right),

with viscosity solutions providing a [weak formulation](/page/Weak_formulation) that accommodates singularities and non-smooth evolutions. This approach, developed for [mean curvature](/page/Mean_curvature) motion, extends the classical flow to weak limits and handles topological changes such as merging or pinching. Parametric representations excel in preserving smooth geometry for regular flows without singularities, while [level set](/page/Level_set) methods are advantageous for capturing topological transitions and developing singularities through viscosity solutions.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) Graphical formulations bridge these by restricting to graphs, facilitating analysis via PDE techniques. For numerical implementation of graphical mean curvature flow, [finite difference](/page/Finite_difference) schemes discretize the [quasilinear](/page/Quasilinear) [equation](/page/Equation) on a fixed [grid](/page/Grid) over $ \Omega $, updating $ u $ explicitly or implicitly while enforcing graph constraints to prevent overturning.[](https://www.jstor.org/stable/1971452) ## Well-Posedness ### Existence of Solutions The existence of solutions to the mean curvature flow (MCF) is established through a combination of classical parabolic PDE theory and geometric measure theory, depending on the regularity of the initial data. For smooth, compact initial hypersurfaces, short-time existence of smooth solutions follows from the quasilinear parabolic nature of the evolution equation. Short-time existence is typically proved by linearizing the flow around the initial hypersurface or employing a gauge-fixing technique to render the system strictly parabolic. One standard approach involves representing the evolving hypersurface locally as a graph over the initial surface in a tubular neighborhood, transforming the MCF into a quasilinear parabolic PDE for the graph function, to which standard existence theorems for such equations apply.[](https://www.uni-regensburg.de/assets/mathematik/winterschool2016/bmcf-lect1.pdf) An alternative method uses the DeTurck trick, originally developed for Ricci flow, which modifies the MCF by adding a diffeomorphism term to eliminate the reparametrization invariance, making the principal symbol uniformly elliptic and allowing application of the classical theory for parabolic systems.[](https://arxiv.org/pdf/1104.4409) Specifically, for a smooth immersion $\phi_0: M \to \mathbb{R}^{n+1}$ of a compact n-manifold $M$, there exists $T > 0$ and a smooth family of immersions $\phi(\cdot, t): M \to \mathbb{R}^{n+1}$ for $t \in [0, T)$ satisfying $\partial_t \phi = \mathbf{H}$ (the mean curvature vector) with $\phi(\cdot, 0) = \phi_0$, and the solution depends smoothly on the initial data in the $C^\infty$-topology.[](https://www.felixschulze.eu/images/mcf_notes.pdf) Higher-order regularity estimates ensure the solution remains smooth on the existence interval. These derive from Schauder theory applied to the MCF system, yielding interior and boundary estimates for derivatives of the second fundamental form $A$. If $\|A\|^2$ is bounded by a constant $C_0$ on $M \times [0, T)$, then for any $k \geq 0$, there exists $C_k$ depending on $n$, the initial data, and $C_0$ such that $\|\nabla^k A\|^2 \leq C_k$ on $M \times [0, T)$. Local versions of these estimates hold in space-time balls, providing $C^{k,\alpha}$-bounds on $A$ under suitable scaling assumptions on $\|A\|^2$.[](https://www.felixschulze.eu/images/mcf_notes.pdf) For initial data that may develop singularities or lack smoothness, weak solutions are constructed using the framework of varifolds. Brakke flows provide a notion of [integral](/page/Integral) varifold flows where [the mean curvature](/page/Mean_curvature) is a [Radon measure](/page/Radon_measure), satisfying a monotonicity [inequality](/page/Inequality) for the Gaussian [mass ratio](/page/Mass_ratio) $\Theta(x, t) = \frac{1}{|B_r(x)|} \int_{B_r(x)} e^{-|y-x|^2/4t} d\mu_t(y)$ (with $\mu_t$ the varifold [mass](/page/Mass)), ensuring $\frac{d}{dt} \Theta(x, t) \geq 0$ [almost everywhere](/page/Almost_everywhere). Such flows exist globally [in time](/page/In_Time) for arbitrary initial compact varifolds and capture the large-scale behavior of smooth MCF, including [singularity](/page/Singularity) formation.[](https://www.felixschulze.eu/images/mcf_notes.pdf) Long-time existence of smooth solutions requires control on the geometry of the flow. If the second fundamental form remains bounded, $\|A\| \leq C$ for some constant $C$ on the maximal existence interval $[0, T_{\max})$, then higher-order estimates from parabolic regularity imply $T_{\max} = \infty$, yielding a global smooth solution. This criterion is particularly useful for convex hypersurfaces, where curvature bounds follow from maximum principles. ### Uniqueness Theorems Uniqueness for smooth solutions to the mean curvature flow is established using the maximum principle applied to the difference of two potential solutions. For complete immersed submanifolds in Riemannian manifolds of arbitrary codimension, a strong uniqueness theorem holds without requiring bounded second fundamental forms, resolving a longstanding open problem.[](https://arxiv.org/abs/math/0703694) This result applies to smooth evolutions where the flow remains regular, ensuring that any two solutions starting from the same initial data coincide for all future times.[](https://intlpress.com/site/pub/files/_fulltext/journals/cag/2007/0015/0003/CAG-2007-0015-0003-a001.pdf) In the context of singularities, Huisken's monotonicity formula provides integral uniqueness for tangent flows, implying that rescaled limits at singular points are unique under suitable conditions such as mean convexity.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-31/issue-1/Asymptotic-behavior-for-singularities-of-the-mean-curvature-flow/10.4310/jdg/1214444099.full) For compact, embedded, mean convex hypersurfaces, every tangent flow is unique, with early results tracing back to analyses of convex cases.[](https://mike-law.github.io/files/ThesisMCF.pdf) This monotonicity ensures that blow-up behaviors, such as convergence to self-shrinkers, are determined uniquely by the asymptotic structure near the singularity.[](https://arxiv.org/abs/1107.4643) For graphical mean curvature flows over fixed domains, uniqueness follows from comparison principles derived from parabolic theory, which prevent crossings and enforce that solutions remain graphs.[](https://arxiv.org/abs/2110.12026) This holds for entire graphs with locally [Lipschitz](/page/Lipschitz) initial data, particularly in rotationally symmetric cases or when properness and bounds on the second fundamental form are imposed, guaranteeing a unique evolution.[](https://par.nsf.gov/servlets/purl/10420371) Weak solutions in the sense of Brakke flows, formulated via varifold theory with Gaussian weighting, exhibit [uniqueness](/page/Uniqueness) under entropy conditions or for mean convex flows, where the [entropy](/page/Entropy) functional's monotonicity selects a [canonical](/page/Canonical) representative.[](https://web.stanford.edu/~ochodosh/MCFnotes.pdf) [Integral](/page/Integral) Brakke flows with unit density are regular and unique in such settings, aligning with smooth flows away from singularities.[](https://nedelen.science.nd.edu/brian-mcf-notes.pdf) However, uniqueness fails in certain non-compact or singular settings without additional assumptions, as demonstrated by computed examples where smooth initial data leads to fattening phenomena and multiple possible evolutions in $\mathbb{R}^3$.[](https://www.semanticscholar.org/paper/A-Computed-example-of-nonuniqueness-of-mean-flow-in-Angenent-Ilmanen/39607519689b00fa4b89ca6a5f21011bcefe5e5b) In non-compact cases, such as unbounded hypersurfaces, energy arguments are needed to recover [uniqueness](/page/Uniqueness), but counterexamples persist without growth controls.[](https://arxiv.org/abs/1709.00253) ## Intrinsic Properties ### Monotonicity Formulas One of the key tools in analyzing the mean curvature flow is Huisken's monotonicity formula, which provides a non-decreasing quantity under the rescaled evolution. For a mean curvature flow $M_t$ in $\mathbb{R}^{n+1}$, consider the rescaled mean curvature flow where parabolically rescaling around a space-time point $(p, T)$ with $r = \sqrt{2(T-t)}$ yields hypersurfaces $\tilde{M}_s$ for $s > 0$. The monotonicity formula states that the Gaussian-weighted area functional satisfies

\frac{d}{ds} \int_{\tilde{M}_s} e^{-|y|^2 / (4s)} , d\tilde{\mu}_s \geq 0,

with equality holding if and only if $\tilde{M}_s$ is a [self-similar solution](/page/Self-similar_solution), such as a self-shrinker. This formula was established by Gerhard Huisken in his analysis of singularity formation. The derivation begins with the evolution equation for the area element under [mean curvature flow](/page/Mean_curvature_flow), $\partial_t d\mu_t = -H^2 d\mu_t$, where $H$ is the mean curvature scalar. For the rescaled flow, one computes the time derivative of the weighted [integral](/page/Integral) $\Theta(p, r, t) = (4\pi r^2)^{-n/2} \int_{M_t} e^{-|x-p|^2 / (4r^2)} d\mu_t$. Using [integration by parts](/page/Integration_by_parts) on the ambient space and the evolution of the second fundamental form, the derivative simplifies to $\partial_r \Theta \geq 0$, leveraging the parabolic [maximum principle](/page/Maximum_principle) to control higher-order terms. This yields the monotonicity after a [change of variables](/page/Change_of_variables). The implications of this formula are profound for understanding singularities. As $t \to T$, the monotonicity ensures weak convergence of rescaled flows to self-shrinkers in the sense of varifolds, providing a tangent cone structure at singular points. The functional $\Theta$ behaves like an entropy, bounding the geometry and enabling estimates on the singular set. Additionally, under parabolic rescaling, the formula preserves multiplicity, meaning that if the initial flow has integer multiplicity, the limit self-shrinker inherits the same multiplicity. These properties facilitate the study of singularity models without detailed resolution of the flow. A related entropy functional, introduced by Colding and Minicozzi, is defined as

\lambda(M_t) = \sup_{p \in \mathbb{R}^{n+1}, \tau > 0} (4\pi \tau)^{-n/2} \int_{M_t} e^{-|x-p|^2 / (4\tau)} , d\mu_t.

This [entropy](/page/Entropy) is nonincreasing under the mean curvature flow and achieves equality in the limit for self-shrinkers, providing a tool for analyzing generic singularities and stability.[](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf) ### Preservation of Geometric Quantities Under mean curvature flow, certain intrinsic and extrinsic geometric properties of the evolving [hypersurface](/page/Hypersurface) are preserved or evolve in predictable ways, reflecting the flow's role as a geometric [heat equation](/page/Heat_equation) that [smooth](/page/Smooth)s while maintaining key structural features. For [smooth](/page/Smooth) solutions without singularities, the [topology](/page/Topology) of an [embedded](/page/Embedded) [hypersurface](/page/Hypersurface) is preserved throughout the [evolution](/page/Evolution). This follows because the flow generates a family of [smooth](/page/Smooth) embeddings, ensuring no topological changes occur as long as the solution remains regular. Mean convexity is also preserved under the [flow](/page/Flow). If the [initial](/page/Initial) hypersurface satisfies $ H \geq 0 $ everywhere (with respect to the inward unit [normal](/page/Normal)), then $ H(t) \geq 0 $ for all $ t $ in the maximal existence interval. This preservation arises from the parabolic evolution equation for the [mean curvature](/page/Mean_curvature),

\frac{\partial}{\partial t} H = \Delta H + |A|^2 H,

where $ \Delta $ is the Laplace-Beltrami [operator](/page/Operator) and $ |A|^2 $ is the squared norm of the second fundamental form. By the strong [maximum principle](/page/Maximum_principle) for parabolic equations, the minimum value of $ H $ cannot decrease and remains non-negative if initially so.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg.1984.v20.n1.a3.full) The enclosed volume $ V(t) $ by a closed hypersurface evolves monotonically. Specifically,

\frac{dV}{dt} = -\int_{M_t} H , d\mu_t,

where $ d\mu_t $ is the area element on the hypersurface $ M_t $ at time $ t $. Since $ H \geq 0 $ for mean convex initial data (or more generally if the flow is inward), this integral is non-positive, leading to a non-increasing volume that decreases unless the hypersurface is minimal ($ H = 0 $).[](https://eudml.org/doc/152982) Rotational symmetry is preserved if present initially. For a hypersurface that is rotationally symmetric around an axis, the mean curvature and unit normal remain symmetric under the flow, as the evolution equation is invariant under rotations in the ambient space. Thus, the solution retains this symmetry for as long as it exists smoothly.[](https://link.springer.com/article/10.1007/s00209-008-0333-6) The center of mass of the enclosed region also evolves explicitly. For a closed [hypersurface](/page/Hypersurface) $ M_t = \partial \Omega_t $ in $ \mathbb{R}^{n+1} $, the center of mass $ \bar{x}(t) = \frac{1}{V(t)} \int_{\Omega_t} x \, dV $ satisfies

\frac{d \bar{x}}{dt} = -\frac{1}{V(t)} \int_{M_t} H \nu , d\mu_t,

where $ \nu $ is the outward unit normal. This indicates that the center moves with a [velocity](/page/Velocity) given by the negative average of the [mean curvature](/page/Mean_curvature) vector over the surface, reflecting the inward motion driven by the flow.[](https://math.jhu.edu/~js/Math745/ilmanen.mcflow.pdf) ## Specific Cases ### Planar Curves Mean curvature flow restricted to immersed [curves](/page/Curve) in the [Euclidean plane](/page/Euclidean_plane) is known as the [curve shortening flow](/page/Curve-shortening_flow), where a [curve](/page/Curve) $\gamma(t)$ evolves by moving each point in the direction of its unit normal $\mathbf{N}$ with speed equal to its [curvature](/page/Curvature) $\kappa$. The evolution equation is given by

\frac{\partial X}{\partial t} = \kappa \mathbf{N},

where $X(s,t)$ parameterizes the [curve](/page/Curve) by [arc length](/page/Arc_length) $s$ at each time $t$.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-23/issue-1/The-heat-equation-shrinking-convex-plane-curves/10.4310/jdg/1214439902.pdf) For a closed convex curve in the plane, Gage and Hamilton established that the flow exists for a finite time interval $[0, T)$, during which the curve remains smooth and convex while shrinking self-similarly to a point at time $T$. The length $L(t)$ of the evolving curve decreases monotonically according to

\frac{dL}{dt} = -\int_{\gamma(t)} \kappa^2 , ds.

By Gage's isoperimetric inequality, $\int_{\gamma(t)} \kappa^2 \, ds \geq 4\pi^2 / L(t)$ for simple closed curves, with equality for circles.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-23/issue-1/The-heat-equation-shrinking-convex-plane-curves/10.4310/jdg/1214439902.pdf) This implies $\frac{dL}{dt} \leq -4\pi^2 / L(t)$, so $L(t)^2 \leq L(0)^2 - 8\pi^2 t$, and the maximal [existence](/page/Existence) time satisfies $T \leq L(0)^2 / (8\pi^2)$. For an initial [circle](/page/Circle), equality holds throughout, and the curve shrinks to a point in exactly this time while remaining circular. In general, the [curvature](/page/Curvature) becomes asymptotically constant, and the curve develops a round tip as it approaches [extinction](/page/Extinction).[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-23/issue-1/The-heat-equation-shrinking-convex-plane-curves/10.4310/jdg/1214439902.pdf) A prominent example of a non-compact solution is the Grim Reaper [curve](/page/Curve), a translating [soliton](/page/Soliton) that preserves its shape while moving upward at unit speed. Parameterized as $X(s,t) = (s, t - \log \cos s)$ for $s \in (-\pi/2, \pi/2)$, its [curvature](/page/Curvature) equals the vertical component of the translation velocity, satisfying the flow equation exactly. This eternal solution illustrates asymptotic behavior for certain initial data and serves as a model for [singularity](/page/Singularity) formation in more complex flows.[](https://www.math.utoronto.ca/roberth/pde2/curve_shortening_flow.pdf) ### Surfaces in Euclidean Space The mean curvature flow for a compact immersed or [embedded](/page/Embedded) surface in [Euclidean](/page/Euclidean) 3-space $\mathbb{R}^3$ evolves the surface $\Sigma_t$ such that each point moves in the direction of the inward unit normal with speed equal to the [mean curvature](/page/Mean_curvature) $H$, where $H = (\kappa_1 + \kappa_2)/2$ and $\kappa_1, \kappa_2$ are [the principal](/page/The_Principal) curvatures at that point.[](https://www.math.toronto.edu/roberth/papers/lectures_mcf21.pdf) This parabolic evolution equation smooths the surface locally while reducing its area monotonically, but global behavior depends strongly on the [initial topology](/page/Initial_topology) and geometry. For compact surfaces of genus zero (topological spheres), the long-time behavior under mean curvature flow typically involves smoothing and convergence to a round sphere before the surface shrinks to a point in finite time, provided the initial surface is embedded and mean convex.[](https://www.math.toronto.edu/roberth/papers/lectures_mcf21.pdf) This asymptotic roundness arises from the preservation of convexity and the monotonicity of certain geometric quantities, such as the Hawking mass, which drives the surface toward sphericity. In contrast, for embedded tori (genus one), the flow often develops neckpinch singularities in finite time before complete extinction, where the surface pinches off at narrow regions, leading to topological changes if the flow is continued weakly.[](https://arxiv.org/pdf/2006.06118) These singularities manifest as the surface forming a thin [neck](/page/Neck) that collapses, typically resulting in the torus breaking into multiple [spherical](/page/Sphere) components.[](https://web.stanford.edu/~ochodosh/MCFnotes.pdf) The density theorem characterizes the structure at such singularities: the asymptotic Gaussian density of the rescaled flow at a singular point is an [integer](/page/Integer) representing the multiplicity of the blow-up limit, which is often a self-shrinking [cylinder](/page/Cylinder) or [sphere](/page/Sphere).[](https://link.springer.com/article/10.1007/BF01192093) Numerical simulations of mean curvature flow for non-spherical initial surfaces frequently reveal the formation of spherical tips at regions of high curvature connected by nearly cylindrical necks, which narrow rapidly and lead to pinch-off singularities.[](https://epubs.siam.org/doi/10.1137/22M1531919) These computations highlight the qualitative dynamics, such as the rapid development of high-curvature regions and the approach to self-similar shrinking models near singularities.[](https://iopscience.iop.org/article/10.1088/1361-6544/ac15a9) ### Higher-Dimensional Spheres Under the mean curvature flow, a round [sphere](/page/Sphere) $ S^n(r(0)) $ embedded in $ \mathbb{R}^{n+1} $ with initial [radius](/page/Radius) $ r(0) > 0 $ evolves by homothetic contraction, maintaining its spherical shape at every time $ t \geq 0 $. The [mean curvature](/page/Mean_curvature) of such a [sphere](/page/Sphere) is $ H = n / r(t) $, where $ r(t) $ denotes the [radius](/page/Radius) at time $ t $, directed inward along the unit [normal](/page/Normal).[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-3/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214444099.full) The evolution of the radius satisfies the [ordinary differential equation](/page/Ordinary_differential_equation) derived from the flow equation $ \partial_t X = -H \vec{\nu} $, yielding

r(t)^2 = r(0)^2 - 2 n t

for $ 0 \leq t < T $, where $ T = r(0)^2 / (2n) $ is the finite extinction time at which the sphere shrinks to a point.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-3/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214444099.full) This explicit solution highlights the self-similar nature of the flow: the evolving sphere is a homothetic shrinking solution, and upon rescaling by $ \sqrt{(T-t)/T} $, the limit as $ t \to T^- $ approaches a stationary round sphere of unit radius under the rescaled mean curvature flow.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-3/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214444099.full) For initial data that are strictly convex closed hypersurfaces in $ \mathbb{R}^{n+1} $ (with $ n \geq 2 $), Huisken's theorem establishes that the flow remains smooth until extinction and asymptotically approaches a round sphere as $ t \to T^- $, with the rescaled hypersurface converging in the $ C^\infty $-topology to the standard unit sphere.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-3/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214444099.full) ## Convexity Aspects ### Evolution of Convex Hypersurfaces Under mean curvature flow, an initially convex hypersurface $M_0$ in $\mathbb{R}^{n+1}$ remains convex for all $t \in [0, T)$, where $T$ is the maximal time of existence. This preservation follows from the evolution equation satisfied by the support function $s(x, t) = \sup_{y \in M_t} \langle y, x \rangle$, which obeys a parabolic inequality ensuring the second derivatives remain positive, thus maintaining strict convexity.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) As the flow progresses toward the extinction time $T < \infty$, the convex hypersurface $M_t$ exhibits asymptotic roundness. In the case of planar curves ($n=1$), $M_t$ becomes circular,[](https://www.ams.org/journals/tran/1986-296-02/S0002-9947-1986-0853812-3/S0002-9947-1986-0853812-3.pdf) while in higher dimensions ($n \geq 2$), it approaches a sphere in the $C^\infty$-topology as $t \to T^-$.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) This roundness is quantified by pinching estimates on the principal curvatures $\kappa_i$, where the ratios $\kappa_{\max}/\kappa_{\min} \to 1$ uniformly, implying the hypersurface evolves toward an umbilic shape. Huisken's theorem provides a specific bound on the diameter: $D(t) \leq D(0) (1 - t/T)^{1/2}$, reflecting the accelerating contraction near extinction.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) The extinction time $T$ admits an upper bound $T \leq V(0) / (n c_n)$, where $V(0)$ is the initial enclosed [volume](/page/Volume) and $c_n > 0$ is a dimension-dependent constant arising from the monotonic decrease of [volume](/page/Volume) under the [flow](/page/Flow). Post-extinction, the rescaled [flow](/page/Flow) converges weakly to the origin in the sense of measures, confirming that the [hypersurface](/page/Hypersurface) shrinks to a single point without developing non-round singularities.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) ### Extinction and Asymptotic Behavior For smooth, compact, convex hypersurfaces in [Euclidean space](/page/Euclidean_space), the mean curvature flow remains smooth and extinguishes completely in finite time $T < \infty$ without developing singularities. This follows from the comparison principle: the hypersurface is enclosed within a sphere, which shrinks to a point in finite time under the flow, implying the inner hypersurface vanishes no later than the enclosing sphere.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) Near the extinction time $T$, the asymptotic behavior is analyzed via rescaling. Consider the rescaled hypersurfaces $\lambda(t) (M_{T-t} - c(t))$, where $\lambda(t) = 1/\sqrt{2(T-t)}$ is the parabolic rescaling factor and $c(t)$ is the center of mass. For convex flows, this rescaled flow converges smoothly to the origin (a round point self-shrinker) as $t \to T^-$. Self-shrinkers are hypersurfaces $\Sigma$ satisfying

\mathbf{H} = \frac{\langle x, \nu \rangle}{2} \nu,

where $\mathbf{H}$ is the mean curvature vector, $\nu$ is the unit normal, and $x$ is the position vector; in the convex case, the limit is the degenerate point solution modeling smooth contraction to a point.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-31/issue-1/Asymptotic-behavior-for-singularities-of-the-mean-curvature-flow/10.4310/jdg/1214444099.pdf)[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-20/issue-1/Flow-by-mean-curvature-of-convex-surfaces-into-spheres/10.4310/jdg/1214438998.full) To continue beyond classical flows in non-convex settings, Brakke flows provide a weak formulation using varifolds, preserving monotonicity of mass and allowing passage through singularities while satisfying an integral mean curvature inequality.[](https://kenbrakke.com/papers/downloads/motionbook.pdf) ## Advanced Phenomena ### Singularity Formation In mean curvature flow, singularities occur when the curvature becomes unbounded at a finite time $T$, marking the breakdown of the smooth evolution of the hypersurface. These singularities can be classified geometrically as neckpinch models, where the hypersurface develops a narrow cylindrical region that pinches off, tip or spherical models resembling shrinking spheres at the ends of fingers or protrusions, or more complex structures involving multiple components or higher-genus topologies.[](https://www.math.utoronto.ca/roberth/papers/haslhofer_icm2026.pdf)[](https://www.maths.usyd.edu.au/u/haotianw/mcf_type2_survey.pdf) Singularities are further categorized by the rate of curvature blow-up. Type I singularities satisfy $|A| \leq C / \sqrt{T - t}$ near the singular time $T$, where $A$ is the second fundamental form and $C$ is a constant depending on the initial data; these are generic for mean convex flows in low dimensions, as they align with the expected scaling from the evolution equations.[](https://link.springer.com/article/10.1007/s005260050113)[](https://arxiv.org/pdf/2210.00419) In contrast, Type II singularities exhibit $|A| \sqrt{T - t} \to \infty$ as $t \to T^-$, indicating faster blow-up; they are rare and typically arise in non-convex settings or higher codimension, with examples including certain ancient solutions or flows in non-standard geometries.[](https://www.maths.usyd.edu.au/u/haotianw/mcf_type2_survey.pdf)[](https://arxiv.org/pdf/2210.00419) Blow-up analysis provides insight into the structure of singularities through parabolic rescaling, where the flow is zoomed in spatio-temporally around the singular point, yielding a tangent flow that is either an ancient solution (defined for all past times) or a self-shrinking soliton. These tangent flows have multiplicity one, as proven in 2024, resolving Ilmanen's longstanding conjecture.[](https://arxiv.org/abs/2312.02106) This rescaling leverages monotonicity properties to ensure convergence to a smooth limit in appropriate topologies. The theorem of Colding and Minicozzi establishes that, for generic initial data in dimensions up to three, all singularities are Type I, implying that non-generic perturbations avoid the exotic Type II behavior.[](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf) Specific examples illustrate these phenomena. In two dimensions, the Angenent oval demonstrates a neckpinch singularity, where a figure-eight shaped curve evolves to form a degenerate cylindrical pinch before breaking into two components.[](https://eudml.org/doc/153879) For surfaces in $\mathbb{R}^3$, Huisken and Sinestrari analyzed mean convex flows, showing that singularities form as neckpinches or spherical tips, with the blow-up limits being cylinders or spheres, respectively.[](https://link.springer.com/article/10.1007/s005260050113) ### Regularity and Rescaling The ε-regularity theorem for mean curvature flow provides conditions under which solutions remain smooth in spacetime regions where certain energy quantities are controlled. Specifically, if the integral of the squared norm of the second fundamental form over a parabolic ball, ∫_{B_R(x) × (t-1,t)} |A|^2 dμ, is sufficiently small for some ε > 0 depending on the dimension, then the solution is smooth on the smaller region B_{R/2}(x) × (t-1,t).[](https://arxiv.org/pdf/1102.4800) This result, analogous to ε-regularity for minimal surfaces, ensures that singularities cannot form abruptly without prior growth in curvature energy, and it holds for smooth, properly immersed hypersurfaces evolving by mean curvature in Euclidean space.[](https://arxiv.org/pdf/1102.4800) A related tool is the Gaussian density introduced by Huisken, defined for a mean curvature flow M_t in ℝ^{n+1} as Θ(M_t, x, r) = (4π(T-t))^{-n/2} ∫_{M_t} e^{-|y-x|^2 / (4(T-t))} dμ_t(y), where T is the maximal time and μ_t is the area measure.[](https://link.springer.com/article/10.1007/BF01192093) This density is monotonically non-decreasing under backward parabolic rescaling and satisfies lim sup_{r→0} Θ(M_t, x, r) ≤ 1 at regular [spacetime](/page/Spacetime) points (x,t), while at singular points it equals a positive [integer](/page/Integer) multiple of the [density](/page/Density) of the [tangent cone](/page/Tangent_cone).[](https://link.springer.com/article/10.1007/BF01192093) The monotonicity follows from Huisken's monotonicity formula for the Gaussian-weighted area, which bounds the [density](/page/Density) and controls the [geometry](/page/Geometry) near potential singularities.[](https://link.springer.com/article/10.1007/BF01192093) Rescaling techniques analyze singularity formation by considering sequences of blow-ups near spacetime points approaching a singular time T. For a sequence of points (x_j, t_j) with t_j → T and rescaling factors λ_j = 1/√(T - t_j), the rescaled flows λ_j (M_{t_j + s/λ_j^2} - x_j) converge, in the sense of varifolds, to a non-trivial [limit](/page/Limit) that is a [smooth](/page/Smooth), self-shrinking [solution](/page/Solution) (self-shrinker) satisfying [H](/page/H+) = (x^⊥)/ (2√(-s)) for s < 0, where [H](/page/H+) is the [mean curvature](/page/Mean_curvature) vector and x^⊥ its normal component.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-31/issue-1/Asymptotic-behavior-for-singularities-of-the-mean-curvature-flow/10.4310/jdg/1214444099.full) This convergence holds for compact, embedded hypersurfaces with non-negative [mean curvature](/page/Mean_curvature), and the limit is unique up to translation, modeling the asymptotic behavior at type I singularities.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-31/issue-1/Asymptotic-behavior-for-singularities-of-the-mean-curvature-flow/10.4310/jdg/1214444099.full) Partial regularity theorems establish that singularities occupy a small portion of the spacetime. For integral Brakke flows (weak solutions in the varifold sense), the spacetime singular set has parabolic [Hausdorff measure](/page/Hausdorff_measure) zero, meaning the flow is [smooth](/page/Smooth) almost everywhere in space and time. In the mean-convex case, the structure of the singular set is further refined: it admits a [stratification](/page/Stratification) where the top-dimensional strata consist of [smooth](/page/Smooth) points, and lower strata satisfy quantitative Gaussian volume estimates, Vol(T_r(S_j)) ≤ C r^{n+1 - j + 2 - ε} for tubular neighborhoods, with the singular set having [Hausdorff codimension](/page/Codimension) at least 7 in dimensions n ≥ 7 due to the absence of singular self-shrinkers of lower [codimension](/page/Codimension).[](https://www.math.utoronto.ca/roberth/papers/CheegerHaslhoferNaber_QuantitativeStratificationMCF.pdf) This [codimension-7](/page/Codimension) phenomenon mirrors Allard-Almgren regularity for stationary varifolds and applies to the [tangent](/page/Tangent) cones of the flow, ensuring that generic [hypersurface](/page/Hypersurface) flows develop singularities only along sets of controlled low dimension.[](https://www.math.utoronto.ca/roberth/papers/CheegerHaslhoferNaber_QuantitativeStratificationMCF.pdf)

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