Mean curvature flow

What it is

A family of hypersurfaces $M_t \subset \mathbb R^{n+1}$ moves by mean curvature if each point moves with velocity equal to the mean curvature vector, $\partial_t x = \vec H = -H\nu$. This is the gradient flow of area. It is a quasilinear parabolic equation for the position: spheres shrink homothetically, cylinders shrink in their round directions only, and minimal surfaces stay put.

Throughout, $\nu$ is the outward unit normal, $h_{ij}$ is the second fundamental form, positive on a convex hypersurface, $A$ is the same tensor seen as a shape operator with eigenvalues $\lambda_1\le\dots\le\lambda_n$, and $H=g^{ij}h_{ij}$ is the sum of the principal curvatures, not their average. So a sphere of radius $r$ has $H=n/r$.

Three formulas make this concrete:

The first holds because $\Delta_g x = \vec H$ on any hypersurface, so each coordinate function solves the heat equation on the moving surface. The third is the flow of a graph $x_{n+1}=u(x',t)$. As with Ricci flow, the geometric equation is only weakly parabolic, because tangential reparametrisation changes nothing. Written as a normal graph over the initial surface it becomes a strictly parabolic scalar equation, which gives short-time existence on closed hypersurfaces.

Huisken (1984) proved that a compact convex hypersurface shrinks to a point, becoming round as it does. In 1990 he found the monotonicity formula that drives the whole theory: Gaussian-weighted area, centred at a future space-time point, never increases. Consequently every blow-up of a singularity (a tangent flow) is a self-shrinker, a surface that moves only by scaling (see self-shrinkers and translators). Brakke's varifold flows and the level-set flow let the evolution continue through singularities in a weak sense (see weak and singular flows).

#Two comparison facts

Since $(\partial_t-\Delta)(\lvert x\rvert^2+2nt)=0$, the maximum principle shows that a closed hypersurface inside a ball of radius $R$ is gone by time $R^2/2n$. Choi, Du, Park and Youn (2026) give the sharp version in terms of area: the extinction time of a smooth closed boundary of a domain is at most $\tfrac1{2n}(\operatorname{Area}/\lvert S^n\rvert)^{2/n}$, with equality only for the round sphere.

Disjoint closed hypersurfaces also stay disjoint, and their distance does not decrease. So embeddedness is preserved, and any solution can serve as a barrier for any other. Intrinsic flows have no ordering of this kind.

Worked examples

• Sphere. A round $S^n$ of radius $r_0$ has $r(t)=\sqrt{r_0^2-2nt}$ and disappears at $T=r_0^2/2n$. The one with $T=0$ has radius $\sqrt{2n}$ at $t=-1$.
• Cylinder. On $S^k\times\mathbb R^{n-k}$ only the round factor moves: $r(t)=\sqrt{r_0^2-2kt}$, so the flow collapses onto an $(n-k)$-plane.
• Grim reaper. The curve $y=t-\log\cos x$ on $(-\tfrac\pi2,\tfrac\pi2)$ translates at unit speed, as substituting into the graph equation shows.
• Bowl soliton. The rotationally symmetric, entire, convex translator in higher dimensions, the extrinsic counterpart of the Bryant soliton.
• Angenent's doughnut. An embedded torus that shrinks self-similarly.
• Dumbbell. Grayson (1989) showed that a dumbbell with a thin enough neck pinches at the neck while the bells are still large, so non-convex hypersurfaces need not shrink to points. This is a neckpinch.

The evolution equations

Huisken's 1984 computation gives, in the sign convention above,

The second line uses Simons' identity $\Delta h_{ij}=\nabla_i\nabla_j H+Hh_{il}h^l{}_j-\lvert A\rvert^2h_{ij}$. This is the same reaction–diffusion shape as curvature under Ricci flow, and the consequences are similar:

• On a closed hypersurface with $H>0$, $\min H$ does not decrease, so $H>0$ is preserved. The condition $h_{ij}\ge\varepsilon Hg_{ij}$ is preserved too, by Hamilton's maximum principle for tensors.
• The inequality $\frac{d}{dt}\max\lvert A\rvert^2\le 2\max\lvert A\rvert^4$ forces $\max\lvert A\rvert^2\ge \frac{1}{2(T-t)}$ if $T$ is the first singular time. Huisken showed that $\lvert A\rvert$ must blow up there.
• By the Gauss equation, $R_{ij}=Hh_{ij}-h_{il}h^l{}_j$ in Euclidean space, so the induced metric satisfies $\partial_t g_{ij}=-2R_{ij}-2h_{il}h^l{}_j$. So the two flows are related by analogy, not by reduction.

Whether $H$ itself must blow up at the first singular time is harder. Li and Wang proved it for closed embedded surfaces in $\mathbb R^3$ (arXiv:1608.02832). Lotay and Oliveira (2026) prove it at nondegenerate neck pinches of Lagrangian flow of surfaces.

Why it matters

For anyone learning Ricci flow, mean curvature flow is a Rosetta stone. Huisken's monotonicity plays the role of Perelman's reduced volume and $\mathcal W$-entropy, necks and caps appear in both, and Huisken–Sinestrari's surgery for 2-convex hypersurfaces was developed in parallel with Perelman's surgery for 3-manifolds. Ideas have flowed in both directions. Colding–Minicozzi's entropy, the classification of ancient solutions, and Bamler–Kleiner's methods all cross between the two flows, and many researchers work on both. That is why its papers appear on this site as wider flows.

It also matters by itself. It proves topological classifications of hypersurfaces, such as the 2-convex theorem below. Minimal surfaces are its fixed points. Through the level-set method it is a working engineering tool for image segmentation and moving interfaces, and through phase fields it is the sharp-interface limit of Allen–Cahn models.

Convex hypersurfaces: Huisken's theorem

The curve case $n=1$ is due to Gage and Hamilton, with Grayson removing convexity for embedded curves (curve-shortening flow). The dumbbell shows that Grayson's theorem has no analogue for $n\ge 2$.

The proof follows the arc of Hamilton's 1982 theorem for Ricci flow. First, $h_{ij}\ge\varepsilon Hg_{ij}$ is preserved. Next comes the central pinching estimate: there are $\sigma>0$ and $C$ with $\lvert A\rvert^2-\tfrac1nH^2\le CH^{2-\sigma}$, proved by Stampacchia iteration on $(\lvert A\rvert^2-\tfrac1nH^2)H^{\sigma-2}$ with Simons' identity supplying the good term. So the trace-free part of $A$ is lower order where the curvature is large. A gradient estimate then gives $H_{\max}/H_{\min}\to 1$, and the rescaled flow converges exponentially.

Monotonicity and the shape of singularities

#Huisken's formula

Let $\Phi_{(x_0,t_0)}(x,t)=(4\pi(t_0-t))^{-n/2}\exp\bigl(-\lvert x-x_0\rvert^2/4(t_0-t)\bigr)$ be the backward heat kernel of $\mathbb R^n$ evaluated in $\mathbb R^{n+1}$. For a flow with polynomial area growth and $t<t_0$,

The right side vanishes exactly when the flow is a self-shrinker about $(x_0,t_0)$. The limit of the left integral as $t\nearrow t_0$ is the Gaussian density of the flow at that point. It equals 1 at regular points and exceeds 1 at singular ones. White's local regularity theorem (2005) makes this quantitative: Gaussian density ratios within $\varepsilon(n)$ of 1 on a parabolic ball give a curvature bound on a smaller ball.

#Tangent flows

Rescale parabolically about a singular point, $M^\lambda_s=\lambda(M_{t_0+s/\lambda^2}-x_0)$, and let $\lambda\to\infty$ along a sequence. By the formula, the limit is self-similar, $\sqrt{-s}\,\Sigma$. By the combined work of Huisken, Ilmanen and White this is true in the weak sense in general, where $\Sigma$ may a priori be singular or carry multiplicity. Ilmanen showed that for surfaces in $\mathbb R^3$ the support of $\Sigma$ is smooth, and Han and Wang (2026) extend this to higher dimensions under a total curvature bound $\sup_t\int_{M_t}\lvert A\rvert^n\,d\mu<\infty$. The limit could also depend on the sequence: Colding and Minicozzi (2013) proved uniqueness at cylindrical singularities using Łojasiewicz inequalities, and Schulze had done so at closed multiplicity-one shrinkers (arXiv:1107.4643).

#Type I and Type II

A singularity is Type I if $\sup_{M_t}\lvert A\rvert^2\le C/(T-t)$ and Type II otherwise. Huisken's 1990 argument shows that a Type I flow rescaled by $(T-t)^{-1/2}$ subconverges smoothly to a shrinker at a blow-up point. Type II singularities occur, and Angenent and Velázquez (1997) constructed degenerate neckpinches of this kind. A different rescaling is then needed. For mean-convex flows, Huisken and Sinestrari (1999) proved the convexity estimate $\lambda_1\ge-\eta H-C_\eta$ for every $\eta>0$, so limits at high curvature are convex. With Hamilton's Harnack inequality for mean curvature flow, Type II limits are convex translators, possibly times a Euclidean factor.

#Entropy and stability

Colding and Minicozzi define the entropy $\lambda(\Sigma)=\sup_{x_0,t_0>0}\,(4\pi t_0)^{-n/2}\int_\Sigma e^{-\lvert x-x_0\rvert^2/4t_0}$. Huisken's formula makes it non-increasing along the flow, and it is invariant under rigid motions and dilations. In $\mathbb R^3$ the plane has entropy 1, the sphere $4/e\approx1.47$, and the cylinder $\sqrt{2\pi/e}\approx1.52$.

The operator $L=\Delta-\tfrac12\langle x,\nabla\,\cdot\rangle+\lvert A\rvert^2+\tfrac12$ is the second variation of Gaussian area on a shrinker. Dilations and translations give the eigenfunctions $LH=H$ and $L\langle v,\nu\rangle=\tfrac12\langle v,\nu\rangle$. If $H$ changes sign it is not the bottom eigenfunction, so there is an unstable direction beyond those trivial ones. If $H$ does not change sign, their classification of mean-convex shrinkers, which removes the curvature bound from Huisken's 1993 theorem, says that $\Sigma$ is a generalized cylinder.

Mean-convex flows and surgery

A flow with $H>0$ moves monotonically inward, and the theory becomes much stronger. White (2000, 2003) proved that the singular set has parabolic Hausdorff dimension at most $n-1$, and that tangent flows are multiplicity-one spheres or generalized cylinders.

Andrews (2011) gave a short maximum principle proof of noncollapsing: if every point of $M_0$ has an interior sphere of radius $\alpha/H$, this persists. It parallels Perelman's $\kappa$-noncollapsing. Haslhofer and Kleiner (2013) rebuilt the structure theory on this estimate, with shorter proofs of the derivative, convexity and cylindrical estimates, a global convergence theorem, and a structure theorem for ancient solutions.

A cylindrical estimate, $\lvert A\rvert^2-\tfrac1{n-1}H^2\le\eta H^2+C_\eta$, shows that where the curvature is large the surface is close to a neck or is convex. So long necks can be found, cut and capped, and the discarded pieces are recognised topologically. Brendle and Huisken treated mean-convex surfaces in $\mathbb R^3$ by a new inscribed radius estimate, and Haslhofer–Kleiner gave a proof for all $n\ge 2$.

Flows through singularities

#Weak solutions

Brakke's varifold flows (1978) allow singular surfaces and multiplicity, but are far from unique. The level-set flow of Evans and Spruck and of Chen, Giga and Goto (1991) is unique but can fatten, developing an interior. Fattening does happen for embedded initial data: Ilmanen and White construct genus-$g$ surfaces in $\mathbb R^3$, for large $g$, that fatten at the first singular time (arXiv:2406.18703).

#Multiplicity one

Ilmanen conjectured that tangent flows of embedded surface flows have multiplicity one, so that no two sheets collapse onto the same limit. Bamler and Kleiner proved this in 2023 for flows of surfaces in $\mathbb R^3$, for every blow-up limit. Combined with results of Brendle and of Choi, Haslhofer, Hershkovits and White, it follows that the level-set flow of an embedded 2-sphere does not fatten, and that evolving embedded 2-spheres is well posed in a natural class of singular solutions. Compare singular Ricci flow in dimension three.

#Ancient solutions and the mean-convex neighbourhood

Brendle and Choi (2017) proved that the bowl is the only noncompact ancient solution in $\mathbb R^3$ that is strictly convex and noncollapsed, solving the analogue of a question of Perelman about Ricci flow. Angenent, Daskalopoulos and Šešum (2018) proved that closed, noncollapsed, uniformly 2-convex ancient solutions are unique up to translation and scaling; apart from the round sphere that is the rotationally symmetric oval built by White and by Haslhofer–Hershkovits.

The mean-convex neighbourhood conjecture says that a flow with a neck or spherical singularity is mean-convex nearby. Choi, Haslhofer and Hershkovits proved it for surfaces in $\mathbb R^3$ through a classification of low-entropy ancient flows, and the same authors with White did so for neck singularities in all higher dimensions. Bamler and Lai (2025) treat all cylindrical singularities. If the tangent flow is a multiplicity-one cylinder, then nearby the flow is mean-convex, its time-slices are level sets of a continuous function, and nearby tangent flows are cylindrical, with a canonical neighbourhood theorem describing the local models. Their classification of asymptotically cylindrical ancient flows lists ancient ovals, the bowl and flying-wing translators.

Mean curvature flow beside Ricci flow

	Mean curvature flow	Ricci flow
Curvature equation	$\partial_t H=\Delta H+\lvert A\rvert^2H$	$\partial_t R=\Delta R+2\lvert\operatorname{Ric}\rvert^2$
First convergence theorem	convex $\Rightarrow$ round point (Huisken 1984)	positive Ricci in dimension 3 $\Rightarrow$ round (Hamilton 1982)
Monotone quantity	Gaussian area, entropy	reduced volume, $\mathcal W$-entropy
Noncollapsing	interior sphere of radius $\alpha/H$	$\kappa$-noncollapsing
Shrinking models	self-shrinkers	gradient shrinking solitons
Translating model	bowl soliton	Bryant soliton
Compact ancient models	ancient ovals	Perelman's ovals
Surgery	2-convex, Huisken–Sinestrari	dimension 3, Perelman
Well posed through singularities	embedded $S^2$, Bamler–Kleiner and others	dimension 3, Bamler–Kleiner

Key results

Years give first appearance, as a preprint where there is one.

Year	Result	Who
1984	Closed convex hypersurfaces shrink to round points	Huisken
1989–91	Entire graphs exist for all time	Ecker–Huisken
1990	Monotonicity formula; Type I blow-ups are shrinkers	Huisken
1999	Convexity estimate; Type II limits are translators	Huisken–Sinestrari
2000–03	Structure of mean-convex singularities	White
2005	Local regularity from Gaussian density	White
2009	Surgery for 2-convex hypersurfaces, $n\ge 3$	Huisken–Sinestrari
2009	Entropy; stable shrinkers are generalized cylinders	Colding–Minicozzi
2011	Noncollapsing by the maximum principle	Andrews
2013	Uniqueness of cylindrical tangent flows	Colding–Minicozzi
2017–18	Convex noncompact and 2-convex closed ancient flows	Brendle–Choi; Angenent–Daskalopoulos–Šešum
2020–23	Generic surface flows avoid many singularities	Chodosh, Choi, Mantoulidis, Schulze
2023	Multiplicity one for surfaces in $\mathbb R^3$	Bamler–Kleiner
2025	Mean-convex neighbourhood at all cylindrical singularities	Bamler–Lai

Where it stands

Singularities of generic flows are now well understood. Colding and Minicozzi showed that the only stable self-shrinkers are spheres, cylinders and planes, so these are the singularities that survive perturbation. Chodosh, Choi, Mantoulidis and Schulze (2020) proved that generic closed surfaces in $\mathbb R^3$ avoid asymptotically conical and non-spherical compact singularities, and Chodosh, Choi and Schulze (2023) excluded the remaining multiplicity-one non-generic tangent flows. The key tool is existence and uniqueness of ancient flows on one side of a shrinker. Bamler and Kleiner then proved Ilmanen's multiplicity-one conjecture for surfaces in $\mathbb R^3$, removing a long-standing obstacle to the weak theory and completing Huisken's genericity conjecture there. In $\mathbb R^4$, Chodosh, Choi, Mantoulidis and Schulze also showed that generic low-entropy closed hypersurfaces stay smooth until they disappear in a round point.

The questions have moved to finer structure. The following are recent preprints.

• How many singularities? Daniels-Holgate, Hershkovits and Székelyhidi (2026) show that generic closed surfaces in $\mathbb R^3$ and generic closed mean-convex hypersurfaces meet only finitely many singularities, all nondegenerate, answering generic Morse regularity conjectures of Ilmanen and Colding–Minicozzi–Pedersen in those cases.
• Morse theory of the flow. Bamler, Schulze and Wang (2026) show that a compact flow with multiplicity-one cylindrical tangent flows admits a smooth Morse resolution, with no nondegeneracy assumption. Sun and Xue (2026) study stability of nondegenerate cylindrical singularities, and Choi, Daskalopoulos and Šešum (2026) prove that nondegenerate neckpinches are structurally stable.
• Topology across singularities. Chen (2026) constructs closed flows with arbitrarily large genus increase in any Riemannian 3-manifold, giving counterexamples to Ilmanen's genus-reduction conjecture. Diederen (2026) builds a flow from a sphere to $S^{p-1}\times S^q$ through an isolated singularity.

#Open directions

As far as we know, these are open:

• generic singularities and multiplicity one for hypersurfaces with $n\ge 3$ that are not mean-convex;
• the behaviour of flows through non-generic singularities, and which shrinkers actually occur;
• higher codimension, where a stable self-shrinking Möbius bundle in $\mathbb R^4$ shows that the hypersurface classification of stable shrinkers does not carry over, while Naff's planarity estimate, completed by Lynch (2026) in dimensions 5 and 6, makes high-curvature regions of quadratically pinched flows asymptotically codimension one;
• Lagrangian mean curvature flow, including the Thomas–Yau conjecture;
• free-boundary flows, where Ghosh (2026) has uniqueness of half-cylindrical tangent flows and Bao and Haslhofer (2026) have an avoidance principle in arbitrary domains;
• volume-preserving variants.