Singularities

What it is

Hamilton showed that a Ricci flow on a closed manifold can be continued as long as its curvature stays bounded. So a finite singular time $T$ is exactly a time when $\sup|\operatorname{Rm}|$ blows up. Singularities are sorted by speed:

• Type I: $|\operatorname{Rm}| \le C/(T-t)$, the rate of a shrinking sphere or cylinder.
• Type II: anything faster.

The standard picture is the neckpinch. A dumbbell-shaped sphere develops a thin neck that shrinks to a point while the two ends stay smooth. Angenent and Knopf gave rigorous examples with precise asymptotics. Degenerate neckpinches, in which the cap itself collapses, give Type II singularities.

To see a singularity, rescale around points of large curvature and take a limit. Hamilton's compactness theorem provides the limit once a lower bound on the injectivity radius is available, and Perelman's non-collapsing theorem supplies that bound. The limits are ancient solutions. At Type I singularities they are gradient shrinking solitons (Naber; Enders–Müller–Topping).

A round sphere shrinks to a point everywhere at once, while a neckpinch is singular only on one cross-sectional sphere. The neckpinch forces new ideas, because the rest of the manifold is still there and its topology is still unknown.

Why it matters

The proof of the Poincaré and geometrization conjectures is, at its core, a complete understanding of singularities in dimension three. Perelman showed that every high-curvature region is a neck, a cap or a closed component of known topology, and that is what makes surgery possible. In higher dimensions, the lack of such a list is the main reason Ricci flow cannot yet be used as freely. So far surgery there needs strong curvature conditions, such as positive isotropic curvature.

Rates and Hamilton's four types

The curvature obeys $\partial_t|\operatorname{Rm}|^2 \le \Delta|\operatorname{Rm}|^2 + C_n|\operatorname{Rm}|^3$. The maximum principle turns this into a lower bound at any finite singular time:

So no singularity is slower than Type I. The round sphere shows the rate is attained. Under $\partial_t g = -2\operatorname{Ric}$, a round $S^n$ of radius $r$ has $r(t)^2 = 2(n-1)(T-t)$, so its sectional curvature is $1/(2(n-1)(T-t))$. The same formula gives the radius of the sphere factor of a shrinking cylinder $S^n\times\mathbb{R}$.

Hamilton's 1995 survey sorts flows on a maximal interval into four classes, by whether the time interval is finite and whether the scale-invariant curvature stays bounded. Each class comes with the kind of limit its point picking produces.

Class	Interval	Condition	Limit
Type I	$T<\infty$	$\sup (T-t)\lvert\operatorname{Rm}\rvert<\infty$	ancient
Type IIa	$T<\infty$	$\sup (T-t)\lvert\operatorname{Rm}\rvert=\infty$	eternal
Type IIb	$T=\infty$	$\sup t\lvert\operatorname{Rm}\rvert=\infty$	eternal
Type III	$T=\infty$	$\sup t\lvert\operatorname{Rm}\rvert<\infty$	immortal

The last two are not singularities in time at all. They describe how the flow behaves as $t\to\infty$, which is the subject of long-time existence. On this page "Type II" means Type IIa.

The neckpinch in equations

#The rotationally symmetric system

Take $g = \varphi(x,t)^2dx^2 + \psi(x,t)^2g_{S^n}$ on $(-1,1)\times S^n$ with $n\ge 2$, closed off at the ends to make $S^{n+1}$, and let $s$ be arc length. The flow reduces to

At an interior minimum of $\psi$ we have $\psi_s=0$ and $\psi_{ss}\ge 0$, so $\tfrac{d}{dt}\psi_{\min}^2 \ge -2(n-1)$. If the neck pinches at $T$, integrating gives $\psi_{\min}(t)^2 \le 2(n-1)(T-t)$. The neck is never wider than the cylinder that would vanish at the same time. Warped products are the testing ground because the problem becomes a parabolic system in one space variable.

#The nondegenerate neckpinch

Simon (2000) gave the first rigorous local singularities, on noncompact warped products $\mathbb{R}\times_f S^n$. Angenent and Knopf (2004) gave the first on compact manifolds, for an open set of metrics on $S^{n+1}$ that are symmetric under rotation and reflection, have positive scalar curvature, positive Ricci curvature on the polar caps, positive sectional curvature on planes tangent to the spheres, and a sufficiently thin neck. The singularity occurs only on the central sphere $\{0\}\times S^n$, and the diameter stays bounded.

Their later paper gives the shape. With

one has, as $\tau\to\infty$,

Further out, for small $s$, the profile is

So parabolic blow-ups at the neck converge to the cylinder $S^n\times\mathbb{R}$, and the correction away from the cylinder is only logarithmic in $T-t$. The limit metric at time $T$ has $\psi/s\to 0$ at the pinch: it is sharper than any cone. Nondegenerate neckpinches are Type I.

#The degenerate neckpinch

Now move the pinch to a pole, so that the neck and the cap beyond it vanish together. Gu and Zhu (2008) proved that Type II singularities form for suitable rotationally symmetric metrics on spheres, without determining the rate. Angenent, Isenberg and Knopf (2012) supplied rates. For each integer $k\ge 3$ and each dimension at least three there are compact rotationally symmetric solutions with

the maximum attained at a pole. For odd $k$ the pinch forms at one pole and the volume stays positive at $T$. For even $k$ the solution is reflection symmetric, the pinch forms at both poles and the volume goes to zero.

These flows show that one singularity can have more than one model, depending on the scale. Rescaling at the pole at the Type II rate gives the steady Bryant soliton. Rescaling about the smallest neck at the parabolic rate gives the shrinking cylinder.

Blow-up analysis

#Rescaling and point picking

If $g(t)$ is a Ricci flow, so is $g_k(s) = Q_k\,g(t_k + s/Q_k)$. Choose $(x_k,t_k)$ with $Q_k = |\operatorname{Rm}(x_k,t_k)|\to\infty$. The rescaled flows live on time intervals that reach back to $-Q_kt_k\to-\infty$, so any limit is ancient. A limit needs curvature bounds on growing parabolic neighbourhoods, and point picking arranges them. For Type II, Hamilton chooses points that nearly maximise $(T_k-t)|\operatorname{Rm}|$ for $T_k\nearrow T$, which makes the limit eternal.

#Why the limit exists, and the cigar

Compactness needs an injectivity radius bound at the base points. This was the gap in Hamilton's programme. The feared Type II model in dimension three was the product of Hamilton's cigar soliton with a line, an eternal solution of exactly the kind point picking produces. It would look like a thin tube with a cap, which cannot be cut like a neck.

Perelman's first preprint closes the gap with his $\mathcal W$-entropy. A flow on a closed manifold on a finite time interval is κ-noncollapsed below some scale: if $|\operatorname{Rm}|\le r^{-2}$ on a parabolic ball of radius $r$, the ball has volume at least $\kappa r^n$. This inequality is scale invariant, so blow-up limits are κ-noncollapsed at every scale. Cigar times a line is not, so it never appears.

#Type I singularities are shrinkers

Naber (2007) showed that the dilation of a Type I singularity is a shrinking soliton. Enders, Müller and Topping (2010) proved this at every singular point, with a nontrivial limit. Let $g(t)$ be a complete Type I flow and $p$ a Type I singular point, meaning there are $p_i\to p$ and $t_i\nearrow T$ with $|\operatorname{Rm}(p_i,t_i)|\ge c/(T-t_i)$. For any $\lambda_j\to\infty$, the flows $\lambda_j\,g(T+t/\lambda_j)$ subconverge to a nonflat gradient shrinking soliton in canonical form:

The tool is Perelman's reduced volume based at the singular time. It is monotone, and it is constant only on shrinkers, so the limit inherits a constant value.

The same paper shows that the natural definitions of the singular set agree: points with no neighbourhood of bounded curvature, Type I singular points, and points where scalar curvature blows up at the Type I rate. There is also a gap: the density at $T$, defined by the limit of the reduced volume, lies in $(0,1]$, and if it exceeds $1-\eta_n$ the point is regular. The authors compare this with White's regularity theorem for mean curvature flow.

Singularity models in dimension three

The Hamilton–Ivey estimate forces nonnegative curvature on every blow-up limit in dimension three, because pinching improves as curvature grows. So singularity models are κ-solutions: complete ancient flows with bounded nonnegative curvature on each slice, κ-noncollapsed at all scales, and not flat.

#The complete list

• Round shrinking spheres and their quotients.
• Shrinking cylinders $S^2\times\mathbb{R}$ and their quotients.
• The Bryant soliton, a rotationally symmetric steady soliton on $\mathbb{R}^3$ that opens like a paraboloid.
• Perelman's ancient solution on $S^3$, often called the ancient oval, and its quotients.

Brendle proved the noncompact half. The compact half is due to Brendle, Daskalopoulos and Šešum (2020), building on Angenent, Brendle, Daskalopoulos and Šešum. Brendle's 2022 survey presents the two together as a complete classification of singularity models in dimension three. The list says what can occur; it does not assert that every entry occurs. The steady case is settled too: a nonflat κ-noncollapsed steady gradient soliton in dimension three is the Bryant soliton. Perelman showed that a two-dimensional κ-solution is a shrinking round sphere or its $\mathbb{Z}_2$ quotient.

#Canonical neighbourhoods

The classification alone is not enough, because surgery needs a statement about the manifold before any limit is taken. That is the canonical neighbourhood theorem. Fix $\varepsilon,\kappa>0$. There is $r_0>0$ such that, for a Ricci flow on a closed 3-manifold with normalised initial data that is κ-noncollapsed below scale $r_0$, every point with $t\ge 1$ and $Q=R(x,t)\ge r_0^{-2}$ has a parabolic neighbourhood of size about $(\varepsilon Q)^{-1/2}$ which, after scaling by $Q$, is $\varepsilon$-close to a piece of a κ-solution. The proof rescales a failing sequence and extracts a limit using bounded curvature at bounded distance.

The structure of κ-solutions then turns this into geometry. Each high-curvature point lies in an $\varepsilon$-neck, an $\varepsilon$-cap or a closed component of known type.

#The manifold at the singular time

Perelman's second preprint describes the whole manifold at $T$. Let $\Omega$ be the open set where $R$ stays bounded as $t\to T$. On $\Omega$ the metrics converge smoothly to a limit, $R$ is proper on the limit, and each end of $\Omega$ lies in an $\varepsilon$-horn: a tube with cross-sectional spheres shrinking to zero as it runs to the singular set. Deep in a horn, points are centres of strong $\delta$-necks, which have stayed close to a shrinking cylinder for a definite backward time. Those are the spheres along which surgery cuts.

Dimension four and above

#Bamler's structure theory

Bamler's 2020 theorem says that non-collapsed limits of Ricci flows are smooth away from a set of codimension at least four in the parabolic sense. Tangent flows at every point are gradient shrinking solitons, possibly with a singular set of codimension at least four. The theory also gives a stratification of the singular set and local $L^p$ bounds on $|\operatorname{Rm}|$ for $p<2$. Its foundations are Nash entropy and $\mathbb F$-convergence.

At the first singular time of a closed flow, Bamler constructs a singular time-slice, a complete metric space (distances may be infinite) built from conjugate heat flows concentrating as $t\nearrow T$. Every sequence of spacetime points approaching $T$ subconverges to a point of this slice. Each such point is regular or has the flow nearby asymptotic to a gradient shrinking soliton. There is a gap here too: a point whose pointed Nash entropy at $T$ is at least $-\varepsilon_n$ is regular. In dimension four the singular spaces of tangent flows are smooth orbifolds with isolated singularities, and either $R>0$ or the orbifold is $\mathbb{R}^4/\Gamma$. The paper also gives a backwards version of pseudolocality, which propagates curvature control backward in time from a ball with a volume lower bound.

#Examples that differ from dimension three

• Negative Ricci curvature in limits. Máximo (2012) proved a conjecture of Feldman, Ilmanen and Knopf: their shrinking soliton on the tautological line bundle over $\mathbb{CP}^1$ is the limit of blow-ups of a Type I singularity on a closed manifold. So blow-up limits on closed 4-manifolds need not have nonnegative Ricci curvature, and the Hamilton–Ivey picture has no direct analogue.
• Models that are not shrinkers. Appleton (2019) showed that a class of $U(2)$-invariant flows on $TS^2$ forms Type II singularities modelled on the Ricci-flat Eguchi–Hanson space. The possible blow-up limits there are Eguchi–Hanson, the flat cone $\mathbb{R}^4/\mathbb{Z}_2$, the $\mathbb{Z}_2$ quotient of the 4-dimensional Bryant soliton and the cylinder $\mathbb{R}\times\mathbb{RP}^3$.
• Every asymptotically conical shrinker occurs. Stolarski (2022) showed that any asymptotically conical gradient shrinking soliton models a finite-time singularity on some closed manifold, with no symmetry or Kähler assumption.

#The Kähler case

On Kähler manifolds the flow reduces to a scalar equation and singularities connect to birational geometry. Jian and Song posted a Type I estimate for the scalar curvature at finite-time singularities on compact Kähler manifolds in 2026. Also in 2026, three preprints proved a Type I curvature bound for volume-collapsing finite-time singularities on compact Kähler surfaces: Cifarelli, Conlon, Hallgren and Zhang, Xu and Zhang, and Xu and Zhou. The first two state that, combined with earlier results, this makes every finite-time singularity on a compact Kähler surface Type I.

Through the singularity

In dimension three there are two routes past $T$. Surgery depends on choices of scale. Kleiner and Lott replaced it with singular Ricci flows, obtained as surgery scales go to zero, and Bamler and Kleiner proved that these flows are unique and depend continuously on the initial metric. That verifies a conjecture of Perelman, and it led to the generalized Smale conjecture.

In dimension four nothing comparable is known in general. Bamler's survey conjectures that a closed Riemannian 4-manifold has a Ricci flow through singularities in which topology changes along cylinders or cones, with isolated orbifold points allowed in time-slices.

Conical singularities are a case with theorems. Gianniotis and Schulze built smooth flows out of metrics with isolated conical points whose cones are nonnegatively curved. Kim and Ozuch, in a 2026 preprint, show that the Feldman–Ilmanen–Knopf blowdown soliton is strictly dynamically stable. On a closed 4-manifold the metrics whose first singularity is modelled on it form a $C^{14}$-open set, nonempty exactly when the manifold contains a sphere of self-intersection $\pm1$, and the flow continues by blowing that sphere down.

Comparison with mean curvature flow

Mean curvature flow has a parallel blow-up theory, with shrinkers as its Type I models and necks as its basic singularity, and its generic behaviour is now better understood. Choi, Daskalopoulos and Šešum prove that nondegenerate neckpinches are stable under $C^2$ perturbation. Daniels-Holgate, Hershkovits and Székelyhidi show that a generic compact surface in $\mathbb{R}^3$ meets only finitely many singularities. We know of no statement of this strength for Ricci flow neckpinches. One difference is basic: a hypersurface visibly breaks in its ambient space, while Ricci flow has none.

Key results

Years are of the arXiv posting for papers with an arXiv record here, otherwise of publication.

Year	Result	Who
1982	Curvature blows up at any finite singular time on a closed manifold	Hamilton
1995	Compactness theorem; classification of singularity types	Hamilton
2000	Local singularities on noncompact warped products	Simon
2002–03	Noncollapsing; canonical neighbourhoods; horns; surgery	Perelman
2004–05	Neckpinches on $S^{n+1}$, then precise asymptotics	Angenent, Knopf
2007	Type I blow-ups are shrinking solitons	Naber
2008	Existence of Type II singularities on spheres	Gu, Zhu
2010	Nontrivial shrinkers at every Type I singular point	Enders, Müller, Topping
2012	Degenerate neckpinches at rates $(T-t)^{-2+2/k}$	Angenent, Isenberg, Knopf
2012	Type I blow-ups in dimension 4 with negative Ricci curvature	Máximo
2014–17	Singular Ricci flows in dimension 3, then their uniqueness	Kleiner, Lott; Bamler, Kleiner
2019	Eguchi–Hanson singularities	Appleton
2018–20	3D κ-solutions classified	Brendle; Brendle, Daskalopoulos, Šešum
2020	Structure theory in all dimensions	Bamler
2022	Closed flows realising asymptotically conical shrinkers	Stolarski
2025	Strong uniqueness of cylindrical tangent flows	Fang, Li

Where it stands

Bamler's structure theory (2020) extends much of Perelman's picture to all dimensions. At the first singular time, tangent flows are gradient shrinking solitons, possibly singular on a set of codimension at least four, and there is a stratification of the singular set. The missing piece is the list of possible shrinkers. Even in dimension four it is not known which singularity models occur, and continuing the flow through singularities in general dimensions is open. The feed below tracks both problems.

Two questions are live:

• Which models occur. Gradient shrinkers in low dimensions are the natural list. Munteanu and Wang showed that most noncompact 4-dimensional ones have cylindrical or conical ends, as Bamler's survey puts it. The survey's working conjecture is that most singularity models are gradient shrinking solitons, possibly degenerate and possibly singular in codimension at least four. Appleton's examples are why it says "most", and the survey records that it is unknown whether the last two of his limits occur as singularity models.
• Whether the model at a point is unique. A tangent flow is defined along a subsequence, so different sequences could give different limits. Fang and Li (2025) prove strong uniqueness at cylindrical tangent flows, through a Łojasiewicz inequality for the pointed $\mathcal W$-entropy, and Li, Yan and Zhang (2026 preprint) do the same when one tangent flow is a smooth asymptotically conical shrinker. For the Kähler setting, see the analytic minimal model program, where singularities correspond to birational operations. Symmetric examples, where neckpinches were first proved rigorously, are collected under rotational symmetry and warped products.