Ricci flow

What it is

Start with a Riemannian manifold $(M, g_0)$ and let the metric move by $\partial_t g = -2\operatorname{Ric}(g)$ with $g(0) = g_0$. In harmonic coordinates $\operatorname{Ric}_{ij} = -\tfrac12 \Delta g_{ij} + (\text{lower order terms})$, so the flow behaves like a heat equation for the metric. Directions of positive curvature contract, directions of negative curvature expand, and bumps in the geometry tend to diffuse away. The first chapter of the Guidebook's Ricci flow book builds this picture from spheres and cylinders.

The equation has a scaling symmetry: if $g(t)$ is a solution, so is $\lambda\, g(t/\lambda)$ for any $\lambda>0$. Distances scale by $\sqrt\lambda$, time by $\lambda$, and curvature by $\lambda^{-1}$. Every blow-up argument in the subject relies on this.

The equation is only weakly parabolic: it is invariant under diffeomorphisms, so it cannot be strictly parabolic. Hamilton (1982) proved short-time existence and uniqueness on closed manifolds with the Nash–Moser inverse function theorem. DeTurck (1983) found a much simpler route. Fix a background connection $\tilde\Gamma$ and add a Lie-derivative term:

This Ricci–DeTurck flow is strictly parabolic, and pulling back by a family of diffeomorphisms recovers Ricci flow. Those diffeomorphisms solve a harmonic map heat flow, a link that also drives uniqueness proofs. Shi (1989) extended existence to complete manifolds with bounded curvature, and Chen and Zhu proved uniqueness in that class. Without a curvature bound both questions are harder; see boundaries and non-compact manifolds.

Curvature itself satisfies a reaction–diffusion equation, so the maximum principle preserves many curvature conditions, and the reaction term decides whether they improve or blow up. The formulas are in How curvature evolves.

Why it matters

Hamilton introduced the flow to prove that a closed 3-manifold with positive Ricci curvature carries a metric of constant positive curvature: the volume-normalised flow converges to one. He then laid out a program: run the flow on any 3-manifold, understand the singularities, cut them out, and read off the topology. Perelman completed that program in 2002–03, proving the Poincaré and geometrization conjectures.

Everything else on this site grows from that story: solitons, entropy, surgery, the sphere theorems and the Kähler theory. The flow also became a template. Results about one PDE turned into results about topology, and later into results about whole families of metrics.

Worked examples

Einstein metrics. If $\operatorname{Ric}(g_0) = \lambda g_0$, then $g(t) = (1-2\lambda t)g_0$. A round unit $S^n$ has $\lambda = n-1$ and shrinks to a point at time $T = \tfrac{1}{2(n-1)}$. A hyperbolic metric of curvature $-1$ has $\lambda = -(n-1)$ and expands forever. A flat or Ricci-flat metric does not move. These are the fixed points up to scaling.

Cylinders and products. Products evolve factor by factor. On $S^{n-1}\times\mathbb R$ with $n\ge 3$, the sphere factor shrinks and the line stays put:

which becomes singular at $T = \tfrac{1}{2(n-2)}$. This shrinking cylinder is the local model of a neck.

Surfaces. In dimension two $\operatorname{Ric} = \tfrac R2 g$, so the flow is conformal: $\partial_t g = -Rg$. On a closed surface the normalised flow converges to a constant-curvature metric from any starting metric (Hamilton 1988; Chow 1991 for the sphere). The surface case and its link to uniformization are worked through in Guidebook chapter 5A.5.

The cigar. Hamilton's cigar soliton is an explicit solution on $\mathbb R^2$ that exists for all time:

Each time slice is isometric to the others. At $t=0$ the scalar curvature is $R = 4/(1+x^2+y^2)$, and far from the origin the metric looks like a cylinder of radius 1.

Symmetric settings. On homogeneous spaces the flow reduces to an ODE. Under rotational symmetry it reduces to a PDE in one space variable, which is where neckpinches were first proved rigorously.

How curvature evolves

Under $\partial_t g = -2\operatorname{Ric}$ the volume form and the scalar curvature satisfy

The Ricci tensor evolves by the Lichnerowicz Laplacian, $\partial_t R_{ij} = \Delta_L R_{ij}$, where (with the sign convention that $R_{ijji}>0$ on the round sphere, and indices raised with $g$)

For the whole curvature tensor, regard $\operatorname{Rm}$ as a symmetric operator on 2-forms, normalised to be the identity on the unit sphere. In Uhlenbeck's moving frame,

where $\operatorname{Rm}^\#$ is a quadratic expression built from the Lie algebra structure of $\mathfrak{so}(n)$. The derivations are in Guidebook chapters 11A.2 and 11A.3.

#What the maximum principle gives

Hamilton's tensor maximum principle reduces the question "is this curvature condition preserved?" to the ODE $\dot{\operatorname{Rm}} = 2(\operatorname{Rm}^2+\operatorname{Rm}^\#)$: a closed convex $O(n)$-invariant set of curvature operators that the ODE preserves is preserved by the PDE. See chapter 11A.7 and curvature pinching.

• Scalar curvature. Since $|\operatorname{Ric}|^2 \ge R^2/n$, the minimum of $R$ is nondecreasing and satisfies $\tfrac{d}{dt}R_{\min} \ge \tfrac 2n R_{\min}^2$. On a closed manifold this gives $R \ge -\tfrac{n}{2t}$ for every initial metric, and if $R_{\min}(0)>0$ then the flow must end by time $\tfrac{n}{2R_{\min}(0)}$. The round sphere attains this bound.
• Dimension three. Write $\lambda\ge\mu\ge\nu$ for the eigenvalues of $\operatorname{Rm}$ in the normalisation above, so they are sectional curvatures and $R = 2(\lambda+\mu+\nu)$. The ODE is

From this Hamilton obtained that positive Ricci curvature is preserved and that its trace-free part becomes small relative to $R$: $|\operatorname{Ric} - \tfrac R3 g|^2 \le C R^{2-\delta}$. A gradient estimate then yields the 1982 theorem.

• Hamilton–Ivey pinching. Also in dimension three, where a sectional curvature is very negative, $R$ is larger still by a logarithmic factor. So rescaled limits at singularities have nonnegative sectional curvature. See the card on this estimate.

#Smoothing and Harnack estimates

Shi's estimates say that a curvature bound controls all derivatives after a short time. If $|\operatorname{Rm}|\le K$ on $[0,1/K]$, then $|\nabla^m\operatorname{Rm}| \le C(n,m)\,K\,t^{-m/2}$ on that interval (chapter 11A.5). This is what makes smooth blow-up limits possible.

Hamilton's Harnack inequality (1993) applies to complete solutions with bounded, nonnegative curvature operator. Its traced form says that for every vector $V$,

Taking $V=0$ shows that $tR$ is nondecreasing at each point, and on an ancient solution that $\partial_t R\ge 0$. See Harnack inequalities.

Special solutions

A Ricci soliton is a metric that evolves only by scaling and diffeomorphism. It satisfies

and is shrinking, steady or expanding according to $\lambda>0$, $\lambda=0$ or $\lambda<0$. When $X = \nabla f$ the equation becomes $\operatorname{Ric}+\nabla^2 f = \lambda g$, a gradient soliton. The resulting flow is $g(t) = (1-2\lambda t)\varphi_t^* g$, where the diffeomorphisms $\varphi_t$ are generated by a rescaled $X$. Round spheres and cylinders are shrinkers, the cigar is a steady soliton, and the Bryant soliton is a rotationally symmetric steady soliton on $\mathbb R^3$.

An ancient solution is one that exists on $(-\infty, T)$. Rescaling near a singularity produces one, so these are the natural singularity models (the guide has the details). Perelman's κ-solutions are ancient, nonflat, κ-noncollapsed at all scales and have bounded nonnegative curvature on each time slice. In dimension three they are now classified:

• Noncompact: Brendle (2018) proved that such a solution is a family of shrinking cylinders, a quotient of one, or the Bryant soliton.
• Compact, on $S^3$: Brendle, Daskalopoulos and Šešum (2020) proved that it is a family of shrinking round spheres or the Type II ancient solution constructed by Perelman.

Einstein metrics and solitons also raise a stability question: does a flow that starts nearby return? That is the subject of stability and uniqueness, with the spectrum of Perelman's operator $-4\Delta+R$ covered under eigenvalues.

Singularities

On a closed manifold, if the maximal time $T$ is finite then $\sup_M|\operatorname{Rm}|\to\infty$ as $t\nearrow T$ (Hamilton). Long-time existence collects the sharper criteria. Singularities are sorted by rate. They are Type I if $\sup_{M\times[0,T)}(T-t)|\operatorname{Rm}|<\infty$, the rate of a shrinking sphere or cylinder, and Type II otherwise.

The basic picture is the neckpinch. Take a dumbbell-shaped metric on $S^{n+1}$, $n\ge2$, with a thin neck. The neck shrinks faster than the ends and pinches in finite time. Angenent and Knopf proved this for rotationally symmetric metrics, with precise asymptotics. The neck is asymptotic to a shrinking cylinder and the singularity is Type I. Degenerate neckpinches give Type II singularities. See the card on neckpinches and chapter 11B.6.

Blow-up analysis is the main method. Pick points $(x_i,t_i)$ with $\lambda_i = |\operatorname{Rm}|(x_i,t_i)\to\infty$ and consider $g_i(t) = \lambda_i\, g(t_i + t/\lambda_i)$. Hamilton's compactness theorem gives a limit once curvature is bounded and the injectivity radius at the base points is bounded below. The limit is an ancient solution (chapter 11B.4). At Type I singularities, suitable blow-up limits are nontrivial gradient shrinking solitons (Naber; Enders–Müller–Topping). The singularities guide follows this story further.

Perelman's toolkit

Perelman's first preprint added the ingredients Hamilton's program lacked.

Entropy. Perelman's functional is $\mathcal F(g,f) = \int_M(R+|\nabla f|^2)e^{-f}\,dV$. If $g$ moves by Ricci flow and $e^{-f}dV$ solves the conjugate heat equation, then

So the flow is, up to diffeomorphism, a gradient flow, and there are no nontrivial steady or expanding breathers. The scale-dependent $\mathcal W$-entropy does the same job for shrinkers: it is constant exactly on gradient shrinking solitons.

No local collapsing. On a closed flow over a finite time interval, for each finite scale $\rho$ there is $\kappa>0$ with the following property: if $r\le\rho$ and $|\operatorname{Rm}|\le r^{-2}$ on the parabolic ball $B(x,t,r)\times[t-r^2,t]$, then $\operatorname{Vol}B(x,t,r)\ge\kappa r^n$. This supplies the injectivity radius bound that compactness needs, and it rules out the product of the cigar with a line as a singularity model. See κ-noncollapsing.

Reduced length and volume. Perelman's $\mathcal L$-length is a space-time distance, and the reduced volume built from it is monotone along the flow. It gives a second proof of non-collapsing, and it shows that blow-down limits of κ-solutions and blow-ups at Type I singularities are shrinkers. The conjugate heat kernel, which Perelman also used, is central to the modern theory.

Pseudolocality. A region that is almost Euclidean at the initial time stays controlled for a short time, whatever happens far away. See the card.

Canonical neighbourhoods. In dimension three, every point of sufficiently large scalar curvature has a neighbourhood that, after rescaling, is close to a piece of a κ-solution: a neck, a cap or a closed component of known topology. See the theorem's card.

The three-dimensional theorems

Surgery. Perelman's second preprint constructs Ricci flow with surgery. Just before a singular time, cut along necks of small scale, discard the high-curvature pieces, whose topology is known, glue in caps modelled on a standard solution, and continue. The parameters are chosen so that canonical neighbourhoods persist after surgery and surgery times do not accumulate.

Finite extinction and the Poincaré conjecture. Perelman's third preprint proves that on a closed oriented 3-manifold whose prime decomposition contains no aspherical factors, the flow with surgery becomes extinct in finite time. The proof follows the area of minimal discs and a regularised curve shortening flow, developed in chapter 10A.12. Colding and Minicozzi gave a separate proof of extinction. Undoing the surgeries shows that such a manifold is a connected sum of spherical space forms and copies of $S^2\times S^1$, and simple connectivity leaves only $S^3$.

Geometrization. When the flow does not go extinct, rescale by $t$ and study large times. The thick part converges to hyperbolic pieces of finite volume, and the collapsed part is a graph manifold. Perelman deferred the collapsing statement to a separate paper. It was worked out by Shioya–Yamaguchi, by Morgan and Tian, by Kleiner–Lott and by Bessières, Besson, Boileau, Maillot and Porti.

Written accounts. Full accounts were written by Kleiner and Lott, Morgan and Tian, and Cao and Zhu. In 2026 preprints reported Lean formalizations of the Poincaré and geometrization proofs.

Beyond surgery. Surgery depends on choices. Kleiner and Lott showed that as the surgery parameter tends to zero the flows subconverge to a singular Ricci flow. Bamler and Kleiner proved its uniqueness and continuous dependence on the initial metric, which verifies a conjecture of Perelman. Applied to families of metrics, this led to the generalized Smale conjecture: for a spherical space form or a hyperbolic 3-manifold the isometry group is homotopy equivalent to the diffeomorphism group. The case of $S^3$ was proved earlier by topological methods by Hatcher, and the hyperbolic case by Gabai.

Higher dimensions

Sphere theorems. Hamilton (1986) proved that a closed 4-manifold with positive curvature operator is diffeomorphic to $S^4$ or $\mathbb{RP}^4$. Böhm and Wilking (2006) extended this to every dimension: on a closed manifold with 2-positive curvature operator, the normalised flow converges to constant sectional curvature. Their family of invariant cones is a general construction for such results. Brendle and Schoen (2007) proved the same conclusion for strictly pointwise 1/4-pinched metrics, using the fact that positive isotropic curvature is preserved in all dimensions.

Surgery with curvature assumptions. Hamilton (1997) introduced surgery in dimension four, under positive isotropic curvature, and the 4-manifolds guide follows the classification that came after. A 2026 preprint by Brendle and Tsiamis extends the classification to all dimensions $n\ge5$.

Bamler's structure theory. Without curvature assumptions, Bamler's 2020 papers start from heat kernel and Nash entropy bounds, and build a compactness theory for a weak notion of flow. Their main structural result is the following: non-collapsed limits of Ricci flows are smooth away from a set of parabolic codimension at least 4, and tangent flows at every point are gradient shrinking solitons, possibly with a singular set of codimension at least 4. As an application, he describes singularity formation at the first singular time and gives a thick-thin decomposition for immortal flows. See the cards on Nash entropy and on codimension-4 structure.

Blow-ups are not always shrinkers. Bamler's survey records Appleton's class of $U(2)$-invariant 4-dimensional flows, in which blow-up limits include the Ricci-flat Eguchi–Hanson metric. Hence shrinkers describe tangent flows, not every rescaling limit, and they may be singular or degenerate. A 2026 preprint of Kim and Ozuch shows that the blowdown soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in a strong local sense, and that the flow can then be continued by blowing down a sphere of self-intersection $\pm1$.

Key results

Year	Result	Who
1982	Closed 3-manifolds with $\operatorname{Ric}>0$ flow to constant curvature	Hamilton
1983	Short-time existence by gauge fixing	DeTurck
1986	Closed 4-manifolds with positive curvature operator are $S^4$ or $\mathbb{RP}^4$	Hamilton
1988, 1991	The normalised flow on a closed surface converges to constant curvature	Hamilton; Chow
1989	Existence on complete manifolds with bounded curvature; derivative estimates	Shi
1993	Harnack inequality for nonnegative curvature operator	Hamilton
1997	Surgery on 4-manifolds with positive isotropic curvature	Hamilton
2002–03	Entropy, non-collapsing, canonical neighbourhoods, surgery, finite extinction	Perelman
2006	Written accounts of Perelman's proofs	Kleiner–Lott; Morgan–Tian; Cao–Zhu
2006	2-positive curvature operator implies space form, in all dimensions	Böhm–Wilking
2007	Strictly pointwise 1/4-pinched metrics flow to constant curvature	Brendle–Schoen
2014, 2017	Singular Ricci flow in dimension 3: existence, then uniqueness and stability	Kleiner–Lott; Bamler–Kleiner
2017 onward	Generalized Smale conjecture by Ricci flow	Bamler–Kleiner
2018, 2020	Classification of 3D κ-solutions: noncompact, then compact on $S^3$	Brendle; Brendle–Daskalopoulos–Šešum
2020	Structure theory of non-collapsed limits in all dimensions	Bamler

Where a paper first appeared as a preprint, the year is that of the preprint.

Neighbouring flows

• Kähler–Ricci flow. The flow preserves the Kähler condition and reduces to a scalar parabolic Monge–Ampère equation. Its singularities correspond to birational operations in the analytic minimal model program.
• Mean curvature flow. A hypersurface moves by its mean curvature. Huisken's monotonicity formula there plays the role that Perelman's monotone quantities play here, and methods move between the two theories.
• Yamabe flow. The conformal flow $\partial_t g = -Rg$. On surfaces it is the same equation as Ricci flow.
• Generalized and $G_2$ flows. Generalized Ricci flow couples the metric to a closed 3-form, and Laplacian flow evolves $G_2$-structures.
• Physics. Perelman's first preprint notes that the flow has been discussed in quantum field theory as an approximation to the renormalization group equation of the 2-dimensional nonlinear σ-model.
• Discrete versions. Combinatorial and graph Ricci flows borrow the idea, but their theorems are separate from the smooth theory.

Where it stands

In dimension three the theory is essentially complete. Kleiner and Lott constructed a canonical flow through singularities, and Bamler and Kleiner proved it is unique and depends continuously on the initial metric. That turned Ricci flow into a tool for studying whole families of metrics, and led to proofs of the generalized Smale conjecture. The long-time behaviour is also understood in detail: Bamler proved that only finitely many surgeries occur and that curvature then decays like $1/t$ (details here).

In higher dimensions the frontier is understanding singularities without special curvature assumptions. Bamler's structure theory gives a higher-dimensional analogue of Perelman's picture, but the analytic description does not yet come with a list of models or a way to continue the flow.

Open questions include:

• Which singularity models occur in dimension four? Classifying gradient shrinking solitons is unfinished even there, and Bamler's survey calls the general task intimidating.
• Can the flow be continued through singularities? Bamler's survey conjectures that a closed 4-manifold has a Ricci flow through singularities in which topology changes along cylinders or cones and time slices may have isolated orbifold points.
• What would such a flow prove? The same survey considers it unlikely to reach the smooth 4-dimensional Poincaré conjecture, and names the 11/8 conjecture as a more feasible target.
• Must scalar curvature blow up at a finite singular time? It does in dimension three, at Type I singularities (Enders–Müller–Topping; a 2026 preprint drops the global Type I assumption) and for Kähler–Ricci flow (Z. Zhang). In general this is open, as far as we know. Other partial results are integral curvature bounds under bounded scalar curvature, including a 2026 preprint reaching dimension 7.
• How rough can the initial data be? Weak and singular flows and limit spaces study flows from nonsmooth metrics, such as the recent smoothing of polyhedral spaces.