In the mathematical field of differential geometry, the **Ricci flow** ( /ˈriːtʃi/ , Italian: [ˈrittʃi] ), sometimes also referred to as **Hamilton's Ricci flow**, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation; however, it exhibits many phenomena not present in the study of the heat equation. Many results for Ricci flow have also been shown for the mean curvature flow of hypersurfaces.

- Mathematical definition
- Normalized Ricci flow
- Existence and uniqueness
- Convergence theorems
- Corollaries
- Possible extensions
- Li–Yau inequalities
- Examples
- Constant-curvature and Einstein metrics
- Ricci solitons
- Relationship to uniformization and geometrization
- Singularities
- Blow-up limits of singularities
- Type I and Type II singularities
- Singularities in 3d Ricci flow
- Singularities in 4d Ricci flow
- Relation to diffusion
- Recent developments
- See also
- Applications
- General context
- Notes
- References
- Textbooks
- External links

The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard S. Hamilton, who used it to prove a three-dimensional sphere theorem (Hamilton 1982). Following Shing-Tung Yau's suggestion that the singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture, Hamilton produced a number of results in the 1990s which were directed towards its resolution. In 2002 and 2003, Grigori Perelman presented a number of new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's method (Perelman 2002, Perelman 2003a). He was awarded a Fields medal in 2006 for his contributions to the Ricci flow, which he declined to accept.

Hamilton and Perelman's works are now widely regarded as forming a proof of the Thurston conjecture, including as a special case the Poincaré conjecture, which had been a well-known open problem in the field of geometric topology since 1904. However, many of Perelman's methods rely on a number of highly technical results from a number of disparate subfields within differential geometry, so that the full proof of the Thurston conjecture remains understood by only a very small number of mathematicians. The proof of the Poincaré conjecture, for which there are shortcut arguments due to Perelman and to Tobias Colding and William Minicozzi, is much more widely understood (Perelman 2003b, Colding & Minicozzi 2005). It is regarded as one of the major successes of the mathematical field of geometric analysis.

Simon Brendle and Richard Schoen later extended Hamilton's sphere theorem to higher dimensions, proving as a particular case the differentiable sphere conjecture from Riemannian geometry, which had been open for over fifty years (Brendle & Schoen 2009).

On a smooth manifold *M*, a smooth Riemannian metric *g* automatically determines the Ricci tensor Ric^{g}. For each element *p* of *M*, *g*_{p} is (by definition) a positive-definite inner product on the tangent space *T*_{p}*M* at *p*; if given a one-parameter family of Riemannian metrics *g*_{t}, one may then consider the derivative ∂/∂t*g*_{t}, evaluated at a particular value of *t*, to assign to each *p* a symmetric bilinear form on *T*_{p}*M*. Since the Ricci tensor of a Riemannian metric also assigns to each *p* a symmetric bilinear form on *T*_{p}*M*, the following definition is meaningful.

- Given a smooth manifold
*M*and an open real interval (*a*,*b*), a "Ricci flow" assigns to each*t*∈(*a*,*b*) a Riemannian metric*g*_{t}on*M*such that

The Ricci tensor is often thought of as an average value of the sectional curvatures, or as an algebraic trace of the Riemann curvature tensor. However, for the analysis of the Ricci flow, it is extremely significant that the Ricci tensor can be defined, in local coordinates, by an algebraic formula involving the first and second derivatives of the metric tensor. The specific character of this formula provides the foundation for the existence of Ricci flows; see the following section for the corresponding result.

Let *k* be a nonzero number. Given a Ricci flow *g*_{t} on an interval (*a*,*b*), consider *G*_{t}=*g*_{kt} for *t* between *a*/*k* and *b*/*k*. Then

So, with this very trivial change of parameters, the number −2 appearing in the definition of the Ricci flow could be replaced by any other nonzero number. For this reason, the use of −2 can be regarded as an arbitrary convention, albeit one which essentially every paper and exposition on Ricci flow follows. The only significant difference is that if −2 were replaced by a positive number, then the existence theorem discussed in the following section would become a theorem which produces a Ricci flow that moves backwards (rather than forwards) in parameter values from initial data.

The parameter *t* is usually called "time," although this is as part of standard terminology in the mathematical field of partial differential equations, rather than as physically meaningful terminology. In fact, in the standard quantum field theoretic interpretation of the Ricci flow in terms of the renormalization group, the parameter *t* corresponds to length or energy rather than time.^{ [1] }

Suppose that *M* is a compact smooth manifold, and let *g*_{t} be a Ricci flow for *t*∈(*a*,*b*). Define Ψ:(*a*,*b*)→(0,∞) so that each of the Riemannian metrics Ψ(t)*g*_{t} has volume 1; this is possible since *M* is compact. (More generally, it would be possible if each Riemannian metric *g*_{t} had finite volume.) Then define *F*:(*a*,*b*)→(0,∞) by

Since Ψ is positive-valued, *F* is a bijection onto its image (0,*S*). Now the Riemannian metrics *G*_{s}=Ψ(*F*^{−1}(*s*))*g*_{F−1(s)}, defined for parameters *s*∈(0,*S*), satisfy

This is called the "normalized Ricci flow" equation. Thus, with an explicitly defined change of scale Ψ and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The reason for doing this is that the major convergence theorems for Ricci flow can be conveniently expressed in terms of the normalized Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form.

Let be a smooth closed manifold, and let *g*_{0} be any smooth Riemannian metric on . Making use of the Nash–Moser implicit function theorem, Hamilton (1982) showed the following existence theorem:

- There exists a positive number
*T*and a Ricci flow*g*_{t}parametrized by*t*∈(0,*T*) such that*g*_{t}converges to*g*_{0}in the*C*^{∞}topology as*t*decreases to 0.

He showed the following uniqueness theorem:

- If and are two Ricci flows as in the above existence theorem, then for all

The existence theorem provides a one-parameter family of smooth Riemannian metrics. In fact, any such one-parameter family also depends smoothly on the parameter. Precisely, this says that relative to any smooth coordinate chart (*U*,φ) on *M*, the function is smooth for any *i*,*j*=1,...,*n*.

Dennis DeTurck subsequently gave a proof of the above results which uses the Banach implicit function theorem instead.^{ [2] } His work is essentially a simpler Riemannian version of Yvonne Choquet-Bruhat's well-known proof and interpretation of well-posedness for the Einstein equations in Lorentzian geometry.

As a consequence of Hamilton's existence and uniqueness theorem, when given the data (*M*,*g*_{0}), one may speak unambiguously of *the* Ricci flow on *M* with initial data *g*_{0}, and one may select *T* to take on its maximal possible value, which could be infinite. The principle behind virtually all major applications of Ricci flow, in particular in the proof of the Poincaré conjecture and geometrization conjecture, is that, as *t* approaches this maximal value, the behavior of the metrics *g*_{t} can reveal and reflect deep information about *M*.

Complete expositions of the following convergence theorems are given in Andrews & Hopper (2011) and Brendle (2010).

Let (

M,g_{0}) be a smooth closed Riemannian manifold. Under any of the following three conditions:

- M is two-dimensional
- M is three-dimensional and
g_{0}has positive Ricci curvature- M has dimension greater than three and the product metric on (
M,g_{0}) × ℝ has positive isotropic curvaturethe normalized Ricci flow with initial data

g_{0}exists for all positive time and converges smoothly, as t goes to infinity, to a metric of constant curvature.

The three-dimensional result is due to Hamilton (1982). Hamilton's proof, inspired by and loosely modeled upon James Eells and Joseph Sampson's epochal 1964 paper on convergence of the harmonic map heat flow,^{ [3] } included many novel features, such as an extension of the maximum principle to the setting of symmetric 2-tensors. His paper (together with that of Eells−Sampson) is among the most widely cited in the field of differential geometry. There is an exposition of his result in Chow, Lu & Ni (2006 , Chapter 3).

In terms of the proof, the two-dimensional case is properly viewed as a collection of three different results, one for each of the cases in which the Euler characteristic of M is positive, zero, or negative. As demonstrated by Hamilton (1988), the negative case is handled by the maximum principle, while the zero case is handled by integral estimates; the positive case is more subtle, and Hamilton dealt with the subcase in which *g*_{0} has positive curvature by combining a straightforward adaptation of Peter Li and Shing-Tung Yau's gradient estimate to the Ricci flow together with an innovative "entropy estimate". The full positive case was demonstrated by Bennett Chow (1991), in an extension of Hamilton's techniques. Since any Ricci flow on a two-dimensional manifold is confined to a single conformal class, it can be recast as a partial differential equation for a scalar function on the fixed Riemannian manifold (*M*, *g*_{0}). As such, the Ricci flow in this setting can also be studied by purely analytic methods; correspondingly, there are alternative non-geometric proofs of the two-dimensional convergence theorem.

The higher-dimensional case has a longer history. Soon after Hamilton's breakthrough result, Gerhard Huisken extended his methods to higher dimensions, showing that if *g*_{0} almost has constant positive curvature (in the sense of smallness of certain components of the Ricci decomposition), then the normalized Ricci flow converges smoothly to constant curvature. Hamilton (1986) found a novel formulation of the maximum principle in terms of trapping by convex sets, which led to a general criterion relating convergence of the Ricci flow of positively curved metrics to the existence of "pinching sets" for a certain multidimensional ordinary differential equation. As a consequence, he was able to settle the case in which M is four-dimensional and *g*_{0} has positive curvature operator. Twenty years later, Christoph Böhm and Burkhard Wilking found a new algebraic method of constructing "pinching sets," thereby removing the assumption of four-dimensionality from Hamilton's result (Böhm & Wilking 2008). Simon Brendle and Richard Schoen showed that positivity of the isotropic curvature is preserved by the Ricci flow on a closed manifold; by applying Böhm and Wilking's method, they were able to derive a new Ricci flow convergence theorem (Brendle & Schoen 2009). Their convergence theorem included as a special case the resolution of the differentiable sphere theorem, which at the time had been a long-standing conjecture. The convergence theorem given above is due to Brendle (2008), which subsumes the earlier higher-dimensional convergence results of Huisken, Hamilton, Böhm & Wilking, and Brendle & Schoen.

The results in dimensions three and higher show that any smooth closed manifold M which admits a metric *g*_{0} of the given type must be a space form of positive curvature. Since these space forms are largely understood by work of Élie Cartan and others, one may draw corollaries such as

- Suppose that
*M*is a smooth closed 3-dimensional manifold which admits a smooth Riemannian metric of positive Ricci curvature. If*M*is simply-connected then it must be diffeomorphic to the 3-sphere.

So if one could show directly that any smooth closed simply-connected 3-dimensional manifold admits a smooth Riemannian metric of positive Ricci curvature, then the Poincaré conjecture would immediately follow. However, as matters are understood at present, this result is only known as a (trivial) corollary of the Poincaré conjecture, rather than vice versa.

Given any n larger than two, there exist many closed n-dimensional smooth manifolds which do not have any smooth Riemannian metrics of constant curvature. So one cannot hope to be able to simply drop the curvature conditions from the above convergence theorems. It could be possible to replace the curvature conditions by some alternatives, but the existence of compact manifolds such as complex projective space, which has a metric of nonnegative curvature operator (the Fubini-Study metric) but no metric of constant curvature, makes it unclear how much these conditions could be pushed. Likewise, the possibility of formulating analogous convergence results for negatively curved Riemannian metrics is complicated by the existence of closed Riemannian manifolds whose curvature is arbitrarily close to constant and yet admit no metrics of constant curvature.^{ [4] }

Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality."^{ [5] }

- Let
*M*be a smooth manifold, and let*g*_{t}be a solution of the Ricci flow with*t*∈(0,*T*) such that each*g*_{t}is complete with bounded curvature. Furthermore, suppose that each*g*_{t}has nonnegative curvature operator. Then, for any curve γ:[*t*_{1},*t*_{2}]→*M*with [*t*_{1},*t*_{2}]⊂(0,*T*), one has

Perelman (2002) showed the following alternative Li–Yau inequality.

- Let
*M*be a smooth closed*n*-manifold, and let*g*_{t}be a solution of the Ricci flow. Consider the backwards heat equation for*n*-forms, i.e. ∂/∂*t*ω+Δ^{g(t)}ω=0; given*p*∈*M*and*t*_{0}∈(0,*T*), consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as*t*increases to*t*_{0}. Then, for any curve γ:[*t*_{1},*t*_{2}]→*M*with [*t*_{1},*t*_{2}]⊂(0,*T*), one has

- where ω=(4π(
*t*_{0}-t))^{-n/2}*e*^{−f}dμ_{g(t)}.

Both of these remarkable inequalities are of profound importance for the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li-Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem." The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity models," which are Ricci flows on new three-dimensional manifolds. Owing to the Hamilton–Ivey estimate, these new Ricci flows have nonnegative curvature. Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time. This is a powerful result that allows many further arguments to go through. In the end, Perelman shows that any of his singularity models is asymptotically like a complete gradient shrinking Ricci soliton, which are completely classified; see the previous section.

See Chow, Lu & Ni (2006 , Chapters 10 and 11) for details on Hamilton's Li–Yau inequality; the books Chow et al. (2008) and Müller (2006) contain expositions of both inequalities above.

Let (*M*,*g*) be a Riemannian manifold which is Einstein, meaning that there is a number λ such that Ric^{g}=λ*g*. Then *g*_{t}=(1-2λ*t*)*g* is a Ricci flow with *g*_{0}=*g*, since then

If *M* is closed, then according to Hamilton's uniqueness theorem above, this is the only Ricci flow with initial data *g*. One sees, in particular, that:

- if λ is positive, then the Ricci flow "contracts"
*g*since the scale factor 1-2λ*t*is less than 1 for positive*t*; furthermore, one sees that*t*can only be less than 1/2λ, in order that*g*_{t}is a Riemannian metric. This is the simplest examples of a "finite-time singularity." - if λ is zero, which is synonymous with
*g*being Ricci-flat, then*g*_{t}is independent of time, and so the maximal interval of existence is the entire real line. - if λ is negative, then the Ricci flow "expands"
*g*since the scale factor 1-2λ*t*is greater than 1 for all positive*t*; furthermore one sees that*t*can be taken arbitrarily large. One says that the Ricci flow, for this initial metric, is "immortal."

In each case, since the Riemannian metrics assigned to different values of *t* differ only by a constant scale factor, one can see that the normalized Ricci flow *G*_{s} exists for all time and is constant in *s*; in particular, it converges smoothly (to its constant value) as *s*→∞.

The Einstein condition has as a special case that of constant curvature; hence the particular examples of the sphere (with its standard metric) and hyperbolic space appear as special cases of the above.

Ricci solitons are Ricci flows that may change their size but not their shape up to diffeomorphisms.

- Cylinders
**S**^{k}×**R**^{l}(for*k ≥ 2*) shrink self similarly under the Ricci flow up to diffeomorphisms - A significant 2-dimensional example is the
**cigar soliton**, which is given by the metric (*dx*^{2}+*dy*^{2})/(*e*^{4t}+*x*^{2}+*y*^{2}) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons. - An example of a 3-dimensional steady Ricci soliton is the
**Bryant soliton**, which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations. A similar construction works in arbitrary dimension. - There exist numerous families of Kähler manifolds, invariant under a
*U(n)*action and birational to*C*, which are Ricci solitons. These examples were constructed by Cao and Feldman-Ilmanen-Knopf. (Chow-Knopf 2004)^{n}

A **gradient shrinking Ricci soliton** consists of a smooth Riemannian manifold (*M*,*g*) and *f*∈*C*^{∞}(*M*) such that

One of the major achievements of Perelman (2002) was to show that, if *M* is a closed three-dimensional smooth manifold, then finite-time singularities of the Ricci flow on *M* are modeled on complete gradient shrinking Ricci solitons (possibly on underlying manifolds distinct from *M*). In 2008, Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu completed the classification of these solitons, showing:

- Suppose (
*M*,*g*,*f*) is a complete gradient shrinking Ricci soliton with dim(*M*)=3. If*M*is simply-connected then the Riemannian manifold (*M*,*g*) is isometric to , , or , each with their standard Riemannian metrics.

This was originally shown by Perelman (2003a) with some extra conditional assumptions. Note that if *M* is not simply-connected, then one may consider the universal cover and then the above theorem applies to

There is not yet a good understanding of gradient shrinking Ricci solitons in any higher dimensions.

The Ricci flow was utilized by Richard S. Hamilton (1981) to gain insight into the geometrization conjecture of William Thurston, which concerns the topological classification of three-dimensional smooth manifolds.^{ [6] } Hamilton's idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. Then, by placing an *arbitrary* metric *g* on a given smooth manifold *M* and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for *M*. Suitable canonical forms had already been identified by Thurston; the possibilities, called **Thurston model geometries**, include the three-sphere **S**^{3}, three-dimensional Euclidean space **E**^{3}, three-dimensional hyperbolic space **H**^{3}, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes.) Hamilton's idea was that these special metrics should behave like fixed points of the Ricci flow, and that if, for a given manifold, globally only one Thurston geometry was admissible, this might even act like an attractor under the flow.

Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of *positive* Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time.) This doesn't prove the full geometrization conjecture, because the most difficult case turns out to concern manifolds with *negative* Ricci curvature and more specifically those with negative sectional curvature.

Indeed, a triumph of nineteenth-century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

Note that the term "uniformization" suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" suggests placing a geometry on a smooth manifold. *Geometry* is being used here in a precise manner akin to Klein's notion of geometry (see Geometrization conjecture for further details). In particular, the result of geometrization may be a geometry that is not isotropic. In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

The Ricci flow does not preserve volume, so to be more careful, in applying the Ricci flow to uniformization and geometrization one needs to *normalize* the Ricci flow to obtain a flow which preserves volume. If one fails to do this, the problem is that (for example) instead of evolving a given three-dimensional manifold into one of Thurston's canonical forms, we might just shrink its size.

It is possible to construct a kind of moduli space of n-dimensional Riemannian manifolds, and then the Ricci flow really does give a * geometric flow * (in the intuitive sense of particles flowing along flowlines) in this moduli space.

Hamilton showed that a compact Riemannian manifold always admits a short-time Ricci flow solution. Later Shi generalized the short-time existence result to complete manifolds of bounded curvature.^{ [7] } In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time. These singularities are curvature singularities, which means that as one approaches the singular time the norm of the curvature tensor blows up to infinity in the region of the singularity. A fundamental problem in Ricci flow is to understand all the possible geometries of singularities. When successful, this can lead to insights into the topology of manifolds. For instance, analyzing the geometry of singular regions that may develop in 3d Ricci flow, is the crucial ingredient in Perelman's proof the Poincare and Geometrization Conjectures.

To study the formation of singularities it is useful, as in the study of other non-linear differential equations, to consider blow-ups limits. Intuitively speaking, one zooms into the singular region of the Ricci flow by rescaling time and space. Under certain assumptions, the zoomed in flow tends to a limiting Ricci flow , called a **singularity model**. Singularity models are ancient Ricci flows, i.e. they can be extended infinitely into the past. Understanding the possible singularity models in Ricci flow is an active research endeavor.

Below, we sketch the blow-up procedure in more detail: Let be a Ricci flow that develops a singularity as . Let be a sequence of points in spacetime such that

as . Then one considers the parabolically rescaled metrics

Due to the symmetry of the Ricci flow equation under parabolic dilations, the metrics are also solutions to the Ricci flow equation. In the case that

- ,

i.e. up to time the maximum of the curvature is attained at , then the pointed sequence of Ricci flows subsequentially converges smoothly to a limiting ancient Ricci flow . Note that in general is not diffeomorphic to .

Hamilton distinguishes between **Type I and Type II singularities** in Ricci flow. In particular, one says a Ricci flow , encountering a singularity a time is of Type I if

- .

Otherwise the singularity is of Type II. It is known that the blow-up limits of Type I singularities are gradient shrinking Ricci solitons.^{ [8] } In the Type II case it is an open question whether the singularity model must be a steady Ricci soliton—so far all known examples are.

In 3d the possible blow-up limits of Ricci flow singularities are well-understood. By Hamilton, Perelman and recent^{[ when? ]} work by Brendle, blowing up at points of maximum curvature leads to one of the following three singularity models:

- The shrinking round spherical space form
- The shrinking round cylinder
- The Bryant soliton

The first two singularity models arise from Type I singularities, whereas the last one arises from a Type II singularity.

In four dimensions very little is known about the possible singularities, other than that the possibilities are far more numerous than in three dimensions. To date the following singularity models are known

- The 4d Bryant soliton
- Compact Einstein manifold of positive scalar curvature
- Compact gradient Kahler-Ricci shrinking soliton
- The FIK shrinker
^{ [9] } - The Eguchi–Hanson space
^{ [10] }

Note that the first three examples are generalizations of 3d singularity models. The FIK shrinker models the collapse of an embedded sphere with self-intersection number -1.

To see why the evolution equation defining the Ricci flow is indeed a kind of nonlinear diffusion equation, we can consider the special case of (real) two-manifolds in more detail. Any metric tensor on a two-manifold can be written with respect to an **exponential isothermal coordinate chart** in the form

(These coordinates provide an example of a conformal coordinate chart, because angles, but not distances, are correctly represented.)

The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan. Take the ** coframe field **

so that metric tensor becomes

Next, given an arbitrary smooth function , compute the exterior derivative

Take the Hodge dual

Take another exterior derivative

(where we used the **anti-commutative property** of the exterior product). That is,

Taking another Hodge dual gives

which gives the desired expression for the Laplace/Beltrami operator

To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe:

From these expressions, we can read off the only independent ** Spin connection one-form**

where we have taken advantage of the anti-symmetric property of the connection (). Take another exterior derivative

This gives the **curvature two-form**

from which we can read off the only linearly independent component of the Riemann tensor using

Namely

from which the only nonzero components of the Ricci tensor are

From this, we find components with respect to the **coordinate cobasis**, namely

But the metric tensor is also diagonal, with

and after some elementary manipulation, we obtain an elegant expression for the Ricci flow:

This is manifestly analogous to the best known of all diffusion equations, the heat equation

where now is the usual Laplacian on the Euclidean plane. The reader may object that the heat equation is of course a linear partial differential equation—where is the promised *nonlinearity* in the p.d.e. defining the Ricci flow?

The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking . So if is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation. This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate. But if our hot plate is finite in size, and has no boundary where heat can be carried off, we can expect to *homogenize* the temperature, but clearly we cannot expect to reduce it to zero. In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.

The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher-dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that **neckpinch singularities** will form on an evolving *n*-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time . In certain cases, such neckpinches will produce manifolds called Ricci solitons.

For a 3-dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

Kähler metrics remain Kähler under Ricci flow, and so Ricci flow has also been studied in this setting, where it is called "Kähler-Ricci flow."

- ↑ Friedan, D. (1980). "Nonlinear models in 2+ε dimensions".
*Physical Review Letters*(Submitted manuscript).**45**(13): 1057–1060. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057. - ↑ DeTurck, Dennis M. (1983). "Deforming metrics in the direction of their Ricci tensors".
*J. Differential Geom*.**18**(1): 157–162. doi: 10.4310/jdg/1214509286 . - ↑ Eells, James, Jr.; Sampson, J.H. (1964). "Harmonic mappings of Riemannian manifolds".
*Amer. J. Math*.**86**: 109–160. doi:10.2307/2373037. JSTOR 2373037. - ↑ Gromov, M.; Thurston, W. (1987). "Pinching constants for hyperbolic manifolds".
*Invent. Math*.**89**(1): 1–12. doi:10.1007/BF01404671. - ↑ Li, Peter; Yau, Shing-Tung (1986). "On the parabolic kernel of the Schrödinger operator".
*Acta Math*.**156**(3–4): 153–201. doi: 10.1007/BF02399203 . S2CID 120354778. - ↑ Weeks, Jeffrey R. (1985).
*The Shape of Space: how to visualize surfaces and three-dimensional manifolds*. New York: Marcel Dekker. ISBN 978-0-8247-7437-0.. A popular book that explains the background for the Thurston classification program. - ↑ Shi, W.-X. (1989). "Deforming the metric on complete Riemannian manifolds".
*Journal of Differential Geometry*.**30**: 223–301. doi: 10.4310/jdg/1214443292 . - ↑ Enders, J.; Mueller, R.; Topping, P. (2011). "On Type I Singularities in Ricci flow".
*Communications in Analysis and Geometry*.**19**(5): 905–922. arXiv: 1005.1624 . doi:10.4310/CAG.2011.v19.n5.a4. S2CID 968534. - ↑ Maximo, D. (2014). "On the blow-up of four-dimensional Ricci flow singularities".
*J. Reine Angew. Math*.**2014**(692): 153171. arXiv: 1204.5967 . doi:10.1515/crelle-2012-0080. S2CID 17651053. - ↑ Appleton, Alexander (2019). "Eguchi-Hanson singularities in U(2)-invariant Ricci flow". arXiv: 1903.09936 [math.DG].

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In mathematics, the **Poincaré conjecture** is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

In the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measure the failure of second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is *flat*, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

In mathematics, **Thurston's geometrization conjecture** states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In Riemannian geometry, the **sectional curvature** is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 2. The sectional curvature *K*(σ_{p}) depends on a two-dimensional linear subspace σ_{p} of the tangent space at a point *p* of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ_{p} as a tangent plane at *p*, obtained from geodesics which start at *p* in the directions of σ_{p}. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

**Shing-Tung Yau** is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.

In mathematics and especially differential geometry, a **Kähler manifold** is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

**Richard Streit Hamilton** is Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. He made foundational contributions to the theory of the Ricci flow and its use in the resolution of the Poincaré conjecture and geometrization conjecture in the field of geometric topology.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called **harmonic** if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy. As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.

**Tian Gang** is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis.

In differential geometry, a **Kähler–Einstein metric** on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be **Kähler–Einstein** if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

**Huai-Dong Cao** is a Chinese-American mathematician. He is the A. Everett Pitcher Professor of Mathematics at Lehigh University. He is known for his research contributions to the Ricci flow, a topic in the field of geometric analysis.

In mathematics, in the field of differential geometry, the **Yamabe invariant**, also referred to as the **sigma constant**, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe.

In the field of differential geometry in mathematics, **mean curvature flow** is an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly. Except in special cases, the mean curvature flow develops singularities.

In Riemannian geometry, **Schur's lemma** is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.

In differential geometry, a complete Riemannian manifold is called a **Ricci soliton** if, and only if, there exists a smooth vector field such that

**Articles for a popular mathematical audience.**

- Anderson, Michael T. (2004). "Geometrization of 3-manifolds via the Ricci flow" (PDF).
*Notices Amer. Math. Soc*.**51**(2): 184–193. MR 2026939. - Milnor, John (2003). "Towards the Poincaré Conjecture and the classification of 3-manifolds" (PDF).
*Notices Amer. Math. Soc*.**50**(10): 1226–1233. MR 2009455. - Morgan, John W. (2005). "Recent progress on the Poincaré conjecture and the classification of 3-manifolds".
*Bull. Amer. Math. Soc. (N.S.)*.**42**(1): 57–78. doi: 10.1090/S0273-0979-04-01045-6 . MR 2115067. - Tao, T. (2008). "Ricci flow" (PDF). In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.).
*The Princeton Companion to Mathematics*. Princeton University Press. pp. 279–281. ISBN 978-0-691-11880-2.

**Research articles.**

- Böhm, Christoph; Wilking, Burkhard (2008). "Manifolds with positive curvature operators are space forms".
*Ann. of Math. (2)*.**167**(3): 1079–1097. arXiv: math/0606187 . doi:10.4007/annals.2008.167.1079. JSTOR 40345372. MR 2415394. S2CID 15521923. - Brendle, Simon (2008). "A general convergence result for the Ricci flow in higher dimensions".
*Duke Math. J*.**145**(3): 585–601. arXiv: 0706.1218 . doi:10.1215/00127094-2008-059. MR 2462114. S2CID 438716. Zbl 1161.53052. - Brendle, Simon; Schoen, Richard (2009). "Manifolds with 1/4-pinched curvature are space forms".
*J. Amer. Math. Soc*.**22**(1): 287–307. arXiv: 0705.0766 . Bibcode:2009JAMS...22..287B. doi:10.1090/S0894-0347-08-00613-9. JSTOR 40587231. MR 2449060. S2CID 2901565. - Cao, Huai-Dong; Xi-Ping Zhu (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures — application of the Hamilton-Perelman theory of the Ricci flow" (PDF).
*Asian Journal of Mathematics*.**10**(2). MR 2488948. Erratum.- Revised version: Huai-Dong Cao; Xi-Ping Zhu (2006). "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". arXiv: math.DG/0612069 .

- Chow, Bennett (1991). "The Ricci flow on the 2-sphere".
*J. Differential Geom*.**33**(2): 325–334. doi: 10.4310/jdg/1214446319 . MR 1094458. Zbl 0734.53033. - Colding, Tobias H.; Minicozzi, William P., II (2005). "Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman" (PDF).
*J. Amer. Math. Soc*.**18**(3): 561–569. arXiv: math/0308090 . doi:10.1090/S0894-0347-05-00486-8. JSTOR 20161247. MR 2138137. S2CID 2810043. - Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature".
*Journal of Differential Geometry*.**17**(2): 255–306. doi: 10.4310/jdg/1214436922 . MR 0664497. Zbl 0504.53034. - Hamilton, Richard S. (1986). "Four-manifolds with positive curvature operator".
*J. Differential Geom*.**24**(2): 153–179. doi: 10.4310/jdg/1214440433 . MR 0862046. Zbl 0628.53042. - Hamilton, Richard S. (1988). "The Ricci flow on surfaces".
*Mathematics and general relativity (Santa Cruz, CA, 1986)*. Contemp. Math.**71**. Amer. Math. Soc., Providence, RI. pp. 237–262. doi:10.1090/conm/071/954419. MR 0954419. - Hamilton, Richard S. (1993a). "The Harnack estimate for the Ricci flow".
*J. Differential Geom*.**37**(1): 225–243. doi: 10.4310/jdg/1214453430 . MR 1198607. Zbl 0804.53023. - Hamilton, Richard S. (1993b). "Eternal solutions to the Ricci flow".
*J. Differential Geom*.**38**(1): 1–11. doi: 10.4310/jdg/1214454093 . MR 1231700. Zbl 0792.53041. - Hamilton, Richard S. (1995a). "A compactness property for solutions of the Ricci flow".
*Amer. J. Math*.**117**(3): 545–572. doi:10.2307/2375080. JSTOR 2375080. MR 1333936. - Hamilton, Richard S. (1995b). "The formation of singularities in the Ricci flow".
*Surveys in differential geometry, Vol. II (Cambridge, MA, 1993)*. Int. Press, Cambridge, MA. pp. 7–136. doi: 10.4310/SDG.1993.v2.n1.a2 . MR 1375255. - Hamilton, Richard S. (1997). "Four-manifolds with positive isotropic curvature".
*Comm. Anal. Geom*.**5**(1): 1–92. doi: 10.4310/CAG.1997.v5.n1.a1 . MR 1456308. Zbl 0892.53018. - Hamilton, Richard S. (1999). "Non-singular solutions of the Ricci flow on three-manifolds".
*Comm. Anal. Geom*.**7**(4): 695–729. doi: 10.4310/CAG.1999.v7.n4.a2 . MR 1714939. - Bruce Kleiner; John Lott (2008). "Notes on Perelman's papers".
*Geometry & Topology*.**12**(5): 2587–2855. arXiv: math.DG/0605667 . doi:10.2140/gt.2008.12.2587. MR 2460872. S2CID 119133773. - Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv: math/0211159 .
- Perelman, Grisha (2003a). "Ricci flow with surgery on three-manifolds". arXiv: math/0303109 .
- Perelman, Grisha (2003b). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv: math/0307245 .

- Andrews, Ben; Hopper, Christopher (2011).
*The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem*. Lecture Notes in Mathematics.**2011**. Heidelberg: Springer. doi:10.1007/978-3-642-16286-2. ISBN 978-3-642-16285-5. - Brendle, Simon (2010).
*Ricci Flow and the Sphere Theorem*. Graduate Studies in Mathematics.**111**. Providence, RI: American Mathematical Society. doi:10.1090/gsm/111. ISBN 978-0-8218-4938-5. - Cao, H.D.; Chow, B.; Chu, S.C.; Yau, S.T., eds. (2003).
*Collected Papers on Ricci Flow*. Series in Geometry and Topology.**37**. Somerville, MA: International Press. ISBN 1-57146-110-8. - Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2007).
*The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects*. Mathematical Surveys and Monographs.**135**. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3946-1. - Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2008).
*The Ricci Flow: Techniques and Applications. Part II. Analytic Aspects*. Mathematical Surveys and Monographs.**144**. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4429-8. - Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2010).
*The Ricci Flow: Techniques and Applications. Part III. Geometric-Analytic Aspects*. Mathematical Surveys and Monographs.**163**. Providence, RI: American Mathematical Society. doi:10.1090/surv/163. ISBN 978-0-8218-4661-2. - Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2015).
*The Ricci Flow: Techniques and Applications. Part IV. Long-Time Solutions and Related Topics*. Mathematical Surveys and Monographs.**206**. Providence, RI: American Mathematical Society. doi:10.1090/surv/206. ISBN 978-0-8218-4991-0. - Chow, Bennett; Knopf, Dan (2004).
*The Ricci Flow: An Introduction*. Mathematical Surveys and Monographs.**110**. Providence, RI: American Mathematical Society. doi:10.1090/surv/110. ISBN 0-8218-3515-7. - Chow, Bennett; Lu, Peng; Ni, Lei (2006).
*Hamilton's Ricci Flow*. Graduate Studies in Mathematics.**77**. Beijing, New York: American Mathematical Society, Providence, RI; Science Press. doi:10.1090/gsm/077. ISBN 978-0-8218-4231-7. - Morgan, John W.; Fong, Frederick Tsz-Ho (2010).
*Ricci Flow and Geometrization of 3-Manifolds*. University Lecture Series.**53**. Providence, RI: American Mathematical Society. doi:10.1090/ulect/053. ISBN 978-0-8218-4963-7. - Morgan, John; Tian, Gang (2007).
*Ricci Flow and the Poincaré Conjecture*. Clay Mathematics Monographs.**3**. Providence, RI and Cambridge, MA: American Mathematical Society and Clay Mathematics Institute. ISBN 978-0-8218-4328-4. - Müller, Reto (2006).
*Differential Harnack inequalities and the Ricci flow*. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS). doi:10.4171/030. hdl:2318/1701023. ISBN 978-3-03719-030-2. - Topping, Peter (2006).
*Lectures on the Ricci Flow*. London Mathematical Society Lecture Note Series.**325**. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511721465. ISBN 0-521-68947-3. - Zhang, Qi S. (2011).
*Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture*. Boca Raton, FL: CRC Press. ISBN 978-1-4398-3459-6.

- Isenberg, James A. "Ricci Flow" (video). Brady Haran . Retrieved 23 April 2014.

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