In differential geometry, the Ricci flow (//, Italian: [ˈrittʃi]) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat. Heuristically speaking, at every point of the manifold the Ricci flow shrinks directions of positive curvature and expands directions of negative curvature, while simultaneously smoothing out irregularities in the metric. The latter is analogous to the smoothing behavior of the heat equation.
The Ricci flow, named after Gregorio Ricci-Curbastro, was first introduced by Richard S. Hamilton in 1981 and is also referred to as the Ricci–Hamilton flow. It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the differentiable sphere theorem by Simon Brendle and Richard Schoen.
Given a Riemannian manifold with metric tensor , we can compute the Ricci tensor , which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called "time" (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation
The normalized Ricci flow makes sense for compact manifolds and is given by the equation
where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and is the dimension of the manifold. This normalized equation preserves the volume of the metric.
The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However, the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed, then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.)
Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.
- If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is Ricci-flat.
- If the manifold is a sphere (with the usual metric), or more generally an Einstein metric (where Ricci tensor = constant × metric tensor) with positive curvature, then Ricci flow collapses the manifold to a point in finite time. For example, the metric of the n-dimensional sphere of radius 1, after time , will be multiplied by , so the manifold will collapse after time . This shows that the Ricci flow sometimes cannot be continued for all time, instead producing singularities.
- If the manifold is an Einstein manifold with negative curvature, then Ricci flow will expand it.
Ricci solitons are Ricci flows that may change their size but not their shape up to diffeomorphisms.
- Cylinders Sk × Rl (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 (dx2 + dy2)/(e4t + x2 + y2) 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 Cn, which are Ricci solitons. These examples were constructed by Cao and Feldman-Ilmanen-Knopf.
Relationship to uniformization and geometrization
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. 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 S3, three-dimensional Euclidean space E3, three-dimensional hyperbolic space H3, 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. 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.
Blow-up limits of singularities
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 .
Type I and Type II singularities
Otherwise the singularity is of Type II. It is known that the blow-up limits of Type I singularities are gradient shrinking Ricci solitons. 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.
Singularities in 3d Ricci flow
In 3d the possible blow-up limits of Ricci flow singularities are well-understood. By Hamilton, Perelman and recent 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.
Singularities in 4d Ricci flow
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
- The Eguchi–Hanson space
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.
Relation to diffusion
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.)
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
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 applications to the construction of Kähler–Einstein metrics.
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