Ricci flow

In differential geometry, the Ricci flow 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, smoothing out irregularities in the metric.

The Ricci flow was first introduced by Richard 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 Brendle and Schoen.

Mathematical definition

Given a Riemannian manifold with metric tensor \( g_{ij} \), we can compute the Ricci tensor \( R_{ij} \), 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

\( \partial_t g_{ij}=-2 R_{ij}. \)

The normalized Ricci flow makes sense for compact manifolds and is given by the equation

\( \partial_t g_{ij}=-2 R_{ij} +\frac{2}{n} R_\mathrm{avg} g_{ij} \)

where R_\mathrm{avg} is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and n 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.

Examples

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) then Ricci flow collapses the manifold to a point in finite time. If the sphere has radius 1 in n dimensions, then after time t the metric will be multiplied by (1-2t(n-1)), so the manifold will collapse after time 1/2(n-1). More generally, if the manifold is an Einstein manifold (Ricci = constant × metric), then Ricci flow will collapse it to a point if it has positive curvature, leave it invariant if it has zero curvature, and expand it if it has negative curvature.
For a compact Einstein manifold, the metric is unchanged under normalized Ricci flow. Conversely, any metric unchanged by normalized Ricci flow is Einstein.

In particular, this shows that in general the Ricci flow cannot be continued for all time, but will produce singularities. For 3 dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

A significant 2-dimensional example is the cigar soliton solution, 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.

Relationship to uniformization and geometrization

The Ricci flow (named after Gregorio Ricci-Curbastro) was introduced by Richard Hamilton in 1981 in order 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. (A strange and interesting fact is that all closed three-manifolds admit metrics with negative Ricci curvatures! This was proved by L. Zhiyong Gao and Shing-Tung Yau in 1986.) 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" correctly suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" correctly 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 fail 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.
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

\( ds^2 = \exp(2 \, p(x,y)) \, \left( dx^2 + dy^2 \right). \)

(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

\( \sigma^1 = \exp (p) \, dx, \; \; \sigma^2 = \exp (p) \, dy \)

so that metric tensor becomes

\( \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 = \exp(2 p) \, \left( dx \otimes dx + dy \otimes dy \right). \)

Next, given an arbitrary smooth function h(x,y), compute the exterior derivative

\( d h = h_x dx + h_y dy = \exp(-p) h_x \, \sigma^1 + \exp(-p) h_y \, \sigma^2. \)

Take the Hodge dual

\( \star d h = -\exp(-p) h_y \, \sigma^1 + \exp(-p) h_x \, \sigma^2 = -h_y \, dx + h_x \, dy. \)

Take another exterior derivative

d \star d h = -h_{yy} \, dy \wedge dx + h_{xx} \, dx \wedge dy = \left( h_{xx} + h_{yy} \right) \, dx \wedge dy \)

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

\( d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right) \, \sigma^1 \wedge \sigma^2. \)

Taking another Hodge dual gives

\( \Delta h = \star d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right) \)

which gives the desired expression for the Laplace/Beltrami operator

\Delta = \exp(-2 \, p(x,y)) \left( D_x^2 + D_y^2 \right). \)

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

\( d \sigma^1 = p_y \exp(p) dy \wedge dx = -\left( p_y dx \right) \wedge \sigma^2 = -{\omega^1}_2 \wedge \sigma^2 \)
\( d \sigma^2 = p_x \exp(p) dx \wedge dy = -\left( p_x dy \right) \wedge \sigma^1 = -{\omega^2}_1 \wedge \sigma^1. \)

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

\( {\omega^1}_2 = p_y dx - p_x dy. \)

Take another exterior derivative

\( d {\omega^1}_2 = p_{yy} dy \wedge dx - p_{xx} dx \wedge dy = -\left( p_{xx} + p_{yy} \right) \, dx \wedge dy. \)

This gives the curvature two-form

\( {\Omega^1}_2 = -\exp(-2p) \left( p_{xx} + p_{yy} \right) \, \sigma^1 \wedge \sigma^2 = -\Delta p \, \sigma^1 \wedge \sigma^2 \)

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

\( {\Omega^1}_2 = {R^1}_{212} \, \sigma^1 \wedge \sigma^2. \)

Namely

\( {R^1}_{212} = -\Delta p \)

from which the only nonzero components of the Ricci tensor are

\( R_{22} = R_{11} = -\Delta p. \)

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

\( R_{xx} = R_{yy} = -\left( p_{xx} + p_{yy} \right). \)

But the metric tensor is also diagonal, with

\( g_{xx} = g_{yy} = \exp (2 p) \)

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

\( \frac{\partial p}{\partial t} = \Delta p. \)

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

\( \frac{\partial u}{\partial t} = \Delta u \)

where now \( \Delta = D_x^2 + D_y^2 \) 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 p(x,y) = 0. So if p 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.

Recent developments

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 \( t_{0} \) . In certain cases, such neckpinches will produce manifolds called Ricci solitons.

There are many related geometric flows, some of which (such as the Yamabe flow and the Calabi flow) have properties similar to the Ricci flow.