Ricci flow

In differential geometry, the Ricci flow is a process which deforms the metric of a Riemannian manifold in a manner formally analogous to the diffusion of heat.

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

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: ds^2 = exp(2 , p(x,y)) , left( dx^2 + dy^2 ight)

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

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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

Related Topics:
Ricci tensor - Laplace-Beltrami operator - Élie Cartan

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

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so that metric tensor becomes

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: sigma^1 otimes sigma^1 + sigma^2 otimes sigma^2 = exp(2 p) , left( dx otimes dx + dy otimes dy ight)

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

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

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Take the Hodge dual

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

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Take another exterior derivative

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

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

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: d star d h = exp(-2 p) , left( p_{xx} + p_{yy} ight) , sigma^1 wedge sigma^2

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Taking another Hodge dual gives

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: Delta h = star d star d h = exp(-2 p) , left( p_{xx} + p_{yy} ight)

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which gives the desired expression for the Laplace/Beltrami operator

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: Delta = exp(-2 , p(x,y)) left( D_x^2 + D_y^2 ight)

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

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: d sigma^1 = p_y exp(p) dy wedge dx = -left( p_y dx ight) wedge sigma^2 = -{omega^1}_2 wedge sigma^2

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: d sigma^2 = p_x exp(p) dx wedge dy = -left( p_x dy ight) wedge sigma^1 = -{omega^2}_1 wedge sigma^1

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

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: {omega^1}_2 = p_y dx - p_x dy

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Take another exterior derivative

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

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This gives the curvature two-form

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

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

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: {Omega^1}_2 = {R^1}_{212} , sigma^1 wedge sigma^2

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Namely

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: {R^1}_{212} = -Delta p

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from which the only nonzero components of the Ricci tensor are

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: R_{22} = R_{11} = -Delta p

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

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: R_{xx} = R_{yy} = -left( p_{xx} + p_{yy} ight)

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But the metric tensor is also diagonal, with

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

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

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: rac{partial log p}{partial t} = Delta log p

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

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: rac{partial u}{partial t} = Delta u

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where now Delta = D_x^2 + D_y^2 is the usual Laplacian on the Euclidean plane.

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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?

Related Topics:
Linear - Partial differential equation

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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.

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