Ricci Flow - Relation To Diffusion

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.)

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 connection one-form

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.