I'm playing with Ricci flow at the moment. Ricci flow is quite a buzzy(adsbygoogle = window.adsbygoogle || []).push({});

word these days, since it was used recently to prove the Poincarre

conjecture. Since the Poincarre conjecture, as one of the 7 Clay

institute millenium problems, was famously difficult to prove, you'd

think Ricci flow would be something very difficult too. But a discrete

version of it is actually easy enough for me to understand, and even

implement it in a computer program.

The idea of 'discrete Ricci flow', is basically to gradually deform

(flow) a mesh, so that it converges to a correctly curved mesh of a

predetermined manifold. To find out how to deform the mesh locally, you

figure out the local Ricci tensor, and plug it into a formula to stretch

your mesh.

The idea is especially simple 2D surfaces, and beautifully visual. Check

out this:

http://www.cs.sunysb.edu/~vislab/papers/RicciFlow.pdf

Anyway, it struck me that the formula for Ricci flow looks much like the

Einstein field equation. While Ricci flow may be expressed:

d_t g_ij = -2 (R_ij-R_target_ij)

the Einstin equation can be written as:

R g_ij = -2 (R_ij-T_ij)

One way to look at it is to say that Ricci flow can be used to converge

a manifold to a solution of the Einstein equation, by setting

R_target_ij = T_ij -R

One thing I don't understand yet is how to treat the minus signs in the

metric of general relativity. The Ricci flow I'm doing right now has a

positive metric, it is curved space, but not space-time.

So, can we generalize Ricci flow for non-positive metrics? Does it have

any other applications in physics?

Gerard

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# Ricci Flow in Physics

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