Talk:Ricci flow
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[edit] Removed section "Relation to diffusion"
Oops! I noticed an error soon after I wrote this section, but didn't have time to fix it immediately. More than a week later, I still haven't found time to fix it, so I have removed the offending section until I can fix the problem and provide a correct acount. Sorry for the inconvenience---CH (talk) 18:24, 2 August 2005 (UTC)
[edit] Rewrote section "Relation to diffusion"
Yesterday I rewrote this section, following a new eprint by Bakas (see the citation in the article), but Chan Ho caught a new misstatement about the Thurston geometries and says he saw other errors. I have asked him to comment in more detail here before making major changes.---CH (talk) 17:00, 10 August 2005 (UTC)
For usefulness to future editors of the Ricci flow article, here is the conversation from Hillman's talk page:
Nice work on the Ricci flow page. Unfortunately, there are mistakes and some misleading comments. I will help out when I get the chance. For now, I just snipped the statement that the eight Thurston geometries are of constant curvature. They are in general just locally homogenous.
Also, are you Chris Hillman by chance? --Chan-Ho 16:51, August 10, 2005 (UTC)
- Hi, you are right about the Thurston geometries, but can yo clarify on the talk page for Ricci flow what you think the other mistakes are? I did check my computations.---CH (talk) 16:57, 10 August 2005 (UTC)
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- I was really referring to the relation to uniformization and geometrization section. I changed some things in it (check the history and the diffs), but there's still a couple things that need to be reworked. For example, the statement that geometrization is supposed to suggest uniqueness is not really correct. This only works for some geometries in the finite volume case. In the closed 3-manifold case, the Thurston geometry really is unique. If you double-checked the computations, I don't really see a need for me to go through them :-) --Chan-Ho 18:10, August 10, 2005 (UTC)
I don't plan on making major changes; I like the style of Hillman's article, so I will try and preserve that as much as possible while fixing up things here and there. --Chan-Ho 18:14, August 10, 2005 (UTC)
[edit] Perelman
We should add some stuff on Perelman to this page, but I'm not quite sure how to start fitting it in. --Chan-Ho 23:39, August 26, 2005 (UTC)
- I found out fairly recently that Peter Topping is working on a book on Ricci flow (available online here), which incorporates insights from Perelman. In particular, Topping's organization of the topics seems very different than Knopf and Chow and includes for example, an explanation of Ricci flow as a gradient flow. Anyway, interested parties could start incorporating some of this stuff into the article (remember to add Topping to the references!). The gradient flow stuff should be fairly simple to include, at least in a very basic form. Then maybe an explanation of what "Ricci flow with surgery" is. It'll be a while before I can pitch in. --Chan-Ho (Talk) 11:22, 27 April 2006 (UTC)
[edit] neckpinch and soliton
I believe there is a problem with the intro paragraph of the section on recent developments:
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 t0. In certain cases such neckpinches will produce manifolds called Ricci solitons.
Namely, the positive Euler characteristic comment puzzles me. Every closed n-manifold with n odd will have zero Euler characteristic! Certainly n-spheres of that dimension (with a certain family of starting metrics) are known to form neckpinches under Ricci flow...is that what you are thinking of? For n=2, any starting metric on the 2-sphere (which of course has positive Euler char) will smooth out and become constant curvature, so there are no neckpinches here.
Another thing is that the passage implies to me at least that Ricci solitons only result from neckpinches; it might just be my reading, but I think it should be made clearer that soliton is a basic kind of solution independent of neckpinches. --Chan-Ho 12:28, August 30, 2005 (UTC)
