Talk:Ricci flow

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)

How do you get a time derivative of p(x,y) while it is not a function of time ? — Preceding unsigned comment added by 158.251.162.199 (talk) 15:00, 4 December 2015 (UTC)

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? --C S 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)

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 :-) --C S 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. --C S 18:14, August 10, 2005 (UTC)

Perelman

We should add some stuff on Perelman to this page, but I'm not quite sure how to start fitting it in. --C S 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. --C S (Talk) 11:22, 27 April 2006 (UTC)

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. --C S 12:28, August 30, 2005 (UTC)

Hi, I've got my hands full with other stuff and didn't want to think about this bit when I wrote that, and want to deal with it even less now. Can you look up some paper on neckpinches and correct the discussion? TIA---CH (talk) 04:22, 1 September 2005 (UTC)

Lott

The internal link to John Lott directs to wrong person, the matematician can be found at http://www.math.lsa.umich.edu/~lott/ —Preceding unsigned comment added by 193.43.151.5 (talk) 13:18, 19 February 2008 (UTC)

Who is Ricci?

Named after Ricci? Any information about this person? —Preceding unsigned comment added by 72.193.240.57 (talk) 07:19, 22 March 2009 (UTC)

I think that a caveat should be added to this entry. The Ricci flow named after Gregorio Ricci-Curbastro should most definitely be called the Curbastro flow. Not only is it more academically sound (bar calling it the Ricci-Curbastro flow)but it sounds much more cool. Just say it. Curbastro flow - no brainer! Obviously "Ricci flow" has passed into common parlance and is therefore the correct terminology, yet i don't see why an aside cannot be added to this entry with the aim of having the Curbastro flow as an alternative (way more cool alternative) to the Ricci flow as an equally correct mathematical term. Come on people - lets start the change. Vive le revolucion! Djt98 (talk) 22:24, 26 March 2009 (UTC)

Add more examples

I would like to see at least one more example of the Ricci flow applied to more than just a sphere. This would in my humble opinion GREATLY help to show exactly what it does. This will also, in turn, help explain the Poincaré Conjecture (which I think is important). Katana (talk) 04:05, 11 December 2011 (UTC)

I second that request! Also does "sphere" (ex. 2) mean a 2-sphere or a 3-sphere. Finally, could someone give a more detailed concrete example (possibly using the 2-sphere manifold). I have no idea what "the usual metric" is for a sphere. Is there one for a general sphere? Or does it depend on whether it is embedded in a higher space or simply depend on the dimensionality? It is really foolish to write "the usual" in an article which is intended to address a very diverse audience.Abitslow (talk) 12:16, 26 December 2014 (UTC)

Edit request

This stuff is more than a bit over my head. That said, the 3rd section, Relationship to uniformization... contains the following paragraph:
"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 ... This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology." This makes no sense. First is it referring to Richard Hamilton? He was born in 1943 and clearly not doing work in the 19th Century. Second: the first sentence conflates subject and object. I am confused about what is wrong with this paragraph: did the writer mean 20th Century or is a different Hamilton, possibly THE Hamilton (William Rowan Hamilton, b. 1805) referred to? Second the sentence claims that in the proof (of the uniformization theorem) Hamilton demonstrated certain characteristics of Ricci flow. The problem with this is the Uniformization Theorem was proved in 1907 by Poincare and/or P. Koebe. This paragraph needs to be removed or rewritten. As it is, it is a mash-up.Abitslow (talk) 12:08, 26 December 2014 (UTC)

Edit request

Please define symbol of o with x through it.

In the section "Relation to diffusion", it uses a symbol of a circle with an x through it (Shift U in MS Word for Wingdings 2)  . Can you please define it? I think it means "any operation" or "any operator", and thus 1 symbol 2 would cover 1+2, 1-2, 1^2, etc. User: Peter10003 18:00 22 November 2016 (UTC)

Reference

B. Chow and D. Knopf, The Ricci Flow: An Introduction, ser. Mathematical Surveys and Monographs. American Mathematical Society, 2004.

198.129.64.59 (talk) 21:41, 4 May 2016 (UTC)

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Funny

Not long before Grigori Perelman solved the Poincare Conjecture, John Milnor wrote an extensive overview article where he never even mentioned any Ricci flow. Indeed, at the time, before Perelman, nobody dreamed about solving the Conjecture by any Ricci flows. The connection was considered as a curiosa, something like a joke. Young people may not know that at the time John Milnor was considered one of the top specialists on Poincare Conjecture. This whole wikipedia Ricci flow article is very dubious in some historically important aspects. Wlod (talk) 22:53, 5 December 2022 (UTC)

I don't understand. What specifically do you think the article is dubious about? (By the way, your claim "before Perelman, nobody dreamed about solving the Conjecture by any Ricci flows" is definitely incorrect as stated. Maybe you have something else in mind?) Gumshoe2 (talk) 23:58, 5 December 2022 (UTC)