# Tag Info

### Shing-Tung Yau's doubts about Perelman's proof

First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things ...

### Shing-Tung Yau's doubts about Perelman's proof

The question is, unavoidably, somewhat ambiguous. Here are seven points, hopefully rather objective: There are well-known expositions of Perelman's work by Cao & Zhu, Kleiner & Lott, and ...

### Quote by Thurston on the Ricci flow

Most likely, this is just misremembering (or misattribution). In his math writing Thurston did not make any predictions regarding what approach to the Geometrization Conjecture (GC) will be ...
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### Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

Here is a list of some topological and geometric applications: Huisken-Ilmanen used inverse MCF to prove the Riemannian Penrose inequality: https://projecteuclid.org/euclid.jdg/1090349447 Huisken-...
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### Ricci flow preserves almost Kahler condition?

The answer is, in general 'no': Under the Ricci-flow, a metric need not remain compatible with $J$ if $J$ is not integrable, even if the associated $2$-form $\omega$ is assumed closed. I don't see ...
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### Ricci flow and isometry group

I think the phenomenon is much more general: If a sequence of metrics $d_i$ on a compact metric space $X$ converges (pointwise on $X\times X$) to a metric $d$, and if $h$ lies in the intersection of ...
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### How is Ricci flow related to computer graphics?

Perhaps this—and its references both past & future ("cited by 152" subsequent papers)—will help...? Jin, Miao, Junho Kim, Feng Luo, and Xianfeng Gu. "Discrete surface Ricci flow.&...
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### Ricci flow and isometry group

The answer is yes (I'm assuming you are asking about closed manifolds, non-compactness allows for all sort of crazy things to happen, you can check out the work of Topping and collaborators). ...
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### Is there a solution of the Yamabe problem using Ricci flow?

The references provided by Carlo Beenakker solve the problem only in special cases. The article "Global existence and convergence of Yamabe flow" by R. Ye assume -- for example -- conformal flatness. ...
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### Squaring a square and discrete Ricci flow

What does "tangent" mean? If two squares touch only at a vertex, are they tangent? According to the diagram provided, tangent seems to mean that two squares should meet along some (...
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### Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

Mu-Tao Wang (Math. Res. Lett. 2001) showed that any diffeomorphism $f:S^2\to S^2$ is isotopic to an isometry, which was originally shown by Smale (Proc. AMS 1959) Mao-Pei Tsui and Mu-Tao Wang (Comm. ...
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### Quote by Thurston on the Ricci flow

There is a video of Thurston's talk "A discussion on geometrization" from May 7, 2001. In the last part of this talk he speaks about possible approaches to proving geometrization. Starting ...
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Here is a list of literature which I compiled when I taught the course on Ricci flow. Basic differential geometry: Einstein Manifolds (Besse). Riemannian geometry (Gallot S., Hulin D., Lafontaine J....
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### Ricci flow and conformal classes

It's true in dimension $2$ but not in higher dimensions, at least not in general.
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### Is there a solution of the Yamabe problem using Ricci flow?

I presume you are referring to the Yamabe flow approach to the Yamabe problem, which in 2 dimensions reduces to a Ricci flow. Relevant references include The Ricci flow on surfaces (1988) The Ricci ...
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### Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

If you have a look into https://arxiv.org/pdf/1711.05167.pdf , page 4, it's written there: "Conversely, it follows from work of Micallef and Wang [29] that every manifold which is diffeomorphic to a ...
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