29 votes

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 ...
20 votes

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 ...
16 votes

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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14 votes
Accepted

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-...
13 votes
Accepted

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 ...
12 votes
Accepted

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 ...
11 votes
Accepted

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.&...
11 votes

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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11 votes
Accepted

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. ...
11 votes

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 (...
9 votes

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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9 votes

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 ...
  • 28.2k
8 votes

Roadmap to learning about Ricci Flow?

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....
8 votes

Ricci flow and conformal classes

It's true in dimension $2$ but not in higher dimensions, at least not in general.
8 votes

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 ...
7 votes
Accepted

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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7 votes
Accepted

Ricci flow is not a gradient flow for $L^2$-space of metrics

If there were such a functional $\mathcal{F}$, observe that Under Ricci flow the functional would have to decrease. That is, if $\partial_t g(t) = -2 Rc[g]$ then $\partial_t \mathcal{F}(g(t)) \leq ...
7 votes

Self-contained book on Ricci Flow/Geometric Analysis

A quick search on Amazon provides at least three titles that are introductory texts to the topic for graduate students. (1) B. Chow, P. Lu, L. Ni: Hamilton's Ricci Flow, Graduate Studies in ...
7 votes

Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

I think there are two issues at play. MCF of $n$ dimensional hypersurfaces is analogous to $2n$ dimensional Ricci flow (at least for $n=1$ and $n=2$). The Riemann curvature tensor is more ...
  • 1,958
7 votes

Quote by Thurston on the Ricci flow

This 1994 paper by Thurston may or may not be the source you are thinking of, but it is a thoughtful essay that conveys the confidence Thurston had in his conjecture (albeit without referring to ...
7 votes
Accepted

Ricci curvature : beyond heat-like flows

A small number of authors have considered hyperbolic versions of the standard flows, see e.g. "Wave character of metrics and hyperbolic geometric flow" by De-Xing Kong and Kefeng Liu and ...
6 votes
Accepted

Optimal exponent in the Lojasiewicz-Simon gradient inequality

For Inequality (2) in Simon's Theorem 3, a discussion of when the optimal exponent, $\theta=1/2$, is attained can be found (along with many references) in my paper Lojasieicz-Simon gradient ...
6 votes
Accepted

Geometric meaning of Ricci flow

If $g'(t) \neq -2Ric(t)$, it can be anything else. Ricci flow is in some sense a heat kernel applied to the Riemannian metric tensor, uniformizing it over time; if you start modifying the metric ...
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6 votes
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Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow

Remark As @OthisChodosh and @WillieWong have pointed out, the existence of a constant $C_n$ that depends only on the dimension can be proved using only elementary linear algebra. I might as well ...
  • 25.1k
5 votes

Curvature blow up along Ricci flow

Assertion (1) holds; that is, $Ric$ remains bounded as long as the Ricci flow exists. This was proved by N. Sesum (AJM, 2005), and improves the earlier result of R. Hamilton mentioned in the question ...
5 votes
Accepted

Ricci flow and evolution of the shape of drops in spray

For shapes of liquid drops, it is probably not driven by Ricci flow. Fluid interfaces with surface tension is better modeled by mean curvature, going back to Young and Laplace; and there is a lot of ...
  • 31.9k
5 votes

Does Ricci flow depend continuously on the initial metric?

A complete proof of continuous dependence of Ricci flow was published recently by Eric Bahuaud, Christine Guenther & James Isenberg. Let me record their main theorem: Theorem A (Continuous ...
5 votes
Accepted

Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature

I have found doing some calculation that the metric: $$g(t)=\frac{dx^2+dy^2}{e^{-4t}-x^2-y^2}$$ satisfies $\frac{dg(t)}{dt}=-2Ric(t)$ Where $$\frac{dg(t)}{dt}=\frac{4e^{-4t}(dx^2+dy^2)}{(e^{-...
5 votes

Exponential convergence of Ricci flow

See Struwe, Curvature Flows on Surfaces. http://www.numdam.org/item/ASNSP_2002_5_1_2_247_0/, Section 6.2, (particularly equation (64) and surrounding text) where he uses the Kazdan-Warner identity to ...
  • 6,073
4 votes

Self-contained book on Ricci Flow/Geometric Analysis

These books may also be the sort of thing you are after: Peter Topping, Lectures on the Ricci flow Ben Andrews and Christopher Hopper, Ricci Flow in Riemannian Geometry A Complete Proof of the ...

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