Questions tagged [ricci-flow]

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7 answers
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Roadmap to learning about Ricci Flow?

Hello, I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
youngmathperson's user avatar
26 votes
3 answers
3k views

Does Ricci flow depend continuously on the initial metric?

Consider a version of Ricci flow for which short time existence and uniqueness are known, e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
Igor Belegradek's user avatar
24 votes
3 answers
3k views

Is there a combinatorial analogue of Ricci flow?

The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing. However I have ...
Bruce Westbury's user avatar
23 votes
2 answers
5k views

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

[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.] According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of ...
16 votes
4 answers
988 views

Squaring a square and discrete Ricci flow

Is this a theorem? Every $3$-connected planar graph $G$ may be represented as a tiling of a square by squares, one square per node of $G$, with nodes connected in $G$ corresponding to tangent squares....
Joseph O'Rourke's user avatar
15 votes
1 answer
972 views

Ricci curvature : beyond heat-like flows

Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs. There are at least two versions of Ricci curvature in the ...
Mirco A. Mannucci's user avatar
14 votes
2 answers
5k views

What prerequisites do I need to read the book Ricci Flow and the Poincare Conjecture, published by CMI

As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?
user18717's user avatar
  • 351
14 votes
1 answer
386 views

Does the Cheeger constant satisfy a heat-type equation?

It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow. A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of ...
Hollis Williams's user avatar
13 votes
3 answers
2k views

Quote by Thurston on the Ricci flow

I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't ...
Gabe K's user avatar
  • 5,324
13 votes
2 answers
1k views

Is there a solution of the Yamabe problem using Ricci flow?

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
Hollis Williams's user avatar
13 votes
1 answer
1k views

How is Ricci flow related to computer graphics?

I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...
terett's user avatar
  • 1,069
13 votes
0 answers
361 views

Pseudolocality outside of geometric PDE?

In Ricci flow, the pseudolocality theorem says roughly that regularity in some region implies that as time goes on, there is some regularity in a smaller region. The first version is due to Perelman. ...
Tim Carson's user avatar
11 votes
1 answer
764 views

How fast does Ricci flow converge on the three-sphere?

Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the ...
John Pardon's user avatar
  • 18.3k
11 votes
0 answers
480 views

Is there any connection between the Deturck trick and the Uhlenbeck trick?

There are two separate places where ingenious uses of gauge transformations simplify the analysis of Ricci flow considerably.  The Deturck trick is a way to break the diffeomorphism invariance of the ...
Gabe K's user avatar
  • 5,324
10 votes
3 answers
1k views

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

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics ...
Matheus Andrade's user avatar
10 votes
1 answer
467 views

Optimal exponent in the Lojasiewicz-Simon gradient inequality

Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\...
user avatar
9 votes
3 answers
2k views

Ricci flow with surgery in dimension 2

Is it possible to define the Ricci flow with surgery in dimension 2 and use it to classify the surfaces? I know this is overkill, there are simpler ways to classify surfaces, but I would like to ...
Guillaume Brunerie's user avatar
9 votes
3 answers
4k views

The relations between the Perelman's entropy functional and notions of entropy from statistical mechanics

I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of ...
Sepideh Bakhoda's user avatar
9 votes
1 answer
442 views

Ricci flow preserves almost Kahler condition?

I have been unable to find a reference to the following (perhaps too naive) question. Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-...
u184's user avatar
  • 277
9 votes
1 answer
1k views

How submanifolds evolve under Ricci flow?

This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I found online... ...
Renato G. Bettiol's user avatar
8 votes
1 answer
871 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
Joseph O'Rourke's user avatar
7 votes
2 answers
684 views

Ricci flow and isometry group

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...
user82084's user avatar
7 votes
2 answers
607 views

the left hand side of the Ricci flow equation at the initial value

I just started to learn about the Ricci flow and try to understand the Ricci flow evolution equation. It states that a one-parameter family $g_t$, $t\in[0,T)$ of Riemannian metrics on a smooth closed ...
kassandra's user avatar
  • 165
7 votes
2 answers
526 views

Exponential convergence of Ricci flow

I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
Gabe K's user avatar
  • 5,324
7 votes
2 answers
721 views

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

I am reading Ben Andrews book about Ricci flow and at the start of the chapter about Perelman's gradient flow formulation for Ricci flow he says Robert Bryant exposed that there are no functionals ...
L.F. Cavenaghi's user avatar
7 votes
1 answer
919 views

Reverse Ricci Flow and Longtime Existence

The usual Ricci flow and normalized Ricci flow for surfaces are $$ \partial_t g = -2Kg $$ and $$ \partial_t g = -2Kg + 2sg,$$ where $K$ is the Gaussian curvature and $s$ is its average. The latter ...
user35946's user avatar
  • 355
7 votes
1 answer
455 views

"Elliptic" proof that Compact Ricci Solitons are Gradient Ricci Solitons

I'm concerned with the following Proposition: If a compact manifold $M$ satisfies $$Rc + \textstyle\frac{1}{2}\mathcal{L}_Xg = \lambda g $$ where $\lambda$ is a constant (i.e. $M$ is a compact Ricci ...
Brian Klatt's user avatar
7 votes
0 answers
982 views

On Perelman's paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Perelman has written: Fix a closed manifold $M$ with a probability measure $m$, and suppose that our system is ...
Sepideh Bakhoda's user avatar
6 votes
6 answers
2k views

A simple and good reference about solitons

I want to study gradient Ricci solitons. Can anyone help me to find a simple and good reference about solitons and their applications? Thanks
Sepideh Bakhoda's user avatar
6 votes
1 answer
469 views

Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
Jae Ho Cho's user avatar
6 votes
2 answers
531 views

Hamilton-Ivey pinching in dimension 4

I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. ...
Brian Klatt's user avatar
6 votes
2 answers
586 views

Rigorous solution to Ricci Flow on dumbbell $S^3$

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...
Chris Gerig's user avatar
  • 17.1k
6 votes
0 answers
221 views

Curvature decay of Ricci expanders

Let $M$ be a gradient Ricci expander with nonnegative curvature operator. Assume $\Sigma$ is its space of directions at infinity (so $M$ looks like a cone over $\Sigma$). What is the curvature ...
Anton Petrunin's user avatar
6 votes
0 answers
157 views

Strong uniqueness of the Ricci flow

In the paper ``Strong uniqueness of the Ricci flow", Chen proved the following strong uniqueness of the Ricci flow: let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^3$, with ...
Tong's user avatar
  • 183
5 votes
2 answers
1k views

Self-contained book on Ricci Flow/Geometric Analysis

Can someone please tell me whether there is any self-contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self-contained I mean it does not assume that ...
Bingo's user avatar
  • 779
5 votes
1 answer
414 views

Ricci flow negative curvature

We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$. I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
Adrien B's user avatar
5 votes
1 answer
315 views

Curvature blow up along Ricci flow

In the book on Ricci flow by Andrews and Hopper, it has been proved that if Ricci flow on $M$ has a finite time singularity at time $T$ then $\lim_{t \nearrow T} \sup_{x\in M} |Rm(x,t)|=\infty$. I am ...
Bingo's user avatar
  • 779
5 votes
1 answer
203 views

Evolution of $W_+$ and $W_-$ under the Ricci flow

In dimension $4$ the Weyl operator $W$ splits in two parts $$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$ and $$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$ (a) Has there been a study of the evolution ...
I. Tergiakidis's user avatar
5 votes
1 answer
607 views

Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature

I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature. Any help will be appreciated. Thanks!
Onil90's user avatar
  • 823
5 votes
1 answer
257 views

Gradient of solution to heat equation under evolving metric

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
Naruto's user avatar
  • 83
5 votes
1 answer
871 views

Yang-Mills flow, Ricci flow and the holonomy

Is the holonomy group (based at some point) preserved along the Yang-Mills flow/ Ricci flow? (1) For Yang-Mills case, we know that the centralizer of the holonomy $H_x$ is the isotropy group of the ...
Hao Yin's user avatar
  • 527
5 votes
1 answer
228 views

Neckpinch singularity of Ricci flow

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the ...
Hollis Williams's user avatar
5 votes
1 answer
386 views

Negative pinching and Ricci flow

Let $\varepsilon>0$ be sufficiently small. Denote by $\mathrm{Rm}$ and $\mathrm{R}$ the curvature operator and the scalar curvature. Consider the following pinching condition $$\langle\mathrm{Rm}\,\...
ε-δ's user avatar
  • 1,785
5 votes
0 answers
117 views

Metric under Ricci flow on a 2-sphere can be realized by embedding

I am sorry if this is a silly question, but I am new to Ricci flows. Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...
Eduardo Longa's user avatar
5 votes
0 answers
181 views

Converse of Hamilton's Maximum Principle?

The famous maximum principle of Hamilton states the following. Let $C$ be a convex $O(n)$-invariant subset of the space of algebraic curvature operators. Then if it is invariant under the ODE $$ \dot{...
Matthias Ludewig's user avatar
5 votes
0 answers
302 views

Gromov Hausdorff limit and Ricci flow

Let $M$ be a compact, smooth manifold and $\{g(t)\}$ be a family of Riemannian metrics on $M$ evolving under Ricci flow. Suppose the maximal existence time $T$ is finite. To what extent the following ...
Bingo's user avatar
  • 779
5 votes
0 answers
1k views

Prerequisites for reading Gregory Perelman's work

What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture? I am referring to the last three papers here.
Alan's user avatar
  • 1,514
4 votes
1 answer
383 views

Ricci flow and conformal classes

Is it true that the conformal class of the metric is preserved under Ricci flow? I have seen it mentioned in an answer on this site. Is there an easy argument? (This question was asked on MSE but it ...
timur's user avatar
  • 3,292
4 votes
2 answers
2k views

Energy functional

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\...
Sepideh Bakhoda's user avatar
4 votes
1 answer
158 views

One-sided version of the curve-shortening flow

The curve-shortening flow is $$ \frac{\partial C}{\partial t} = \kappa n $$ where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...
André Henriques's user avatar