The ricci-flow tag has no usage guidance.

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### 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 ...

**19**

votes

**3**answers

2k 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 ...

**12**

votes

**7**answers

4k views

### 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 ...

**9**

votes

**1**answer

483 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 ...

**9**

votes

**1**answer

192 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\...

**8**

votes

**3**answers

1k 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 ...

**8**

votes

**2**answers

2k 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?

**8**

votes

**1**answer

438 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 ...

**7**

votes

**1**answer

680 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...
...

**6**

votes

**2**answers

467 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 ...

**6**

votes

**2**answers

242 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. ...

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votes

**0**answers

388 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.

**5**

votes

**5**answers

1k 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

**5**

votes

**1**answer

526 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 ...

**5**

votes

**1**answer

348 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 ...

**5**

votes

**1**answer

293 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}\,...

**5**

votes

**1**answer

633 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 ...

**4**

votes

**1**answer

382 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 ...

**4**

votes

**1**answer

370 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 ...

**3**

votes

**3**answers

2k 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 ...

**3**

votes

**1**answer

182 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 ...

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votes

**2**answers

394 views

### Bryant Soliton is asymptotically cylindrical?

This is my first question in mathoverflow.
I'm now reading Brendle's paper http://arxiv.org/pdf/1203.0270.pdf.
I'm confused about how to check Condition (ii) of asymptotically cylindrical condition ...

**3**

votes

**1**answer

422 views

### What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...

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votes

**2**answers

677 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)=\...

**3**

votes

**1**answer

250 views

### In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms

Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms?
Thanks for your time.

**3**

votes

**1**answer

349 views

### geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...

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votes

**0**answers

100 views

### Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...

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votes

**0**answers

155 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...

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votes

**0**answers

606 views

### On Perelman's paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written:
Fix a closed manifold $M$ with a probability measure $m$, and suppose
that our ...

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votes

**2**answers

335 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 ...

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votes

**2**answers

512 views

### Does normalized Ricci flow on surfaces yield a bundle?

As is well known,
the normalized Ricci flow is defined for all $t>0$ on compact surfaces,
and every metric on a compact surfaces converges to a metric constant curvature if $X \neq S^2$ (at least I ...

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votes

**1**answer

253 views

### Time has dimension $2$ with respect to the Ricci flow scaling

Terence Tao in his lecture notes on Ricci flow has written:
If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the ...

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votes

**1**answer

230 views

### Optimal, conformal diffeomorphisms between two surfaces in 3D

Let $S_1$ and $S_2$ be two smooth, closed surfaces embedded in $\mathbb{R}^3$.
Q. Is there a natural definition of the optimal, conformal diffeomorphism between $S_1$ and $S_2$?
I am imagining $...

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votes

**1**answer

490 views

### Ricci flow as a gradient flow and its Lyapunov function

In study of Ricci flow, for making Ricci flow as a gradient flow I faced $\mathcal{F}(g,f)=\int (R+|\nabla f|^2)e^{-f}$. I know that if we suppose $\frac{df}{dt}=-R$, then $\frac{d}{dt}\mathcal{F}(g,f)...

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votes

**1**answer

397 views

### Ricci flow descending from an universal cover

Reading some of my old notes, I came across a remark, I don't understand. Summarized:
Let $(M^n,g)$ be an open riemannian manifold (in that case it came with sectional curvature $K\ge0$, but I don't ...

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votes

**1**answer

283 views

### Ricci-invariant class of metrics

Imagine that there is a class of Riemannian metrics $\mathcal{R}$ on
3-dimensional manifolds such that
$\mathcal{R}$ is locally finite dimensional; i.e., there are
finite number of real ...

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votes

**0**answers

66 views

### Ricci soliton on contact manifolds

Recently I am studying Ricci flow and its self-similar solution called Ricci soliton. In this respect I found some papers which focuses Ricci soliton in the setting of various kind of contact ...

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**0**answers

102 views

### Sources on evolution of submanifolds subject to Ricci flow

I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...

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vote

**1**answer

401 views

### Possible Error in Chow-Knopfs Ricci Flow Introduction

On page 105 of Chow--Knopfs "Ricci Flow: An Introduction", it reads:
"$r = \int_M R d\mu / \int_M d\mu$ ... is determined by the Euler characteristic $\chi(M^2)$ of the surface, hence is independent ...

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vote

**2**answers

368 views

### Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons".
A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...

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vote

**1**answer

221 views

### On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow
and Geometrization
of 3-Manifolds" book as a definition of canonical neighborhoods have written:...

**1**

vote

**1**answer

135 views

### Long time existence of Ricci flow on compact surfaces of negative curvature

Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative ...

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vote

**0**answers

70 views

### Variational Properties of the Perelman Functional

After reading a bit more about Perelman's entropy and gradient solitons, I came up with a hunch, which I must test. Non-singular solitons can be regarded as critical points of Perelman's entropy, or ...

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vote

**0**answers

118 views

### On the definition on the Ricci flow [closed]

I'm trying to understand how one can define the Ricci flow equation.
First you have to parametrized the set of all Riemannian metrics.
Then you have to define the derivative on this parametrized ...

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vote

**1**answer

303 views

### Ricci flow on non-compact manifold

Suppose $\omega$ defines a Kähler metric on a non-compact complex manifold. Does
the Kähler-Ricci flow equation always have a solution (for small $t$)?

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147 views

### Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary

I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action
$$S(g_{\mu \nu})=\frac{1}{16\pi}\...

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votes

**1**answer

178 views

### Hamilton's Proof of the Tensor Maximum Principle

My questions come from Richard Hamilton's Three-Manifolds with Positive Ricci Curvature paper. I'm trying to work through parts of the paper so I can better understand the Ricci Flow for my research. ...

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**0**answers

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### 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!

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84 views

### Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics.
Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$,
$(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...

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### Derivation of an expression in the Ricci flow on surfaces

Recently I am studying Benett Chow and Dan Knopf's book titled
Ricci flow: An Introduction. In Chapter 5 (Ricci flow on surfaces), I am stuck in a straightforward deduction. Maybe it is very simple, ...