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0answers
78 views

What are the most important papers to read about Ricci flow? [duplicate]

I would like to know, which papers are important to read, if one wants to work about Ricci flow. What papers have to be known by every person conducting research in this field? Can you provide me with ...
3
votes
2answers
145 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. ...
0
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1answer
125 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. ...
0
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0answers
130 views

RG flow and Ricci flow

It looks like the Laplace operator in the nonlinear sigma model (say the Polyakov action) is different from the Laplace-Beltrami operator, how can one get the Ricci flow as a low order approximation ...
5
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1answer
267 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 ...
2
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1answer
188 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 ...
6
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1answer
511 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 ...
-1
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2answers
168 views

On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows: Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...
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0answers
104 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 ...
3
votes
1answer
331 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 ...
1
vote
1answer
297 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 ...
3
votes
2answers
272 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 ...
1
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1answer
198 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 ...
8
votes
1answer
381 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 ...
3
votes
1answer
233 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.
2
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1answer
231 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 ...
3
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0answers
522 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 ...
3
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3answers
765 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 ...
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5answers
610 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
1
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1answer
378 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 ...
3
votes
1answer
228 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 ...
3
votes
2answers
491 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 ...
0
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0answers
134 views

Isoperimetric profile

In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile is defined by ...
0
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0answers
185 views

A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...
3
votes
1answer
251 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, ...
5
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1answer
393 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... ...
4
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1answer
299 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 ...
1
vote
1answer
349 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 ...
2
votes
1answer
272 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 ...
6
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2answers
418 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 ...
1
vote
2answers
477 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 ...
7
votes
2answers
1k 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?
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3answers
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 ...
20
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2answers
1k 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 ...
8
votes
3answers
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 ...
10
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7answers
3k 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 ...