Questions tagged [ricci-flow]

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Metrics $g_1\leqslant g_2$ implies the Ricci flow $g_1(t)\leqslant g_2(t)$?

Let M be a complete,n dimensional Riemannian manifold without boundary. Suppose $g_1,g_2$ are two metrics on M and $g_1\leqslant g_2$. Suppose that there exists $T>0$ such that for $i=1,2$, the ...
mathmetricgeometry's user avatar
1 vote
0 answers
121 views

Reference for example of gradient steady Ricci solitons

Recently I read a paper about Ricci solitons. I quote a paragraph of it here: In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are ...
C.F.G's user avatar
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6 votes
0 answers
158 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
1 vote
1 answer
219 views

Question on $\alpha-$Einstein manifolds

A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such $$\rho=ag+b\alpha\otimes\alpha$$ where $a,b$ are smooth functions on $M$ and $\rho$ is ricci ...
C.F.G's user avatar
  • 4,165
13 votes
0 answers
362 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
7 votes
1 answer
458 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
5 votes
1 answer
316 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
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5 votes
0 answers
303 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
4 votes
1 answer
337 views

Some questions on a paper of Wilking

I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities" (DOI: 10.1515/crelle.2012.018, arXiv:1011....
user94178's user avatar
-2 votes
1 answer
164 views

Ricci flow and evolution of the shape of drops in spray

Several years ago, I was a trainee in a physics lab where I was supposed to study atomisation in sprays (ensemble of liquid drops). As we did observe that the drops tended to adopt a spherical shape ...
Sylvain JULIEN's user avatar
1 vote
1 answer
422 views

Geometric meaning of Ricci flow [duplicate]

What is the geometric meaning, for a metric in function of the time that is a solution of the Ricci flow ($g'(t)=-2Ric(t)$), compared to one that is not? EXPLANATION I'm interested to understand, ...
MathDG's user avatar
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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
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0 votes
0 answers
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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 ...
Farmer's user avatar
  • 109
2 votes
1 answer
264 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 ...
debabrata chakraborty'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
7 votes
2 answers
685 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
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
4 votes
0 answers
150 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 ...
user81712's user avatar
3 votes
0 answers
272 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 ...
debabrata chakraborty's user avatar
3 votes
1 answer
275 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 ...
student's user avatar
  • 39
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
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3 votes
0 answers
150 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 ...
Uche Opara's user avatar
8 votes
1 answer
872 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
1 vote
0 answers
154 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 ...
dede's user avatar
  • 19
2 votes
0 answers
162 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 ...
Uche Opara's user avatar
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
1 vote
1 answer
470 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 $t$)?
Mohawk River's user avatar
4 votes
1 answer
386 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
2 votes
2 answers
676 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 ...
Paul's user avatar
  • 501
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
1 vote
1 answer
489 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. ...
Brian Klatt's user avatar
1 vote
0 answers
467 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 ...
Dan Li's user avatar
  • 11
5 votes
1 answer
387 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
2 votes
1 answer
369 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 $...
Joseph O'Rourke's user avatar
5 votes
1 answer
872 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
0 votes
2 answers
357 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 $TM$...
Sepideh Bakhoda's user avatar
1 vote
0 answers
251 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}\...
Sepideh Bakhoda's user avatar
7 votes
1 answer
921 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
1 vote
1 answer
615 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 ...
user35946's user avatar
  • 355
3 votes
2 answers
1k 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 ...
Naruto Uzumaki's user avatar
1 vote
1 answer
297 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 ...
Sepideh Bakhoda's user avatar
11 votes
1 answer
765 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
3 votes
1 answer
313 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.
Sepideh Bakhoda's user avatar
2 votes
1 answer
326 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 ...
Sepideh Bakhoda'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
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
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
2 votes
1 answer
910 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)...
Sepideh Bakhoda's user avatar
3 votes
1 answer
638 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 ...
Sepideh Bakhoda's user avatar