Questions tagged [ricci-flow]
The ricci-flow tag has no usage guidance.
114
questions
4
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1
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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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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....
-2
votes
1
answer
164
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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 ...
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, ...
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 ...
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!
0
votes
0
answers
137
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 ...
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 ...
13
votes
1
answer
1k
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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
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 ...
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\...
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 ...
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 ...
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 ...
5
votes
0
answers
1k
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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.
3
votes
0
answers
150
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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 ...
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 ...
1
vote
0
answers
154
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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 ...
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 ...
5
votes
2
answers
1k
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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 ...
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 $t$)?
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 ...
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 ...
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. ...
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. ...
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 ...
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
$$\langle\mathrm{Rm}\,\...
2
votes
1
answer
369
views
Optimal, conformal diffeomorphisms between two surfaces in 3D
Let $S_1$ and $S_2$ be two smooth, closed surfaces embedded in $\mathbb{R}^3$.
Q. Is there a natural definition of the optimal, conformal diffeomorphism between $S_1$ and $S_2$?
I am imagining $...
5
votes
1
answer
872
views
Yang-Mills flow, Ricci flow and the holonomy
Is the holonomy group (based at some point) preserved along the Yang-Mills flow/ Ricci flow?
(1) For Yang-Mills case, we know that the centralizer of the holonomy $H_x$ is the isotropy group of the ...
0
votes
2
answers
357
views
On the definition of convergence of a sequence of sections of a bundle
Convergence of a sequence of sections of a bundle is defined as follows:
Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on $TM$...
1
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0
answers
251
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Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary
I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action
$$S(g_{\mu \nu})=\frac{1}{16\pi}\...
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 ...
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 ...
3
votes
2
answers
1k
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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
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 ...
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 ...
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.
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 ...
7
votes
0
answers
982
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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 ...
9
votes
3
answers
4k
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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 ...
6
votes
6
answers
2k
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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
2
votes
1
answer
910
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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)...
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 ...