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Questions tagged [ricci-flow]

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2
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1answer
246 views

Understanding the Hamilton's definition of $\ast$-operation

I'm studying by myself Mean Curvature Flow and I'm trying understand the definition of $\ast$-operation given by Richard Hamilton in the beginning of the section $13$ (page $40$) of his paper "Three-...
0
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0answers
79 views

On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define $$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$ where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
1
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0answers
41 views

Eigenvalues of geometric operators along geometric flows

I have two questions: 1- what is the relation between eigenvalues of geometric operators such as Laplace operator and topology or geometry of a Riemannian manifold?(please give an example if possible)...
0
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0answers
212 views

The Ricci flow and integrable systems

It is well-know that the Ricci flow is defined by the following equation: $$ \frac{\partial g}{\partial t}=-2R(g)$$ where $g$ is the metric and $R(g)$ is the Ricci curvature. Let $g_0$ be a metric and ...
3
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0answers
87 views

Isometries along the normalized Ricci flow

As we know the Ricci flow preserves isometries of the initial manifold along the flow. But I want to know does the normalized Ricci flow preserves isometries of the initial manifold along the flow as ...
6
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0answers
144 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 ...
5
votes
0answers
140 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{...
1
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0answers
66 views

Differentiable dependence on the data for parabolic equations

Let $g_{\lambda}$ be a one parameter family of Riemannian metrics, which are complete and with bounded curvature, on the unit disk, depending smoothly on the parameter $\lambda$. Let $\Delta_{\lambda}$...
6
votes
1answer
297 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 ...
1
vote
0answers
136 views

Classical solutions for parabolic PDE's

I am studying Ricci flow theory and the Ricci flow is not a parabolic equation. But there is some variant of it, called DeTurck-Ricci equation, that happens to be a parabolic PDE. So to argue ...
1
vote
0answers
73 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 ...
4
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0answers
106 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
1answer
169 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 ...
1
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0answers
126 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. ...
5
votes
1answer
258 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 ...
3
votes
1answer
203 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
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0answers
229 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 ...
3
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0answers
167 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". If I may, I wish to clarify a few points where I ...
-2
votes
1answer
146 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 ...
2
votes
1answer
353 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
1answer
177 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 ...
0
votes
0answers
171 views

Technical argument for Ricci flow uniqueness

I'm trying to prove carefully Ricci flow uniqueness with the book Lectures on the Ricci flow by Peter Topping at http://homepages.warwick.ac.uk/~maseq/topping_RF_mar06.pdf . I have trouble with ...
4
votes
1answer
545 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
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0answers
110 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
1answer
235 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 ...
7
votes
1answer
650 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 ...
5
votes
2answers
485 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 ...
11
votes
1answer
321 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\...
3
votes
0answers
130 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
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0answers
208 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 ...
1
vote
1answer
202 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 ...
6
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0answers
643 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.
1
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0answers
116 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 ...
6
votes
1answer
572 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
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0answers
137 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 ...
2
votes
0answers
132 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 ...
4
votes
2answers
844 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 ...
1
vote
1answer
422 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 $t$)?
3
votes
1answer
305 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
2answers
513 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
2answers
345 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
1answer
283 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
0answers
391 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
1answer
338 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
1answer
271 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
1answer
743 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
votes
2answers
241 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 $TM$...
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0answers
182 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}\...
5
votes
1answer
702 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
510 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 ...