Questions tagged [ricci-flow]

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Does the Volume Ratio of a Geodesic Ball for a Complete Riemannian Manifold tend to the volume of a Unit Ball in Euclidean $n$-space?

I am reading Peter Topping's notes on Ricci flow: on page 99 a statement is made which is needed for his proof of a version of Perelman's no local volume collapse theorem, but I am not sure why it ...
2 votes
1 answer
223 views

Principal Symbol for the Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 ...
2 votes
1 answer
237 views

What is the Weak Maximum Principle for Scalars and how is it Derived?

I am currently reading 'Lectures on Ricci Flow' by Peter Topping and I have got to Chapter 3 where he states the 'weak maximum principle for scalars'. Suppose for $t \in [0,T]$ for finite $T$ that $g(...
4 votes
1 answer
331 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-...
1 vote
0 answers
88 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)...
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
6 votes
0 answers
221 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 ...
3 votes
0 answers
134 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 ...
2 votes
0 answers
109 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 $\Delta_{\lambda}$...
24 votes
3 answers
3k 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 ...
5 votes
0 answers
181 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 vote
0 answers
248 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 ...
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 ...
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 ...
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!
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 ...
-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 ...
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 ...
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 ...
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 ...
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 $t$)?
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 ...
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\...
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 ...
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 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.
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 ...
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 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
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 ...
9 votes
1 answer
1k 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 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. ...
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 ...
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. ...
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}\,\...
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 ...
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.
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 ...
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 ...
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 ...
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 $...