Questions tagged [ricci-flow]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
89 views

A problem in Hamilton's ODE-PDE comparison principle

Recent days, I read the ODE-PDE comparison principle of Ricci flow which is origin from the Hamilton's paper Four-manifolds with positive curvature operator. The pictures below are from this paper ...
  • 215
3 votes
0 answers
100 views

Is the normalized Ricci flow real analytic in the time variable?

Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $...
5 votes
0 answers
97 views

Metric under Ricci flow on a 2-sphere can be realized by embedding

I am sorry if this is a silly question, but I am new to Ricci flows. Let $\Sigma \subset \mathbb{R}^3$ be a smoothly embedded sphere, and denote its metric (induced by $\mathbb{R}^3$) by $g$. Suppose ...
2 votes
1 answer
113 views

Curvature estimate in Hamilton's Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold

In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
2 votes
0 answers
61 views

Using Rauch comparison theorem to get a comparison of two metric

Picture below is from Topping's Lectures on the Ricci flow. I've been stuck by the red line about two months. In fact, I asked it on ME two months ago. To describe the problem more precisely, ...
  • 215
3 votes
0 answers
98 views

Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
0 votes
1 answer
57 views

Changing the system of PDE by diffeomorphism and differentiate a composition

This problem comes from the book Hamilton's Ricci flow. Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...
3 votes
0 answers
90 views

Changing the system of PDE by diffeomorphism

This problem comes from the book Hamilton's Ricci flow. Given a smooth functional $f$, and following system. $$\partial_tg_{ij}=-2(R_{ij}+\nabla_i\nabla_jf)$$ If there exist a 1 parameter family of ...
4 votes
0 answers
45 views

Low boundary of $\mathcal W$ function

Picture below is from Topping's Lectures on Ricci flow. I don't understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But ...
  • 215
2 votes
0 answers
85 views

The Ricci curvature is bounded below by scalar curvature

So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
15 votes
1 answer
724 views

Ricci curvature : beyond heat-like flows

Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs. There are at least two versions of Ricci curvature in the ...
3 votes
0 answers
149 views

How to show the upperbound of the Ricci tensor preserved on 3 manifold

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
20 votes
2 answers
4k views

Shing-Tung Yau's doubts about Perelman's proof

[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.] According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of ...
0 votes
1 answer
192 views

Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow

In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
4 votes
0 answers
139 views

Classifying singularities of the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
11 votes
3 answers
1k views

Quote by Thurston on the Ricci flow

I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't ...
  • 4,381
6 votes
2 answers
404 views

Exponential convergence of Ricci flow

I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
  • 4,381
5 votes
1 answer
224 views

Gradient of solution to heat equation under evolving metric

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
  • 51
11 votes
0 answers
334 views

Is there any connection between the Deturck trick and the Uhlenbeck trick?

There are two separate places where ingenious uses of gauge transformations simplify the analysis of Ricci flow considerably.  The Deturck trick is a way to break the diffeomorphism invariance of the ...
  • 4,381
16 votes
4 answers
905 views

Squaring a square and discrete Ricci flow

Is this a theorem? Every $3$-connected planar graph $G$ may be represented as a tiling of a square by squares, one square per node of $G$, with nodes connected in $G$ corresponding to tangent squares....
3 votes
0 answers
65 views

Proving some identities about the time derivative of the k-th covariant derivatives of scalar curvature under normalized Ricci flow on surfaces

I'm trying to prove the following identities (under the normalized Ricci flow on surfaces, on which $\partial_t g = (r-R)g$ holds true, where $r$ denotes the average scalar curvature and has the same ...
4 votes
1 answer
197 views

Neckpinch singularity of Ricci flow

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the ...
13 votes
2 answers
1k views

Is there a solution of the Yamabe problem using Ricci flow?

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
10 votes
3 answers
697 views

Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics ...
0 votes
0 answers
185 views

What exactly does it mean for Hamilton's cigar soliton to have linear volume growth?

In a couple articles I've read lately, I've seen it mentioned that the cigar soliton has linear volume growth. What does this mean? I thought maybe, if you compute the volume of geodesic balls and ...
9 votes
1 answer
388 views

Ricci flow preserves almost Kahler condition?

I have been unable to find a reference to the following (perhaps too naive) question. Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-...
  • 277
1 vote
1 answer
211 views

Geometric flow equations which are second order in time derivative

All examples about geometric flow equations given in Wikipedia's Geometric flow article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order ...
  • 113
2 votes
1 answer
336 views

Type II singularities for 3D Ricci flow

I know that type II singularities of the Ricci flow can exist on closed 3-manifolds (e.g. on $S^3$), but on the other hand it seems to me that ODE comparison combined with Hamilton's tensor maximum ...
  • 121
4 votes
0 answers
164 views

Is there a version of Ricci Flow for Pseudo-Riemannian Metrics?

The Ricci flow deforms a Riemannian metric. I was wondering if there was something very similar which deforms a pseudo-Riemannian metric or if not, is there reason why such a geometric flow cannot ...
2 votes
0 answers
457 views

Proof Of The Poincare Conjecture: An Unofficial Erratum [closed]

We read and checked the detailed proof of the Poincare conjecture. One can find the article (Ricci Flow And The Poincare Conjecture by Morgan and Tian) on arXiv. Since the proof contains some gaps and ...
's user avatar
2 votes
0 answers
114 views

Ricci flow on Riemannian submersions

Let $(P,g) \to (S^2,h)$ be a Riemannian submersion. Let $g(t)$ be the Ricci flow on $P$ with initial condition $g$. Does the induced flow on $S^2$ converges to the round metric on $S^2?$ I could ...
4 votes
0 answers
110 views

Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
1 vote
0 answers
90 views

Ricci flow preserves locally symmetry along the flow

Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
6 votes
1 answer
358 views

Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
0 votes
0 answers
171 views

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
189 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
195 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(...
3 votes
1 answer
317 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-...
  • 425
1 vote
0 answers
72 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)...
3 votes
0 answers
125 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 votes
0 answers
204 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
0 answers
170 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{...
2 votes
0 answers
91 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 $\Delta_{\lambda}$...
7 votes
2 answers
569 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
0 answers
230 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 ...
4 votes
1 answer
608 views

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
112 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 ...
  • 4,065
5 votes
0 answers
140 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 ...
  • 173
1 vote
1 answer
208 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 ...
  • 4,065
3 votes
0 answers
228 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. ...