Questions tagged [ricci-flow]

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3 votes
1 answer
240 views

Ricci flow and curvature

I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not. So my question is if one starts with a metric that has mostly ...
5 votes
1 answer
414 views

Ricci flow negative curvature

We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$. I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
13 votes
0 answers
361 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. ...
28 votes
7 answers
10k views

Roadmap to learning about Ricci Flow?

Hello, I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
2 votes
1 answer
546 views

Ricci flow descending from an universal cover

Reading some of my old notes, I came across a remark, I don't understand. Summarized: Let $(M^n,g)$ be an open riemannian manifold (in that case it came with sectional curvature $K\ge0$, but I don't ...
7 votes
2 answers
526 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 ...
1 vote
0 answers
83 views

Obstruction for a manifold to admit a periodic Ricci flow

Let M be a (compact) smooth manifold. What kind of obstruction exist for M to admit a metric whose Ricci flow is a t-periodic flow?
4 votes
1 answer
158 views

One-sided version of the curve-shortening flow

The curve-shortening flow is $$ \frac{\partial C}{\partial t} = \kappa n $$ where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...
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....
14 votes
1 answer
386 views

Does the Cheeger constant satisfy a heat-type equation?

It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow. A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of ...
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 ...
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}\...
3 votes
1 answer
159 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
1 answer
196 views

The heat equation for complex time

Let $\Delta$ be a Laplacian or an elliptic operator over a manifold, can the heat equation be defined for complex time? Can we define: $$e^{-z \Delta}$$ for $Re(z)>0$ ? Also can the Ricci flow be ...
7 votes
2 answers
721 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 ...
3 votes
0 answers
120 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
117 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
0 answers
105 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, ...
3 votes
0 answers
113 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{...
4 votes
1 answer
625 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 ...
0 votes
1 answer
74 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(...
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 ...
3 votes
0 answers
103 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
56 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 ...
2 votes
0 answers
113 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,...
26 votes
3 answers
3k views

Does Ricci flow depend continuously on the initial metric?

Consider a version of Ricci flow for which short time existence and uniqueness are known, e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
15 votes
1 answer
973 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
157 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 ...
23 votes
2 answers
5k 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
267 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
174 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)} \|\...
13 votes
3 answers
2k 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 ...
2 votes
1 answer
355 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 ...
5 votes
1 answer
257 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 ...
11 votes
0 answers
480 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 ...
16 votes
4 answers
988 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
86 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 ...
5 votes
1 answer
228 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 ...
6 votes
2 answers
586 views

Rigorous solution to Ricci Flow on dumbbell $S^3$

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...
10 votes
3 answers
1k 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 ...
9 votes
1 answer
442 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-...
0 votes
0 answers
260 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 ...
1 vote
1 answer
284 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 ...
4 votes
0 answers
254 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 ...
14 votes
2 answers
5k views

What prerequisites do I need to read the book Ricci Flow and the Poincare Conjecture, published by CMI

As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?
2 votes
0 answers
518 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 ...
2 votes
0 answers
123 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 ...
6 votes
1 answer
469 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 ...
4 votes
0 answers
120 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
103 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 ...