Questions tagged [ricci-flow]

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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 ...
Emmie's user avatar
  • 31
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?
Ali Taghavi's user avatar
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 ...
André Henriques's user avatar
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 ...
Adrien B's user avatar
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 ...
Antoine Balan's user avatar
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 ...
Hollis Williams's user avatar
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 $...
Eduardo Longa's user avatar
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 ...
Eduardo Longa's user avatar
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 \\...
Matheus Andrade's user avatar
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, ...
Enhao Lan's user avatar
  • 165
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{...
Matheus Andrade's user avatar
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(...
James Chiu's user avatar
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 ...
James Chiu's user avatar
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 ...
Enhao Lan's user avatar
  • 165
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,...
James Chiu's user avatar
15 votes
1 answer
972 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 ...
Mirco A. Mannucci's user avatar
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 ...
James Chiu's user avatar
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 $...
Matheus Andrade's user avatar
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)} \|\...
Matheus Andrade's user avatar
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 ...
Gabe K's user avatar
  • 5,324
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 ...
Gabe K's user avatar
  • 5,324
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 ...
Naruto's user avatar
  • 83
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 ...
Gabe K's user avatar
  • 5,324
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....
Joseph O'Rourke's user avatar
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 ...
Matheus Andrade's user avatar
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 ...
Hollis Williams's user avatar
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 ...
Hollis Williams's user avatar
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 ...
Matheus Andrade's user avatar
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 ...
Matheus Andrade's user avatar
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-...
u184's user avatar
  • 277
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 ...
Kirby's user avatar
  • 113
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 ...
srp's user avatar
  • 121
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 ...
Hollis Williams's user avatar
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 ...
user avatar
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 ...
L.F. Cavenaghi's user avatar
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 ...
Jae Ho Cho's user avatar
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 ...
user162551's user avatar
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 ...
Jae Ho Cho's user avatar
0 votes
0 answers
247 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 ...
Hollis Williams's user avatar
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 ...
Hollis Williams's user avatar
2 votes
1 answer
236 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(...
Hollis Williams's user avatar
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-...
George's user avatar
  • 435
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)...
user162551's user avatar
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 ...
user162551's user avatar
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 ...
Anton Petrunin's user avatar
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{...
Matthias Ludewig's user avatar
2 votes
0 answers
108 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 $\Delta_{\lambda}$...
Richard Muniz's user avatar
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 ...
L.F. Cavenaghi's user avatar
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 ...
L.F. Cavenaghi's user avatar