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Questions tagged [ricci-flow]

The Ricci flow on a Riemannian manifold $(M,g)$ is determined by the geometric evolution equation $\partial_t g_{ij} = -2R_{ij}$ where $R_{ij}$ is the Ricci curvature. The Ricci flow is the main ingredient in Perelman's proof of the Poincaré conjecture.

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Positive scalar curvature Einstein manifolds are noncollapsed

I am currently working through some of the exercises in Ricci Solitons in Low Dimensions by Bennett Chow, and I've been stuck on Exercise 1.23. It asks you to prove that positive scalar curvature ...
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Smoothness of Ricci flow solution on a closed interval

In the paper "Deforming the metric on complete Riemannian manifolds" by Wan-Xiong Shi, the author proves the following theorem which I copy verbatim below: Theorem 1.1. Let $(M, g_{ij}(x)$ ...
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A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
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Definition of Ricci flow

My undergraduate thesis is related to the Ricci flow, and I have a number of basic questions. Let $M$ be a smooth manifold. At the start of Chapter 2.3 of Peter Topping's Lectures on the Ricci flow, ...
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Unbounded diameter for the blow up on Ricci Flow

Let $(M,g_0)$ be a compact n-dimensional Riemannian Manifold. Suppose that $g(t)$ is an smooth solution to the Ricci flow $\partial_tg(t)=-2Rc(g_t)$ on a maximal time interval $[0,T)$ of existence, ...
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How to show $\partial_t h =\Delta h -2|M_{ij}|^2 +rh$ in Hamilton's The Ricci flow on surfaces

Pictures below is from Hamilton's The Ricci flow on surfaces ( 237- 262 of this), I want to calculate the red line. First, we consider the 2-dimensional normalized Ricci flow $$ \partial_t g_{ij}=(...
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Good book about PDE on manifold

I have already learn Riemannian Geometry, and know a littile PDE. I want to know which book is suitable for me to learn the PDE on manifold ? For example, I saw that on closed manifold, $\Delta u= f$...
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3.3 Theorem of Hamilton's The Ricci flow on surfaces

First picture below is from Hamilton's The Ricci flow on surfaces ( 237- 262 of this). Near the red line, why $R_{\max}$ can be replaced by $-\varepsilon$ ? In my view, only at start we know $R\le ...
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$L^2$-estimate of $\nabla R$ in $\partial_t R= \Delta R +R^2 -rR$

Consider 2-sphere $(S^2, g)$, and evolute its metric by $$ \partial_t g_{ij} = (r-R) g_{ij} $$ where $r$ is the average of $R$, and $R$ is scalar curvature. Then, there is $$ \partial_t R= \Delta R + ...
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Why does bounded curvature mean that ricci flow has solution?

Picture below is from Hamilton's The Ricci flow on surfaces. In this paper, the author consider the equation $$ \partial_t g_{ij}=(r-R)g_{ij} \tag{1} $$ where $r$ is the average of $R$. From the 4.7 ...
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Problem calculating simmetrized product of Ricci tensor

I was reading the following article, about an example of neckpinching for Ricci flow on $S^{n+1}$, with $n\geq 2$, but I have a problem in the proof of Proposition 5.7. The setting is the following. ...
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Show that $\mathcal L_X g_{ij} = \nabla_i X_j + \nabla_j X_i$

I’m trying to understand equation (1.8) on p. 4 of Chow et al.’s “The Ricci flow: techniques and applications”. There, the authors say that, using indices, the equation $$ -2\operatorname{Rc}(g) = \...
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Is the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed?

I am puzzed that whether the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed, as my understanding of the definition non-collapsed and computation, I believe it is ...
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Ricci flow on a manifold: How does the manifold itself change?

I'm new to this subject, and I'm a bit confused about how the metric changing corresponds to the manifold itself changing. For example, in Topping's note it says: A simple example of a Ricci flow is ...
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How to show that geodesic curvature of isoperimetric set is constant

A few months ago, I read Hamilton's An isoperimetric estimate for the Ricci flow on the two-sphere, which is collected in Collected papers on Ricci flow. For two-sphere, the isoperimetric ratio can be ...
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Why the higher order covariant derivatives of curvature is uniformly bounded?

I'm reading S.Brendle's Ricci Flow and the Sphere Theorem. In page36 and page43, he says using Cor3.3, we can obtain some higher order covariant curvatures are uniformly bounded in $[0,T)$. But Cor3.3 ...
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A tensor is invariant under diffeomorphism.

Consider $(M,g)$ is a Riemannian Manifold (compact). For a smooth function $f(x,t): M \times \mathbb{R} \to \mathbb{R}$, It is known that we can induce a one parameter group of diffeomorphism such ...
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3 votes
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Ricci Flow: The existence of potential of Curvature

For a compact Riemannian Manifold $(M,g)$ without boundary. $R$ as the scalar curvature. And $d\mu$ is the Riemannian volume form. So we can define the average of the scalar curvature $r:= \frac{\...
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Given a homeomorphism between a euclidean simplicial complex and a round 2-sphere, can one construct a CANONICAL homeomorphism?

The side-lengths of the simplicial complex are given. Alternatively, only the angles of the triangles are given, and the triangulated 2-sphere is known up to change of scale. By a canonical ...
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Heat equation of the distance function and Ricci flow

I would like to get more acquainted to with Ricci flow and evolving metrics. Suppose we are given a familly $\{M, g(t)\}_t$ of Riemannian manifolds with time-dependent metrics. We would like to ...
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Hyperbolization theorem for closed, reducible manifolds

Ashenbrenner et al. state the Hyperbolisation theorem as follows Let $N$ be a compact, orientable, irreducible $3$-manifold with empty or toroidal boundary. If $N$ is atoroidal and $π_1(N)$ is ...
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Time derivative of squared distance function under evolving metric

Suppose we have an evolving familly of Riemannian manifolds $\{M, g(t)\}_t$ indexed by time $t$. Typically a flow. We fix a point $x$ in $M$ and consider the following function : $$ f_x(t;y) = d_{g(t)}...
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Help Understanding Perelman's $\mathcal{L}$-geodesic Equation

I'm reading a paper on Perelman's solution to the Poincare conjecture. The paper derives a geodesic equation by minimizing the following $\mathcal{L}$-length of each curve $\gamma(\tau)$, $0<\tau_1 ...
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Reference request: Lorentzian Ricci flow

I have been studying some aspects of Ricci flow, namely existence, uniqueness, finite time extinction, the preservation of curvature bounds via the maximum principle, and the modifications of Ricci ...
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Has it been proven that the Cheeger constant is attainable on surface?

Recently, I have been studying the monotonicity of the Cheeger constant under Ricci flow on surfaces. In fact, I want to use the monotonicity to prove the convergence of Ricci flow on $S^2$, which ...
Enhao Lan's user avatar
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Why Ricci Flow Always Defines a Riemannian Metric As Long as it Exists

From what I understand, the Ricci flow \begin{equation} \frac{\partial g}{\partial t} = -2Ric(g) \end{equation} always defines a Riemannian metric as long as it exists. I know that the Riemannian ...
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Rosenau solutions or Rosenau Metrics in Ricci flow

In the book “Hamilton Ricci flow" written by Bennet Chow, Peng Lu and Lei Ni. The Rosenau solution is $$g = u \cdot h = \frac{\sinh(-t)}{cosh(x) + cosh(t)} (dx^2 + d \theta^2), x \in \mathbb{R}, \...
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Ricci Curvature under uniform scaling

While reading about the Ricci flow, I've ran into the following statement: It is worth pointing out here that the Ricci tensor is invariant under uniform scaling of the metric. This quote was ...
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Exercise on higher order Bernstein type estimates

I was reading Ricci flow: Techniques and Applications (Part - II: Analytic Aspects) by B. Chow et al. In Chapter $14$ section $6$ there is an exercise stating: Exercise $14.24$: Prove higher ...
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To show Perelman $\mathcal{F}$ Energy in Ricci flow is invariant under diffeomorphisms

The following Perelman $\mathcal{F}$ Energy $$\mathcal{F}(g,f) = \int_{M^{n}} (R + \mid \nabla f \mid^{2}) e^{-f} d\mu$$ is is invariant under diffeomorphisms. How to prove it? Or is there any ...
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Explicit expression for $R^\#$ in Hamilton's paper

In Hamilton's $1986$ paper "Four-manifolds with positive curvature operator", the setting is as follows: We identify the two-forms $\Lambda^2$ with the Lie algebra $\mathfrak{so}(n)$. Let $\{...
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How to show $\frac{dL}{dr}=\int k ds$ (in An isoperimetric estimate for the Ricci Flow on the Two-Sphere)?

I am reading Hamilton's An isoperimetric estimate for the Ricci Flow on the Two-Sphere. It is the 9th paper of Collected papers on Ricci flow. For explain my problem, I use the three parts of this ...
Enhao Lan's user avatar
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On the $S^n$, whether smooth Riemannian metric $g$ will become constant (sectional) curvature metric under Ricci flow?

Recently, I read the Hamilton's An isoperimetric estimate for the Ricci flow on the two sphere. He state that Chow proved that Under the Ricci flow on 2-sphere, any metric approaches constant ...
Enhao Lan's user avatar
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3 votes
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Computing derivative of pullback of time-dependent metric

$\qquad$ Following Topping's book on Ricci flow, $X(t)$ be a time-dependent collection of vector fields with associated collection of diffeomorphisms $\psi_t$ defined on a compact, closed manifold $M$....
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What is the mean of barrier sense or viscosity sense?

Pictures below are from Entropy and heat kernel bounds on a Ricci flow background. I don't know what is the mean of barrier sense or viscosity sense. At beginning, I guess they should be the concepts ...
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Maximum principle and Lipschitz continuous

Let $(M,g(t))$ is a Ricci flow. And $$ \Box = \partial_t -\Delta_{g(t)} $$ is heat operator, which coupled to Ricci flow. If $$ \Box u =0 ~~~~~~\text{and }~~~~~~ \Box |\nabla u|\le 0 $$ Then how to ...
Enhao Lan's user avatar
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Parabolic rescaling and time-shift in Theorem 6.2 of Entropy and heat kernel bounds on a Ricci flow background

Picture below are from Entropy and heat kernel bounds on a Ricci flow background. I don't know how to use parabolic rescaling and time-shift to assume $r=1,t=1$. $M$ is a manifold, and $g_t$ is Ricci ...
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Jensen's inequality applied in proof of Theorem 6.2 of Entropy and heat kernel bounds on a Ricci flow background

Picture below is from Entropy and heat kernel bounds on a Ricci flow background. I want to get the red line. For me, the Jensen's inequality is $$ \varphi\left(\int_\Omega f d\mu\right) \le \int_\...
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Notation in pointed Nash Entropy (or conjugate heat kernel measure)

Fisrt and second pictures below is from Entropy and heat kernel bounds on a Ricci flow background, I don't know the mean of $f_{t_0-\tau}$. Acoording to second picture, I guess that $f_{t_0-\tau}$ ...
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What is the mean of the coupled of heat operator coupled with Ricci flow

I have some problem when I read Entropy and heat kernel bounds on a Ricci flow background. In the page 6 (I don't know why I can't upload picture), I don't know what is the mean of the coupled of ...
Enhao Lan's user avatar
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2 votes
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Normalize of steady gradient Ricci soliton

As everyone knows, the steady gradient Ricci soliton is a triple $(M,g,f)$ such that $$ Ric+\nabla^2 f =0 $$ Then, how to normalize it to $$ ~~~~~~~~~~~~~~~~~~~Ric=\nabla^2 f ~~~~~~~~~~~~~~~~~~~~~~~~~...
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Calculation about the process of Ricci flow induced by soliton

In page 2 of An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons, the authors say the blow result. Assum $(M,g)$ is a Riemannian manifold, $f:M\rightarrow \mathbb R$ ...
Enhao Lan's user avatar
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Analogue of one-parameter transformation for time dependent vector field

In the study of solitons of Ricci flow, we need to consider a time dependent vector field $X(t)$ and the generating diffeomorphisms $\psi: \mathbb R\times M \rightarrow M$ (for detail to read the 1.2 ...
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The negative gradient flow of Einstein-Hilbert functional in a fixed conformal class is Yamabe flow

In the book "Hamilton's Ricci flow" written by Bonnett Chow, Peng Lu and Lei Ni, there is an exercise pullzed me: Show that $n \ge 3$, the negative gradient flow of $$E(g)=\int_{\mathcal{M}} ...
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The basic commutation in evolution of Ricci flow

Many references compute the evolution equation in Ricci flow like this, for example compute the evolution of Christoffel symbols $$\partial_{t} \Gamma^{k}_{ij} = {} \frac{1}{2}\partial_{t}(g^{kl}(\...
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How to show the eigenvectors is preserved under a ODE?

Assume $V$ is a 3-dimsional vector space. And $$ \varphi(t):V\rightarrow V $$ is nondegenerate linear map, and smooth respect to $t$. The eigenvalues of $\varphi(t)$ are $$ \lambda_1(t)\le \...
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Evolution of the energy density under the harmonic map heat flow (The Ricci Flow in Riemannian Geometry)

I am reading now the book The Ricci Flow in Riemannian Geometry by Ben Andrews and Christopher Hopper. It has been a while since I did not use pullback bundles and other objects, and they still look ...
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The support function in ODE-PDE comparison principle of Ricci flow

Recent days, I want to understand the ODE-PDE comparison principle of Ricci flow which is origin from the Hamilton's paper Four-manifolds with positive curvature operator. But the Hamilton's paper ...
Enhao Lan's user avatar
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Any differential geometry with metric structure book recommendation

I am currently reading the book 'Application of Ricci flow' by Bennett Chow which mentions the Cheeger-Gromov Convergence and local collapsing stuff by Perelman. I would like to have a better ...
James Chiu's user avatar
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158 views

A 3-dimensional symmetric space implies the eigenvalues of Ricci tensor?

When i try to read Hamilton’s famous thesis “Three-manifolds with positive Ricci curvature” on section 10 page 283, it says this expression will vanish for any symmetric metric: $$(\lambda^3+\mu^3+v^3)...
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