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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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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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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irreducible compoenets in Hamiton's first Ricci flow paper [closed]
In "Three-manifolds with positive Ricci curvature",page 288,11.6 lemma.
I dont konw why $\left\lvert E_{ijk} \right\rvert ^2=\frac{7}{20}\left\lvert \partial_i R\right\rvert ^2$ ,I have used ...
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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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Blowing up of Ricci flow on surfaces, the low bound of scalar curvature means the nonnegative of scalar curvature of limit
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.
In this paper, Hamilton state that "Since the ...
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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 ...
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Isoperimetric ratio and the boundary of curvature under curve shortening (shrinking) flow
I got stuck by the 3.2 Lemma of An isoperimetric estimate for the Ricci flow on the two-sphere, which is collected in Collected papers on Ricci flow.
As picture below, $L$ is the length of $\Lambda^t$....
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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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Bochner's theorem for positive Ricci curvature
I know by Bochner's theorem one has : If $(M,g)$ is a compact , oriented Riemannian manifold and $\text{Ric}_g<0$ , then $\text{Ric}_g(X,X)=0$ iff $X=0$ for any killing vector field $X$ .
Now I ...
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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 ...
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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 ...
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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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Theorem 3.14 of Entropy and heat kernel bounds on a Ricci flow background
Picture below is from Bamler's Entropy and heat kernel bounds on a Ricci flow background. I don't know how to get the red line. The author state it is direct consequence.
In the red line, $H_n$ is ...
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1
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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 ...
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Two calculation in 7th page of Entropy and heat kernel bounds on a Ricci flow background
Picture below is from 7th page of Entropy and heat kernel bounds on a Ricci flow background.
$\Box =\partial_t -\Delta_{g(t)}$ is the heat operator coupled to Ricci flow.
About the red line 1,
by ...
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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
$$
\varphi\left(\int_\Omega f d\mu\right) \le \int_\...
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Notation in pointed Nash Entropy (or conjugate heat kernel measure)
Fisrt and second pictures below is from Entropy and heat kernel bounds on a Ricci flow background, I don't know the mean of $f_{t_0-\tau}$. Acoording to second picture, I guess that $f_{t_0-\tau}$ ...
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What is the mean of the coupled of heat operator coupled with Ricci flow
I have some problem when I read Entropy and heat kernel bounds on a Ricci flow background.
In the page 6 (I don't know why I can't upload picture), I don't know what is the mean of the coupled of ...
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Normalize of steady gradient Ricci soliton
As everyone knows, the steady gradient Ricci soliton is a triple $(M,g,f)$ such that
$$
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, $f:M\rightarrow \mathbb R$ ...
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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
$$
\lambda_1(t)\le \...
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Evolution of the energy density under the harmonic map heat flow (The Ricci Flow in Riemannian Geometry)
I am reading now the book The Ricci Flow in
Riemannian Geometry by Ben Andrews and Christopher Hopper. It has been a while since I did not use pullback bundles and other objects, and they still look ...
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The support function in ODE-PDE comparison principle of Ricci flow
Recent days, I want to understand the ODE-PDE comparison principle of Ricci flow which is origin from the Hamilton's paper Four-manifolds with positive curvature operator. But the Hamilton's paper ...
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Any differential geometry with metric structure book recommendation
I am currently reading the book 'Application of Ricci flow' by Bennett Chow which mentions the Cheeger-Gromov Convergence and local collapsing stuff by Perelman. I would like to have a better ...
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A 3-dimensional symmetric space implies the eigenvalues of Ricci tensor?
When i try to read Hamilton’s famous thesis “Three-manifolds with positive Ricci curvature” on section 10 page 283, it says this expression will vanish for any symmetric metric:
$$(\lambda^3+\mu^3+v^3)...
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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 $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
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Convex function on Riemannian manifold
When I read the 9.5 of Topping's Lectures on the Ricci flow, I have some problem.
Assume $W$ is a vector bundle over manifold $(M,g)$, and $A$ is connection on $W$. $\{e_1,...,e_l\}$ is a frame of $...
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How to show $A^2e_i=-\frac{1}{2} R_A(\cdot, \cdot)e_i$ in Topping's Lectures on the Ricci flow
When I read the 9.5 of Topping's Lectures on the Ricci flow, I have some problem.
Assume $W$ is a vector bundle over manifold $(M,g)$, and $A$ is connection on $W$. $\{e_1,...,e_l\}$ is a frame of $...
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answers
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A basic question about calculation under geodesic frame
I have been recently reading some books related to Ricci flow, and I am confused when it comes to the evolution of Levi-Citita connection. That is: If $(M^n,g)$ Is a closed Riemannian manifold, and we ...
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Using Rauch comparison theorem to get an estimation of two metric.
Picture bottom is from the section 8.4 of Topping's Lectures on the Ricci flow. I use the following Rauch theorem
From the $|Rm|\le 1$, I have sectional curavture $ K \le 1$. Therefore, I have
$$
|Y|...
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A detail of volume ratio bounds in Ricci flow
Picture bottom is from Topping's Lectures on the Ricci flow. I can't get the red line.
First, I have the Bishop's theorem:
where $V^\alpha(r)$ is the volume of a ball of radius $r$ in the
complete ...
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mean curvature flow vs. ricci flow
Both of these seem to describe very similar flows: curve-shortening flows of a high-dimensional hypersurface, which may form singularities, where the flow velocity at any point is normal to the ...
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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 ...
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How to show $\frac{n}{4}\ln [1+\int |\nabla \phi|^2 dV]\le \frac{1}{2\pi}\int |\nabla \phi|^2 dV + C(g,n)$? [closed]
Picture below is from 90th page of Topping's Lectures on the Ricci flow. The integration is take on closed smooth manifold. I can't get the red line.
What I try:
$$
\frac{n}{4}\ln [1+\int |\nabla \...
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answers
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How to use delta function as initial data?
Picture below is from 74th page of Topping's Lectures on the Ricci flow. The (6.4.8) is
$$
-\partial_t u -\Delta u + Ru =0
\tag{6.4.8}
$$
where $R$ is scalar curvature. And the $N$ is
$$
N= -\int_M u \...
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