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 calculation in Ricci soliton

In the paper Cao, Huai-Dong, Geometry of Ricci solitons, Chin. Ann. Math., Ser. B 27, No. 2, 121-142 (2006). ZBL1102.53025, there is a statement saying that; if $g_{ij}$ is a complete gradient Ricci ...
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Does it make sense to call this a Soliton?

I believe a divergence free vector field is called a solenoid. Certainly the unit normal vector of a minimal surface is divergence free, and so too the minimal surface gives rise to a solenoid. That ...
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interpretation of the derivative with respect to time in the Ricci flow equation

I would like to geometrically understand what $ \frac{d}{dt} g (x, t) $ means, since I know that the metric is a tensor of rank two and that it can be derived but not in this way in all the books and ...
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Understanding a calculation of Ricci soliton

Recently, I have read the paper Munteanu, Ovidiu; Sesum, Natasa, On gradient Ricci solitons, J. Geom. Anal. 23, No. 2, 539-561 (2013). ZBL1275.53061. In that paper at the page 543, there is a ...
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Some misunderstanding on Hamilton's paper “Three-fold with positive Ricci curvature”

Now I am reading the Hamilton's paper named "Three-fold with positive Ricci curvature ". In the section 8 " curvature in dimension three ", he introduces a new tensor $Q_{ij}$ and followed by Theorem ...
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Some Confusion with a Computation relating to Gaussian Solitons

I am reading Peter Topping's lecture notes on Ricci flow and got into a slight amount of confusion with a computation relating to the Gaussian soliton. One considers the stationary flow of the flat ...
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Knowing a certain expression for the scalar curvature under the Ricci flow, prove that $g$ is Einstein

I'm working on the paper Rigidity of Min-Max Spheres on 3-manifolds, by Codá and Neves. At a certain point in a proof they get the following situation: $(M,g)$ is a compact riemannian 3-manifold of ...
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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 ...
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Problem Sheets and Exercises for understanding the Ricci Flow

I have recently been attending a seminar on the Ricci flow and the instructor has asked if we could find some problem sheets or exercises related to the Ricci flow which we could go through together ...
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The shrinking sphere example

I am having some issues with the vocabulary employed by authors when they refer to some solutions of the Ricci flow equation. For instance, the shrinking sphere example. It seems odd to me when I ...
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Why are Compact Steady and Expanding Solitons always Einstein Manifolds? [duplicate]

I am reading some lecture notes on Harnack inequalities and the Ricci flow and at one point the author remarks that it is important that the definitions for gradient steady, expanding and shrinking ...
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Does the Volume Ratio in a Complete Riemannian Manifold always tend to the Volume of the Euclidean Unit Ball?

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 ...
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Integral of Classical Entropy

I am reading some notes on Ricci flow by Peter Topping where the author introduces what he calls the 'classical entropy' for a function $u = e^{-f(t)}$: $$N= \int_M u \log(u)dV.$$ Using the fact ...
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Principal Symbol for Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and linearizing an operator to obtain its principal symbol. We have $T \in \: \Gamma(Sym^2 T^{*}M)$ smooth, fixed and positive definite and then ...
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How to calculate the derivative of a scalar curvature for a Ricci flow?

Now we have a homogeneous ricci flow, which means the initial data is homogeneous and then for each $t$, $g(t)$ is still homogeneous. My question is what is $$\text{scal}(g(t))'$$ In a paper(sec 3,...
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Derivative of Square of Covariant Derivative of Riemann Tensor

I am reading some notes on Ricci flow where it is stated that $\frac{\partial}{\partial t}|\nabla Rm|^2 \leq \Delta|\nabla Rm|^2 - 2 |\nabla^2 Rm|^2 + C |Rm||\nabla Rm|^2$. The text states that you ...
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Extinction time of a 2-sphere under Ricci Flow

So I was trying to pick up some differential geometry on my own and decided to try and solve Ricci Flow for a 2-sphere. Unless restrictions are imposed on the system, the surface will collapse and ...
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Derivative of Norm Squared of the Riemann Tensor

I am currently reading some lecture notes on Ricci flow and am not sure how the following identity is derived: $\frac{\partial}{\partial t} \bigg( g^{ij}g^{kl}g^{ab}g^{cd}R_{ikac}R_{jlbd} \bigg) = 2\...
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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(...
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Why is $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$?

Let $X,Y$ be two vector fields. Then the book "Lectures on the Ricci Flow" says $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$. I don't understand how this is the case. The second fundamental form ...
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Curvature of a homogenous manifold.

I was a reading a paper and it seemed to me that in one of the equations the authors used the fact if $M$ is a homogenous Riemannian manifold (i.e., the group of isometries of $M$ act transitively on $...
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Gram-Schmidt with curvature

Can we use Gram-Schmidt to estimate principal directions of curvature at a point p on a manifold? Suppose for any $tV \in T_pM$, we use the exponential map exp(V_t) to get a curve $\gamma: \Re \to M$...
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Calculation using contracted Binachi identity

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Consider the following equation $$\tag{1} s=nc+\Delta f,$$ where $s$ is the scalar curvature, $c$ is any real number and $f\in C^2(M)$. ...
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Normalized Ricci flow

We know that 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 ...
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Short-time existence of Ricci flow

Hamilton and DeTurck's short time existence theorem of the Ricci flow states that if $M^n$ is a smooth closed manifold with a $C^{\infty}$ Riemannian metric $g_0$ on $M$, then the Ricci flow on $M$ ...
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Cylinder to sphere rule - Ricci Flow

I was reading about the Ricci Flow and the author used the following theorem: Let $0< w \leq \infty$, and let $g$ be a metric on the topological cylinder $(-w, w) \times S^n$ of the form $$g = \...
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Ricci Flow Matlab Code / Algorithm

For a while, I am looking at explicit solutions of the Ricci flow: $$\partial_t g=-2Ric[g(t)]$$ $$g(0)=g_0.$$ Now I would like to make things visual. Is there already some code / algorithm available ...
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Ricci flow of the Torus

Let us consider the torus of revolution, say $T$, and consider a local parametrisation \begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \...
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Can Ricci flow develop singularity if metric is bounded?

Suppose I have a Ricci flow $(M, g(t))$ on $[0, T)$ can it develop a singularity if the metrics $g(t)$ are uniformly bounded? i.e $C^{-1}g(0)\leq g(t) \leq Cg(0)$ for all $t\in [0,T)$. In the ...
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Ricci soliton solves Ricci flow

We defined a Ricci soliton as a Riemannian manifold $(M,g_0)$ such that $$\mathrm{Ric}_{g_0}+\frac{1}{2}\mathcal{L}_{X}g_0=\lambda g_0$$ for some constant $\lambda$ and some vector field $X\in TM$, ...
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Roadmap to the geometry of lie groups and Duistermaat ideas.

I'm currently an advanced undergrad studentwith interest in differential geometry and topology, but this term I took a lie groups class and it was fascinating, nevertheless, there seems to be a break ...
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Does any Riemannian manifold admit Ricci flow and Ricci soliton? [closed]

I started studying Ricci flow and Ricci soliton (personally) and I have read some basic content of Ricci flow. I want to know the following: Does any Riemannian manifold admit Ricci flow and ...
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Request for online reference to Hamilton's CBMS conference notes, Hawaii, 1989

Does anyone know where I might find Hamilton's CBMS conference notes, Hawaii, 1989? The notes are cited in a few books, but I haven't had any luck googling for them.
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How to treat a map as a point of boundary?

Picture below is from 160 page of Hamilton, Richard S., Four-manifolds with positive curvature operator, J. Differ. Geom. 24, 153-179 (1986). ZBL0628.53042. I understand $f$ as a map from $M$ to $R^...
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Perelman's lectures [closed]

I can't find any videos of lectures that Grigori Perelman gave for his proof of Poincare's conjecture. For example, the following is a picture of one of his lecture in US. Edit. I was studying his ...
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Ricci Flow on a Sphere of radius r

Let $(M, g)$ be an $n$-sphere of radius $r$ with metric $g$ where $g = r^{2}g_{\mathbb{S}^{n}}$, $g_{\mathbb{S}^{n}}$ being the standard metric on the unit $n$-sphere. It is well known that $Ric_{g}=(...
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How can entropy be concave in time?

At this point in a lecture by Cedric Villani, prof. Villani talks about his work revealing that "if entropy is a concave function of time, then the Ricci curvature is non-negative." But wait, how can ...
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Given a complete noncompact Riemannian manifold with nonnegative Ricci and level set of a Busemann function

It is known that on a complete noncompact Riemannian manifold with nonnegative sectional curvature the Busemann function is convex and hence it's level set is connected. Suppose we replace ...
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Gradient Ricci solitons on surfaces

Ramos and Berstein-Mettler have provided classification of two dimensional gradient Ricci solitons (even admitting singularities). Is the problem of whether all two dimensional solitons are gradient ...
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Implement Euclidean Ricci Flow on 3D surface

I am not majoring in mathematics but right now I'm trying to work on a 3D surface. I want to apply Euclidean Ricci flow to do the conformal mapping using Mathematica, at first with simple ellipsoid ...
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On a particular solution of the Ricci flow

Doing a little bit of calculations I found that the metric: $$g(t)=\frac{dx^2+dy^2}{e^{-4t}-x^2-y^2}$$ satisfies $\frac{dg(t)}{dt}=-2Ric(t)$ Where $$\frac{dg(t)}{dt}=\frac{4e^{-4t}(dx^2+dy^2)}...
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Calculate of linearization of Ricci flow

According to this question , I want to calculate $DE(g_{ij})\widetilde g_{ij}$. If I treat $g^{ij}$ as irrelevant variable ,then I can get the same result as picture below. But I doubt that $g^{ij}$ ...
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How to prove the expanding Ricci soliton is Einstein metric?

As picture below ,I want to prove the expanding Ricci soliton is Einstein metric. I can get $R+\Delta f-n\lambda=0$. Besides ,I want to prove $R+|\nabla f|^2= C$ for some constant $C$. But fail. I ...
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Definition of steady Ricci soliton

Firstly, my English is poor, so I am not sure that which is solution , and the steady solution is the solution of evolution equation in picture below ? Secondly, what is the symmetry group of a ...
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Why do the Ricci and scalar curvature of an Einstein metric have the same sign?

In the picture below, seemly, if the scalar curvature is positive , then ,the Ricci tensor is positive , because I can use norm coordinate make the $g_{ij}\ge 0$. As I know ,$R=g^{ij}R_{ij}$ , I ...
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Does it make sense to have $\operatorname{Ricci}(g)=3$?

Is there a context in which it would make sense to have $\operatorname{Ricci}(g)$ to equal a constant number, such as $3$? I am not sure this makes sense, but apparently it does.
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Generalization of energy of geodesic

As we know , a curve $\gamma$ is geodesic iff $\nabla_\dot{\gamma}\dot{\gamma}=0$. The energy of curve is defined $$ E(t)=\frac{1}{2}\int |\dot{\gamma}|^2 $$ I don't know why define the energy so. I ...
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Local present of $\mathcal{L}$-geodesic

$X=\dot{\gamma}(\tau)$. I want to locally present the $\mathcal{L}$-geodesic , let $X=X^i e_i$, $\{e_i\}$ is the basis in a local coordinate. So, $$ \nabla_XX=X^iX^j_ie_j +X^iX^j\Gamma_{ij}^ke_k \\ ...
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Which is the best book for studying geometric flows?

I have some knowledge about the basics in Riemannian Geometry (I used Do Carmo's and Petersen's books). Now I would like to focus my attention on geometric flows (mostly mean curvature flow and Ricci ...
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Monotonicity of rank of Ricci tensor under Ricci flow

Let $(M,g)$ be Riemannian manifold , $Ric=\{R_{ij}\}$ is the Ricci tensor. Whether the rank of Ricci tensor is monotonic ? In fact ,I want to show it is increase (non-decreasing).