# 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 ...
20 views

### 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 ...
42 views

### 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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### 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 ...