# 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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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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 ...
### 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 ...