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 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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Reference about the existence of Ricci flow with surgery for all positive time

As far as I can see, the Perelman's Ricci flow with surgery on three-manifolds does not prove the existence of Ricci flow with surgery for all positive time. And it is necessary for proving ...
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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 the upperbound of the Ricci tensor preserved

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
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How to show $|Ric|^2 \ge \frac{R^2}{n}$?

$Ric$ is Ricci curvature. Making orthogonal decomposition, there is $$Ric =\mathring{Ric} + \frac{R}{n}g \tag{1}$$ where $R$ is scalar curvature, $g$ is Riemannian metric. I get (1) from the 42th ...
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I want to show $$(\delta S)Z = \delta (S(\cdot, Z))+\frac{1}{2}\langle S, \mathcal L_Zg\rangle \tag{1}$$ where $\delta= -tr_{12}\nabla$ is divergence operator, $\mathcal L$ is Lie derivative, $S\... • 5,140 5 votes 1 answer 67 views How to show$\delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle $? Picture below is from Topping's 37th page of Lectures on the Ricci flow. I try to show $$\delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle$$$\delta$... • 5,140 1 vote 0 answers 30 views How to understand$(\pm*d*\alpha) dV = \pm d(*\alpha)$in Ricci flow Picture below is from Topping's Lectures on the Ricci flow. First, I don't know why $$(\pm*d*\alpha) dV = \pm d(*\alpha),$$ in fact, I don't know what is$*\alpha$and$d*\alpha$. In my opinion,$*$... • 5,140 2 votes 1 answer 43 views A problem on showing$h(X, \operatorname{Ric}(W))= \operatorname{tr} h(\operatorname{Ric}(W,\cdot)\cdot, X)$I want to show$(2.3.11)$, but in my calculation, there is no minus. What I try: from the$(2)$, I have $$h(X,\text{Ric}(W))=h(X, \text{Ric}(W, e_i)e_i)= \text{tr}\space h( X, \text{Ric}(W, \cdot)\... • 5,140 1 vote 1 answer 39 views Show \operatorname{tr} \nabla^2_{X,~\cdot} h(\cdot, W)= -(\nabla \delta h)(X,W) I want to show$$ \operatorname{tr} \nabla^2_{X,~\cdot} h(\cdot, W)= -(\nabla \delta h)(X,W) $$where X,W are tangent vector fields on Riemannian manifold. And$$ \nabla^2_{X,Y} = \nabla _X \nabla_Y ... • 5,140 1 vote 0 answers 37 views Formal adjoint of divergence which is restricted on$\Gamma(\wedge^k T^*M)$I want to show the red line. In fact, in another problem , with many help, I get the formal adjoint of divergence without restrict. Beforehand, I think I can deal the red line if I know how to get the ... • 5,140 1 vote 1 answer 59 views Proof of Proposition 2.3.5 of Topping's Lectures on the Ricci flow When I read the Proposition 2.3.5 of Topping's Lectures on the Ricci flow. I can't understand why the spcial vector fields which satisfy (2.3.13) do not loss generality. In fact, in my view, they ... • 5,140 1 vote 1 answer 96 views Why$(\partial_t \nabla _X \omega)Y =-\omega(\partial_t \nabla_X Y)$means$\partial_t \nabla \omega = \omega * \nabla (\partial _t g)$?$\omega \in \Gamma(T^*M)$,$X,Y\in \Gamma(TM) $,$(M, g(t))$is Riemannian manifold whoes metric depends on$t$. If I have $$(\partial_t \nabla _X \omega)Y =-\omega(\partial_t \nabla_X Y)$$ Why$\...
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Picture bottom is from Topping's Lectures on the Ricci flow (author website, zbMath link). I wanto to show the formal adjoint of $\delta$ is covariant derivative. The definition of formal adjoint is ...
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Question regarding the quadratic curvature tensor

I am studying the evolution of curvature in my study on the Ricci flow, and in The Ricci Flow in Riemannian Geometry by Hopper and Andrews, I came across the (0,4) quadratic curvature tensor defined ...
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Can Ricci flow be used to prove Poincaré’s conjecture for $n=2$?

This question is concerning the conjecture described in this Wikipedia article. The conjecture has been proved for dimension two ($n = 2$). For $n=3$, conjecture is proved using Ricci flow. My ...
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Does scalar curvature is bounded by absolute value of Riemann curvature?

Is it true if the lowest eigenvalue of the Ricci tensor is negative than $Scal \le |Rm|$? What i guess is $$|Rm| \ge \frac{Scal}{n}$$ by using Cauchy-schwarz inequality. However while i read the ...
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Is there any kind of "flow" that describe balloon-like expansion for 2d surfaces constructed via a 1d paths?

I've looked into Ricci Flow and mean curvature flow/curve shortening flow, but I dont think any of these describe the math behind balloon-like expansion (or even simple diffusion) well enough given ...
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What is missing from solution of Poincaré conjecture to Naivier-Stokes equation?

This question has arised from the links of the comments in the question What are the fundamental problem in solving the NAVIER–STOKES equation.. As I understand Ricci flow has been used in solving the ...
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Question on Ricci flow and Einstein Tensor

My knowledge on this topic is solely based on physics books, introductory texts on differential geometry and this one paper $[1]$. One of the formulas of the paper shows what is called the Ricci ...
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Ricci flow as heat flow on Riemannian manifold

I read that Ricci flow is "a nonlinear heat flow for the Riemannian metric". Can someone explain what this means? Nonlinear heat flow has a wikipedia page but I don't understand how this ...
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Mean curvature flow vs. diffusion

Mean curvature flow (MCF) and diffusion-type flows both have smoothing effects on a curve. I can tell there is a deep connection between the two, as seen in the Merriman-Bence-Osher (MBO) numerical ...
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Gradient of solution to heat equation under evolving metric

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
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Book on the geometry of rotationally symmetric riemannian manifolds

I would like to find some references where there are specific computations and properties of rotationally symmetric riemannian manifolds, e.g. spectrum of the laplacian, schrödinger operators, ...
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Proving some identities about the time derivative of the k-th covariant derivatives of scalar curvature under normalized Ricci flow on surfaces

I'm trying to prove the following identities (under the normalized Ricci flow on surfaces, on which $\partial_t g = (r-R)g$ holds true, where $r$ denotes the average scalar curvature and has the same ...
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Motivation of Uhlenbeck’s trick without local computation(Suitable connection defined in space time manifold M $\times \mathbb{R}$)

I am a beginner to learn Ricci flow and my main study reference is Simon Brendle’s “Ricci Flow and the Sphere Theorem”. In section 2.3 of Brendle, he introduce the following: let D be the Levi-Civita ...
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Global Neckpinch Singularity under Ricci Flow

I am aware that in general for the Ricci flow, neckpinches are local singularities (ie. they occur on a compact subset of the manifold). The usual picture is that of a manifold shaped like a dumbbell ...
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