# 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 ...
1 vote
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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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### Formal adjoint of divergence operator

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 ...
1 vote
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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 ...
1 vote
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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 $$. 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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