# Questions tagged [ricci-flow]

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### Is there a solution of the Yamabe problem using Ricci flow?

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
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### Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics ...
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### What exactly does it mean for Hamilton's cigar soliton to have linear volume growth?

In a couple articles I've read lately, I've seen it mentioned that the cigar soliton has linear volume growth. What does this mean? I thought maybe, if you compute the volume of geodesic balls and ...
312 views

### Ricci flow preserves almost Kahler condition?

I have been unable to find a reference to the following (perhaps too naive) question. Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-...
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### Geometric flow equations which are second order in time derivative

All examples about geometric flow equations given in Wikipedia's Geometric flow article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order ...
193 views

### Type II singularities for 3D Ricci flow

I know that type II singularities of the Ricci flow can exist on closed 3-manifolds (e.g. on $S^3$), but on the other hand it seems to me that ODE comparison combined with Hamilton's tensor maximum ...
91 views

### Is there a version of Ricci Flow for Pseudo-Riemannian Metrics?

The Ricci flow deforms a Riemannian metric. I was wondering if there was something very similar which deforms a pseudo-Riemannian metric or if not, is there reason why such a geometric flow cannot ...
365 views

### 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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### Ricci flow on Riemannian submersions

Let $(P,g) \to (S^2,h)$ be a Riemannian submersion. Let $g(t)$ be the Ricci flow on $P$ with initial condition $g$. Does the induced flow on $S^2$ converges to the round metric on $S^2?$ I could ...
94 views

### Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
82 views

### Ricci flow preserves locally symmetry along the flow

Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
232 views

### Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
80 views

### Does the Volume Ratio of a Geodesic Ball for a Complete Riemannian Manifold tend to the volume of a Unit Ball in Euclidean $n$-space?

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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### Understanding the Hamilton's definition of $\ast$-operation

I'm studying by myself Mean Curvature Flow and I'm trying understand the definition of $\ast$-operation given by Richard Hamilton in the beginning of the section $13$ (page $40$) of his paper "Three-...
63 views

### Eigenvalues of geometric operators along geometric flows

I have two questions: 1- what is the relation between eigenvalues of geometric operators such as Laplace operator and topology or geometry of a Riemannian manifold?(please give an example if possible)...
107 views

### Isometries along the normalized Ricci flow

As we know the 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 ...
187 views

### Curvature decay of Ricci expanders

Let $M$ be a gradient Ricci expander with nonnegative curvature operator. Assume $\Sigma$ is its space of directions at infinity (so $M$ looks like a cone over $\Sigma$). What is the curvature ...
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### Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature

I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature. Any help will be appreciated. Thanks!
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### Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics. Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...
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### Ricci soliton on contact manifolds

Recently I am studying Ricci flow and its self-similar solution called Ricci soliton. In this respect I found some papers which focuses Ricci soliton in the setting of various kind of contact ...
761 views

### How is Ricci flow related to computer graphics?

I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...