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Results tagged with ricci-flow
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user 119114
14
votes
1
answer
395
views
Does the Cheeger constant satisfy a heat-type equation?
It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow.
A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of …
- 5,146
3
votes
Accepted
Does the Cheeger constant satisfy a heat-type equation?
Answer to (2) is negative. For a counterexample, consider a $2$-sphere and blow up the radius. The isoperimetric ratio $C_H$ remains the same, but the Cheeger constant $h$ shrinks, essentially becau …
- 5,146
13
votes
2
answers
2k
views
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 pa …
- 839
5
votes
1
answer
260
views
Neckpinch singularity of Ricci flow
I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the …
- 6,001
0
votes
Rigorous solution to Ricci Flow on dumbbell $S^3$
Yes, these pictures have now been made rigorous. Another paper which you might be interested in on this topic is
Simon, M. (2000). A class of Riemannian manifolds that pinch when evolved by Ricci flo …
- 5,146
4
votes
0
answers
283
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 ex …
- 5,146
0
votes
0
answers
287
views
Does the Volume Ratio of a Geodesic Ball for a Complete Riemannian Manifold tend to the volu...
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 hol …
- 5,146
2
votes
1
answer
243
views
Principal Symbol for the Ricci-DeTurck Flow
I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 …
- 2,830
2
votes
1
answer
248
views
What is the Weak Maximum Principle for Scalars and how is it Derived?
I am currently reading 'Lectures on Ricci Flow' by Peter Topping and I have got to Chapter 3 where he states the 'weak maximum principle for scalars'. Suppose for $t \in [0,T]$ for finite $T$ that $g …
- 671