Undergrad Does Ricci Flow Contract a 3-Sphere to Its Center?

[casparov]

TL;DR

does the center of a 3 sphere remain stable with Ricci flow?

I have a a very basic question and a followup question.

1. Consider you have a 3-sphere, Ricci flow says it contracts to a point in finite time. So the manifold contracts to its center, correct?

2. Say you have two 3-spheres that stay tangent to eachother, and you connect a line between the two centers, naively the two (center) points seem to converge with Ricci flow-- is that mathematically valid to show convergence/a limit?

 

[Infrared]

TL;DR Summary: does the center of a 3 sphere remain stable with Ricci flow?

1. Consider you have a 3-sphere, Ricci flow says it contracts to a point in finite time. So the manifold contracts to its center, correct?

No, this doesn't make sense. All that changes during Ricci flow is the metric $g$ according to the PDE $\frac{\partial g_t}{\partial t}=-2\text{Ric}(g_t).$ The actual manifold doesn't change. What is actually meant by your statement is probably that the metric becomes zero in finite time. Animations with the manifold shrinking, bending, etc. are just ways to visualize what changing the metric does (in the sense that the animated manifold with the standard metric is supposed to be isometric to the starting manifold with the evolved metric) but you shouldn't take them too literally.

Unfortunately, between the time of your post and my reply, Richard Hamilton (the inventor of Ricci flow) has passed away.