# Rigorous solution to Ricci Flow on dumbbell $S^3$

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering if some of his "intuitive pictures" can now be made rigorous:

Consider a dumbbell metric on $S^3$, where the neck looks like $S^2\times B^1$ (something like this, except that picture is for $S^2$). The positive curvature at the ends of the dumbbell will cause the metric to contract (rounding out those parts), whereas the neck area should shrink. In particular, the neck should pinch off at some time $t_{pinch}>0$ (topologically a wedge-sum $S^3\vee S^3$). Has this been put into experimental practice? More to the point, Hamilton remarks that there may be a weak solution extending past the pinching moment when the sphere splits into two spheres, although weak solutions haven't been defined for the Ricci flow. To what extent can we currently rigorous this in practice (if at all)?

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I would say that, while not really defining weak solutions, this pinching problem is precisely what Perelman's Ricci flow with surgery is designed to dodge. –  Benoît Kloeckner Dec 23 '12 at 11:02

## 1 Answer

The neckpinch solution on a dumbell has been constructed by Angenant and Knopff. A whole chapter of the book "The Ricci Flow: An Introduction" by Chow and Knopff is devoted to the construction of such a solution.

For the "weak solution" side, it has not been settled yet. Angenant and Knopff has some result in the rotationnaly symmetric case in a preprint that was put on the arXiv something like two years ago.

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Oh OK, I was going to approach those books right after I finished skimming some foundational papers. As for the Angenant-Knopf paper, is it *An Example of Neckpinching for Ricci Flow on $S^{n+1}$* (math.wisc.edu/~angenent/preprints/MRLrevision2.pdf)? I don't see anything about $t\ge t_{pinch}$. –  Chris Gerig Dec 23 '12 at 4:54
See the links here: mathoverflow.net/questions/35584/… –  Ian Agol Dec 23 '12 at 6:25