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Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the extinction time of this flow?

Really, I want to flow for some (hopefully short) time $t$ so that all the pieces of the manifold at time $t$ are almost isometric to the standard metric on $\mathbb S^3$ (up to scaling). I believe it must be well-known that each piece is very close to the standard $\mathbb S^3$ as it becomes extinct, and it is easy to see that the standard metric of radius $r>0$ becomes extinct in time $t\propto r^3$. Thus hopefully it suffices to get a bound on the extinction time.

I know of the papers by Colding--Minicozzi and Perelman. I am hoping their bounds can be improved, since the don't seem good enough for what I want to do. Basically, I hope to avoid knowing things about the "width" of nontrivial homotopy classes in $\mathbb S^3$ (which is what both Colding--Minicozzi and Perelman use). Is it realistic to expect these can be replaced with quantities like the volume, injectivity radius, curvature, etc. of the original metric?

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up vote 9 down vote accepted

I'm not sure how to answer your question - I believe the Perelman or Colding-Minicozzi widths are the only known way to estimate extinction time in general.

At the time of extinction of a component, the sphere might not be round. Consider a dumbbell rotationally-symmetric metric on $S^3$ which develops a neck singularity at finite time. As one increases the width of the neck until it becomes convex, there must be a time in between when the pinch and the extinction occur simultaneously. The singularity at this time might be like a peanut, with neck remaining until the singularity, so it does not approach a round sphere. At the tips of the peanuts, the metric should be approaching a type II singularity, which has rescaled limit a Bryant soliton. However, I think it is believed that this sort of singularity is non-generic.

A special case in which one may estimate the extinction time is when the metric has positive scalar curvature. If the minimum scalar curvature is $R_{min}(0)$ at time $0$, then the solution must go extinct at time $3/(2R_{min}(0))$ by the maximum principle for the evolution of the scalar curvature (see e.g. Prop. 2.1). A similar estimate holds for $\lambda(g_0)$, the minimal eigenvalue of the operators $-4\Delta+R$. Since $\lambda(g_0)\geq R_{min}(0)$, this might give an extinction estimate when $R_{min}$ does not.

During the Ricci flow-with-surgery when $\lambda(g_t) <0$, one has that the (scale-invariant) quantity $Vol(g_t)(-\frac16\lambda(g_t))^{3/2}$ is decreasing with respect to time. One also knows that $\lambda(g_t)$ must approach $0$ at some point for Ricci flow-with-surgery on $S^3$. However, I don't know of any way to show how fast this quantity approaches zero without invoking the width, since one must have some sort of topological input.

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Just a little remark: the singularity model of degenerate neck-pinch might not be Bryant soliton. Actually this was asked by Perelman: does Bryant soliton the only 3D $\kappa$-noncollapsed ancient solution with $Rm>0$? This is still open except in the category of steady soliton. That is why Perelman need to prove that every 3D $\kappa$-ancient solution with $0\leq Rm\leq C$ has a cylindrical end (in the blow-down sense), in this way he can perform the surgery without knowing that singular model is Bryant or not. –  Chih-Wei Chen Apr 12 at 4:53
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