# How fast does Ricci flow converge on the three-sphere?

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?

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.
• 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. Apr 12 '14 at 4:53