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](http://www.ams.org/mathscinet-getitem?mr=2138137) and [Perelman](http://arxiv.org/abs/math.DG/0307245). 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?



How fast does Ricci flow converge on the three-sphere?
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](http://www.ams.org/mathscinet-getitem?mr=2092903). 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](http://arxiv.org/abs/1001.1458)). 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. 


