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?