Estimates for the Extinction Time for the Ricci Flow on Certain 3-manifolds and a Question of Perelman

In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question that Grisha Perelman asked the first author at a dinner in New York City on April 25th of 2003. His question was “what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular, does the flow become extinct in finite time?” He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min–max argument where the minimal of all maximal slices of sweep–outs is a minimal surface; see, for instance, [2]. The idea is then to look at how the area of this min–max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to the solution becoming extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see [9]) answering his original question about finite extinction time. However, even after the appearance of his paper, we still think that our slightly different approach may be of interest, in part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke, and thus our approach is in some respects technically easier. LetM be a smooth closed orientable 3-manifold and let g(t) be a one–parameter family of metrics on M evolving by the Ricci flow, so