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 of Grisha
Perelman asked to 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, [CD].
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 that the solution becomes 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 [Pe1]) answering his original
question about finite extinction time. However, even after the appearance of
his paper, then 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.