l. INTRODUCTION 199By the work of Angenent and one of the authors [13], one may have for example
a "rotationally symmetric" ~P^3 #~P^3 (considered as a quotient of 52 x ~'i.e., as
52 x [O, 1] with each 2-sphere at the ends identified to an ~P^2 ) where we have either
(1) a neckpinch at one end or (2) two simultaneous neckpinches at opposite ends.1.2. Relation of nonsingular solutions to Ricci flow with surgery.
In Ricci flow with surgery, after p erforming surgery, whether topologically es-
sential or not, one continues the solution until the next singularity. One then
repeats this process of forming a singularity, performing surgery, and continuing
the Ricci fl.ow. Hamilton's program and Perelman's implementation, seek to obtain
either
(1) finite-time extinction or
(2) infinite-time existence.
By finite-time extinction, we mean that after some finite time the solution is
empty; i.e., all components contract until their volumes tend to zero. As examples
of this, we note topological spherical space forms contracting to points and products
52 x 51 contracting to circles.
In either case (1) or (2), one hopes and expects to infer the existence of a
geometric decomposition, in the sense of Thurston, of the original closed orientable
3-manifold. The most difficult aspect of Hamilton's program, as well as Perelman's
implementation, is to show that surgery times cannot accumulate. The techniques
of Perelman (see [312] and [313]), building on Hamilton's body of work, address
this important issue.
The subject of this chapter, nonsingular solutions, corresponds to the infinite-
time existence case. Although singular solutions and nonsingular solutions are
complementary, they share a common technique in their study, namely Hamilton's
Cheeger- Gromov-type compactness theorem (see Theorem 3. 10 in Part I).1.3. Outline of the rest of the chapter.
In §2 we present examples and the statement of the main result of this chapter
and the next, that is, Hamilton's theorem that nonsingular solutions of the nor-
malized Ricci fl.ow on closed 3 -manifolds admit geometric decompositions in the
sense of Thurston. We also indicate a very brief outline of the proof; the detailed
proof, including related background material, occupies the rest of the chapter and
the next.
In §3 we separate the study of nonsingular solutions into the so-called posi-
tive, zero, and negative cases according to the asymptotic behavior of Rmin (t). If
a nonsingular solution sequentially coll apses, then it is diffeomorphic to a graph
manifold. So we assume that the solution is noncollapsed. By applying Hamil-
ton's Cheeger- Gromov compactness theorem, we provide a criterion for when M^3
is diffeomorphic to a space form with nonnegative curvature.
In §4 we prove by applying the above criterion that in the positive and zero
cases, M is diffeomorphic to a space form of positive and zero curvature, respec-
tively.
In §5 we prove that in the negative case any sequential limit must be a complete
hyperbolic manifold with finite volume, which may be either compact or noncom-
pact. If the limit is compact, then M itself admits a hyperbolic metric. Otherwise