- DECOMPOSITION THEOREMS AND THE RICCI FLOW 439
In particular, this implies that the boundary of M is a possibly empty col-
lection of tori. Graph manifolds appear in the work of Cheeger and Gromov
as those 3-manifolds admitting an F-structure. Note that the definition of
graph manifold that appears in Cheeger and Gromov's work [7 4] is slightly
different from the one above. To be more precise, a graph manifold in the
sense of Cheeger-Gromov is a closed 3-manifold admitting a decomposition
by (not necessarily incompressible) tori such that each complementary piece
is a Seifert space. It can be shown that if M is a graph manifold in the sense
of Cheeger-Gromov, then it is either a topological graph manifold or else a
connected sum of topological graph manifolds with 82 x 8^1 factors and lens
spaces. At any rate, there are no fake 3-spheres or fake 3-balls embedded
inside a Cheeger-Gromov graph manifold.3. Decomposition theorems and the Ricci fl.ow
Recall that a solution (M^3 , g(t)) , t E [O, T), to the Ricci fl.ow is said
to develop a singularity at time T E (0, oo) if the norm of the Riemann
curvature tensor becomes infinite at some point or points of the manifold as
t /' T. (See Corollary 7.2 of Volume One.) A typical situation in which a
finite time singularity develops is the neckpinch. It is important to note
that the formation of neckpinch singularities may be triggered more by the
(local) nonlinearity of the Ricci fl.ow PDE than by the (global) topology of
the underlying manifold. In any case, here is a heuristic description. (For
precise statements, see Section 5 of Chapter 2 in Volume One, as well as
the recent papers of Angenent and one of the authors [7, 8].) Suppose a
3-manifold M contains a separating 2-sphere. Then under the Ricci fl.ow,
a region homeomorphic to 82 x IR may develop in M such that, as t /' T,
the sectional curvatures become infinite precisely along the hypersurface
identified with 82 x {O}. In this evolution, the geometry of the region
identified topologically with 82 x IR asymptotically approaches the cylinder
82 x IR with its standard product metric.
Hamilton developed a program of applying Ricci fl.ow techniques to gen-
eral 3-manifolds and analyzed the singularities which may arise (see espe-
cially [186], [189], and [190]).^1 Some of Hamilton's ideas are as follows; we
first consider [186]. Via point-picking arguments and assuming an injectiv-
ity radius estimate, one dilates about singularities and takes limits using the
Cheeger-Gromov-type compactness theorem for solutions of the Ricci fl.ow
to obtain so-called singularity models, which are nonfl.at ancient solutions
of the Ricci fl.ow. In dimension 3 these ancient solutions have nonnegative
(^1) Some other papers in which Hamilton developed his program to approach Thurston's
geometrization conjecture by Ricci flow are as follows: characterizing spherical space forms
[178], weak and strong maximum principles for systems [179], ancient 2-dimensional
solutions and surface entropy monotonicity [180] (as used in [186]), matrix Harnack
estimate [181] and its applications to eternal solutions [182], and the compactness theorem
[187]. (These are only partial descriptions that reflect aspects of the papers' relevance to
Hamilton's 3-manifold program.)