198 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS1.1. The flip side-singular solutions and singularity models.
Nonsingular solut ions , by definition, do not form singularities in finite time. In
contrast, much of the study of the Ricci flow, b esides the quest for proving con-
vergence theorems, is focused on understanding the behavior of singular solutions.
This has been one the main focuses of t his book series.
In the case of the Ricci flow on closed 3 -manifolds , the nature of finite-t ime
singularities is more controll ed and restricted than in higher dimensions. R ec all
that a singularity model is defined as a limit of rescalings centered at space-t ime
points approaching the singularity time (see Definition 28.7). In dimension 3, a
finite-time singularity model is a K-solut ion.
All finite-t ime singularit ies of the Ricci flow on closed orientable 3-manifolds ei-
ther shrink to "round points" (in particular, the solution asymptotically approaches
constant curvature as it shrinks and the underlying 3 -manifold is diffeomorphic to
a spherical space form) or exhibit weak neckpinches; i .e., there exists a singular-
ity model which is a round shrinking cylinder or its orientable Z2-quotient. In
the nonspherical space form case, the existence of such weak neckpinches follows
from Hamilton's work together with P erelman's no local coll apsing theorem (see
for example Theorems 9.68 and 9.7 0 in [77]).
Much more strongly and partly based on his results on K-solutions (some of
which are describ ed in Chapters 19 and 20 of Part III), P erelma n proved a canonical
neighborhood theorem (see §12.l of [312]), which gives a ve ry good description of the
high curvature regions of so lutions to the Ricci flow on closed 3 -manifolds. However ,
even in dimension 3, much of the fine structure of singularities is still unknown; for
example, a priori one could conceivably have a Cantor set of 2-spheres collapsing
simultaneously.^2 This possibility also holds for a similar situ ation in the study of
the mean curvature flow (even in dimension 2).
The reader may see §5 of Chapter 2 of Volume One for a discussion of (strong)
rotationally symmetric neckpinches as well as Chapt er 36 of this book for a discus-
sion of rotationally symmetric degenerate n eckpinches, t he latter b eing an example
of a Type Ila singularity. The more general singularity theory together with the un-
derstanding of solut ions in high curvature regions, which are particularly effective
in dimension 3, are developed by Hamilton and Perelman in [138], [142], [312],
and [313].
In the case of degenerate neckpinches , the geometric-topological surgery that
one performs does not ch ange the topological type of the 3-manifold. On the other
hand, in the case of nondegenerate n eckpinches , surgery may change the topological
type of the 3-manifold. When this occurs, we call t he geometric-topological surgery
topologically essential. A topologically essential surgery does one of the following:
(1) It splits the 3-manifold into two disjoint 3-manifolds, neither of which
is diffeomorphic to 53 and whose connected sum is diffeomorphic to the
original 3-manifold.
(2) It produces a new 3-manifold whose connected sum, either with (a) 52 x 5^1
or with (b) ~P^3 , is diffeo morphic to the original 3-manifold. The former
case arises wh en a topological handle is pinched along an 52 , whereas
the latter arises when one has a neckpinch whose singularity model is the
orientable Z 2 -quotient of the round shrinking cylinder.(^2) We thank John Lott for pointing this out to us.