CHAPTER 32
Nonsingular Solutions on Closed 3-Manifolds
Maybe this time is forever ...- From "Father Figure" by George Michael
Beginning with this chapter we give an exposition of Hamilton's visionary paper
[143] and thus consider an important special class of solutions to the normalized
Ricci flow, namely those which both exist for infinite forward time and have uni-
formly bounded curvatures, i.e., nonsingular solutions.
Regarding the role of nonsingular solutions in his program for the Ricci flow
on 3-manifolds, in [143] Hamilton wrote:
" ... one may hope to produce nonsingular solutions after a finite
number of surgeries."To wit, one may hope that for any closed 3-manifold Mand any initial metric g 0 on
M there exists a solution g (t), t E [O, oo), with g (0) = 90 to the volume normalized
Ricci flow with surgery,^1 where the surgery times are only finitely many and where
the solution after the last surgery time is a nonsingular solution to the Ricci flow.
In short, this is Hamilton's program to prove Thurston's geometrization conjecture.
On the other hand, Perelman's work indicates that, in order to prove the ge-
ometrization conjecture, the global in time existence of the unnormalized Ricci flow
with surgery, without necessarily a uniform curvature bound nor necessarily a fi-
nite number of surgeries (an infinite number of surgery times tending to infinity is
allowed), is evidently sufficient; see the second to last paragraph of §1 of Perelman
[313].
Regarding the completeness of Perelman's proofs in [312] and [313], building
on Hamilton's earlier body of work, of the Poincare and geometrization conjectures,
please see Cao and Zhu [49], Kleiner and Lott [161], Morgan and Tian [251], [252],
and Bessieres, Besson, Maillot, Boileau, and Porti [29].
Throughout this chapter we shall assume that the 3-manifolds under consider-
ation are orientable.
- Introduction
In this section we expand upon the above motivation for the study of nonsin-
gular solutions. At the end of this section we give an outline of the chapter.
(^1) For the original definition of 4-dimensiona l Ricci flow with surgery, see [142].
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