202 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS
(B3) and (B4) (Sequential convergence to a flat or hyperbolic manifold)
There exist a sequence of times ti -+ oo and diffeomorphisms </>i : M -+ M such
that the sequence of pulled-back metrics
zero for (B3) or negative for (B4)) sectional curvature metric g 00 on M as i-+ oo.
(B5) (Toroidal decomposition into hyperbolic and graph manifold
pieces) There exists a finite collection of finite-volume hyperbolic 3-manifolds
{ (H'!x, ha)} : 1 and there exist smooth 1-parameter families of embeddings
'l/Ja(t) : Ha -+ Msuch that the pulled-back metrics 'l/Ja(t)*g(t) converge to ha as t -+ oo for each
a. Moreover, M can be decomposed into two time-dependent 3-manifolds M1(t)
and M 2 (t), where Mj(t) and Mj(t') are ambient isotopic^6 in M for all t and t',
j = 1, 2, whose intersection is their mutual boundary and consists of a finite disjoint
union of 2-tori. Geometrically, the metrics pulled back from M 1 (t) converge to the
hyperbolic pieces; i.e.,
aas t -+ oo, while (M 2 (t), g(t)) collapses. Topologically, the boundary tori are all
incompressible, and all of the homomorphisms induced by the inclusion maps
i* : 7f1 (N) -+ 7f1 (P)are injective, whenever we have both
(1) N = M1(t) nM2(t), M1(t), or M2(t),
(2) P = M 1 (t), M 2 (t), or M are such that N c P.
2.3. Brief outline of Hamilton's proof of Theorem 32.2.
The first idea is to separate the study of noncollapsed nonsingular solutions of
the NRF on closed 3-manifolds into cases, depending on the asymptotic behavior
of the minimum scalar curvature. Let Rmin (t) ~ minxEM R (x , t). We have three
cases:
(I) Rmin (t) > 0 at some time t,
(II) Rmin (t) /' 0 as t-+ oo, and
(III) Rmin (t) increases to a negative limit as t-+ oo.
By considering these sequential limits of noncollapsed nonsingular solutions,
one shows that in cases (I) and (II), M^3 admits a metric with constant nonnegative
sectional curvature.
This leaves us with case (III). In this case, unless M itself admits a hyperbolic
metric, all asymptotic limits are complete noncompact hyperbolic 3-manifolds with
finite volume (see Proposition 32.12 below). Corresponding to each hyperbolic limit
1i^3 there is a time-dependent almost hyperbolic piece in M which is immortal,
i.e., defined for all large enough time. The interior of this compact 3-dimensional
submanifold of M is diffeomorphic to 1i and is metrically as close as we like to
the truncation of 1i along constant mean curvature tori with sufficiently small area
A > 0. Importantly, one shows that each of these hyperbolic pieces is incompressible
inM.
(^6) That is, t here exists a continuous 1-parameter family of homeomorphisms Ft of M with
Fo =id and F1(Mj(t)) = Mj(t').