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218 33. NONCOMPACT HYPERBOLIC LIMITS

In particular, int(MA (t)) is diffeomorphic to 7-l and for any k E N there exists

tA,k E [tA, oo) such that MA (t) is an immortal (A, k)-almost hyperbolic piece for

t 2'. tA,k·

In §3 of this chapter we shall first prove the special cases of Propositions 33.5

and 33.6 where the limits (7-l^3 , h) have the minimal number of cusp ends among

all limits. We then prove both of these propositions by induction on the number of

cusp ends and concurrently with the following result. This additional result says

that M can be decomposed along tori into immortal almost hyperbolic pieces and

collapsed pieces (where the injectivity radius is everywhere small).

PROPOSITION 33.8 (Decomposition into hyperbolic and coll apsed pieces). Let

(M^3 , g (t)) be a nonsingular solution satisfying Condition H. Then there exists a fi-

nite collection of complete hyperbolic 3-manifolds with finite volume { (7-l;, ha)} ;:'= 1

such that for any c > 0 sufficiently small, there exist a time Tc < oo, a number

A., E (0, v'3/4] with corresponding truncations (7-la)A, of 7-la, and harmonic em-

beddings

Fa(t): ((7-la)A,,ha)-+ (M,g(t)), t E [T.,,oo),

with the following properties. The maps Fa(t) depend smoothly on t and satisfy

both the CMG boundary conditions and

lfFa(t)*g(t) - hallcL1/•J((7-l.,)A,•h.,) < €.

Here the submanifolds Fa(t)((7-la)AJ are mutually disjoint in M and the maximum

injectivity radius of M - Ua Fa(t) ((7-la)AJ with respect to g(t) is less than c for

all t E [T.,, oo).

1.3. Hyperbolic pieces have incompressible boundary.

A main use of the continuous dependence on time of the decomposition in

Proposition 33.8 is to help prove that boundary tori of immortal almost hyperbolic

pieces in 3-dimensional nonsingular solutions are incompressible in the ambient

3-manifold (see §4 of this chapter for a proof).

PROPOSITION 33.9 (Incompressibility of boundary tori). Let (M^3 , g (t)) be a

nonsingular so lution satisfying Condition H. For any t (sufficiently large) for which

an immortal asymptotically hyperbolic piece M~ (t) C M is defined, its boundary

8MA (t) is incompressible in M. That is, the inclusion map l : 8MA (t) '---+ M

induces an injection

on fundamental groups.

By combining all three of Propositions 32.12 (on the existence of hyperbolic
limits), 33.8, and 33.9 for Case III and Theorem 31.55, we conclude that M may be
decomposed along incompressible tori into graph manifolds and hyperbolic pieces,
which implies that M admits a geometric decomposition in the sense of Thurston.
This completes the proof of Hamilton's Nonsingular Solutions Theorem 32.2
modulo the proofs of Propositions 33.11 and 33.12 below to be given in the next
chapter. Both of these propositions shall be used in the study of harmonic parametri-
zations of almost hyperbolic pieces in §2 below.