1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 233

the fact that the volumes of p(k) (t(k)) (Hi;k) are uniformly bounded from b elow,

and the fact that the volume of (M, g(t)) is finite and co nstant. This co mpletes

the proof of Theorem 33.17. D

3.2. Disjointness of hyperbolic pieces and the stability of all hyper-

bolic limits.

From the proof of the subclaim containing (33.37), one can show that two im-

mortal almost hyperbolic pieces a re either equivalent or are disjoint. We leave it to

the reader to deduce the following lemma essentially from the fact that topologi-

cal ends of finite-volume hyperbolic 3-ma nifolds are hyp erbolic cusps (see Theorem

31.44).

LEMMA 33.19 (Disjointness of almost hyperbolic pieces). There exists A. E

(0 , A], where A is as in (h5), such that for any 0 < Ai ::; A 2 ::; A. there exists

ko EN with th e following properties. Let (M^3 ,g(t)), t E [ti,t2), where ti <

t 2 ::; oo, be a smooth family of closed Riemannian 3-manifolds. If the (H~, ha) are

finite-volume hyperbolic 3-manifolds and Fa (t) : (Ha)Aa --+ M are embeddings with

llFa(t)*g(t) - hal(1-la)Aa llcko((1-la)Aa,ha) '.S :0

f or a= 1, 2 and t E [ti, t2), th en for all t E [ti , t2) either

(1) Fi (t) ((Hi)A 1 ) and F2(t)((H2)A 2 ) are disjoint or

(2) we have the inclusions

As we shall see from the discussion in the rest of this section, in the case of a

nonsingular solution to the NRF satisfying Condition H we m ay refine the above

statement as follows. There exist A E (0 , v'3/4] and k EN such that if MA (t), t E

[tA,k> oo), is an immortal (A, k )-almost hyperbolic piece, then MA ( t) is contained
in a stable asymptotically hyperbolic subma nifold. In particular , MA (t) is actually
a n immortal asymptotically hyperbolic piece. Moreover, any two immortal (A, k)-
almost hyperbolic pieces are either disjoint or equal on their mutual time interval
of existence.
In the remainder of this section, we shall simultaneously prove, by an induction
argument based on the number of cusp ends, all three of Propositions 33.5, 33.6,
and 33.8.
Let (H5,ho) E ..fJIJPo(M, g(t)) ~ ..fltJp(M,g(t)) with x~ E Ho be a pointed
complete noncompact hyp erbolic limit with a minimal number of cusp ends. Let
A. b e as in Lemma 33.19. By Theorem 33.17, there exist To< oo, Ao: [To, oo)--+
(0 , A.], and a corresponding stable asymptotically hyperbolic submanifold (in the
sense of Definition 33 .2)

M6,Aa(t) (t) CM, t E [To, oo).

In the following,

(33.39) c; < inf m axinj h (x )

1-iEi'>IJP(M,g(t)) xE1-l