1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

234 33. NONCOMPACT HYPERBOLIC LIMITS

will be chosen to be sufficiently small. Regarding the complement M -Mo,Ao(t) (t)

of the first stable asymptotically hyperbolic submanifold, we have the following two

alternatives:

Case (Ao). There exists To E [To, oo) such that

(33.40) sup max inj g(t) (x):::; c.

tE(To,oo) xEM-Mo,Ao(t)(t)

That is, for sufficiently large times, the complement of Mo,Ao(t) (t) is €-coll apsed.

So, for large enough t, (M, g (t)) is the union of a stable asymptotically hyperbolic

submanifold and an €-coll apsed piece, whose intersection is comprised of tori. In

this case, Proposition 33.8 is proved. Here, Propositions 33.5 and 33.6 also follow

since 5J1Jp(M,g(t)) is comprised of only one element.

Case (Bo). Otherwise, there exists a sequence { (yi, ti)} with Yi tf. Mo,Ao(t, ) (t i )

and To :::; ti --+ oo such that

inj g(ti) (Yi) > c for all i.

Let 5J1Jp 1 (M, g (t)) denote the space of pointed hyperbolic limits corresponding

to sequences {(xi, ti)} with Xi ti. Mo,Ao(ti) (ti)· Let (1-l{,h1) E 5Jl)p 1 (M,g(t)) be a

limit with the minimal number of cusp ends. In view of (33.39) and by adjusting

the basepoints in the sequence if necessary, we have that there exist x~ E 1-l 1 and

(x}, t}) E U (M - Mo,Ao(t) (t)) x {t}

tE(To,oo)

with inj 9 (tt)(x}) > c and such that { (M, g (t + t}), x})} converges in the C^00

pointed Cheeger-Gromov sense to (1-l 1 ,h1,x~).

We have the following, whose verification shall closely follow the proof of The-

orem 33.17, which in turn shall depend on the proof of Proposition 33 .16.

Claim. Corresponding to ( 1-l{, h 1 , x~) there exist T 1 E [To, oo), a function

A1 : [T1, oo) --+ (0, A*] decreasing to zero, and a stable asymptotically hyperbolic

submanif old

Mi,Ai(t) (t) CM - Mo,Ao(t) (t), t E [T1, oo).

PROOF OF THE CLAIM.

STEP 1. Existence of an immortal almost hyperbolic piece disjoint from

Mo,Ao(t) (t). Let A E (0, A*). Since the sequence { (M, g (t}), x})} converges

to (1-l1, h1, x~) and by (33.19), for each i sufficiently large we have that t; ;:::: To

and that there exists a harmonic diffeomorphism

F1,i: ((1-l1)A, h1l(1i1)J--+ (F1,i((1-l1)A),g (ti))

satisfying the CMC boundary conditions and limi---+oo dh 1 (F 1 -/(x}), x~) = 0. More-

over, by (33.20) we have '

ilim ---+oo llFl ,i * g (ti^1 ) - h 1 I (1i1)A II Ck((1ii)A,h1) --^0

for each k EN U {O}.
Define T 1 ~ t}, where i is to be chosen below. Denote Mi,A(T 1 ) ~ Fi,i((1-l 1 )A)
and F1(T 1 ) ~ Fi,i· In this new notation, we have that

F1(T1): ((1-l1)A, h1l(1ii)J--+ (M1,A(T1), g(T1))