1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 227

By property (1), the sequence of pointed closed Riemannian 3-manifolds

(33.18)

converges in the C^00 pointed Cheeger- Gromov sense to (ti, h, x 00 ). Assume that

A E (0, A] and recall that x 00 E int(H jjJ By Proposition 33 .12, for i sufficiently

large, there exist harmonic embeddings

(33.19)

where Fi (81iA) = 8(Fi (HA)) is a disjoint union of CMC tori, each with area

equal to A, and where (Fi)* (N) is normal to 8(Fi (HA)), all with respect tog (ti)·

Furthermore,

(33.20) t-t Hm llFtg (ti) - hlH. A lick("' rLA, h) = 0

for each k EN U {O} and

Hm dg(t,) (Fi (xoo), Xi) = 0.

i-+oo

In particular, given A E (0, A] and k E N, there exists io = io(A, k) such that for
any i ~ io we have

llFtg (ti) - hlH.JCk(H.A,h):::; ~;

that is, (Fi('fiA), g (ti )) is an (A, k)-almost hyperbolic piece at time k
We first prove that given a hyperbolic limit with a minimal number of cusp
ends and given (A , k), for each i sufficiently large the (A, k)-almost hyperbolic
piece (Fi('fiA), g (ti)) can be smoothly continued to an immortal almost hyperbolic
piece. This is a special case of Proposition 33.5 and the use of Mostow rigidity is
key to its proof.

PROPOSITION 33.16 (Almost hyperbolic pieces with minimal cusp ends are

immortal). Suppose (ti^3 , h) E fJIJp(M,g(t)) is a hyperbolic limit with a minimal

number of cusp ends. Let A E (0, A], where A is as in (h5), and let k E N. Then

there exist:;;= :;;(A, k) EN, submanifolds M~ k (t) c M, and a smooth I-parameter

family of harmonic diffeomorphisms '

(33.21a)

(33.21b)

F;;(t) : (HA, hlH.A) --+ (MA,k(t), g (t)),

MA,k (t,) = F, (HA), F 2 (t 2 ) = F,,

defined fort E [t 2 , oo), where F 2 is defined by (33.19) and where

(i) 8MA,k (t) = F 2 (t) (81iA) is a disjoint union of CMG tori, each with area

equal to A, and F 2 (t)* (N) is normal to 8MA,k (t), all with respect to

g (t), and

(ii)

llF 2 (t)* g (t) - hlH.A llck(H.A,h) :::; ~
for all t E [t 2 , oo).
Hence MA,k (t), t E [t 2 , oo), is an immortal (A, k)-almost hyperbolic piece.

PROOF. Given A E (0 , A], let f, be given by property (3) above and let k (£)EN
be as in Theorem 34. 22 in the next chapter. Then define ko ~max{ k (£), ko}, where
ko is as in Proposition 33.14.^9

(^9) The natural number ko depends only on 1-l and A.