1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 231

STEP 1. Reduction to a claim. Theorem 33.17 is a consequence of the following.

Claim. There exists a subsequence { kr} rEN such that { t(kr) }rEN is a strictly

increasing sequence and such that

(33.36)

for all t E [t(ksl,oo) and 1::::; r < s < oo.

Indeed, assume that the claim is true. Without loss of generality, we may

assume that k 1 satisfies ;; 1 ::::; A. Choose a smooth nonincreasing function A (t),

defined for t sufficiently large, satisfying

2

A(t(kr)) = -- for r ~ 2.

kr-1

Given any t > t(ki), there exists a unique r(t) EN - {1} such that t E (t(kr<•>-^1 l ,

t(kr<t>l]. We then have that A (t) ~ -k r(t)-1^2 and by (33.35) that each harmonic

embedding

p(krc•>-1) (t)/ : (1iA(t)i h)--+ (M, g (t)), t E [t(krc•>-1l, oo),

1-{A(t)

is a -k-r(t)-1^1 - -almost isometry with the image p(krc•>-^1 l (t)I~, rLA(t) (81iA(t)) of the bound-

ary consisting of a disjoint union of almost totally umbillic tori with respect to

g (t). By Proposition 33 .12, for each t > t(k^1 ) sufficiently large, close to the map

p(kr-^1 ) (t)I.,_, (81iA(t)) there exists a unique almost isometric harmonic em-
rLA(•J
bedding

F(t): (HA(t),h)--+ (M,g(t)),

where F (t) (81iA(t)) is a disjoint union of CMC tori, each with area equal to A (t),

and where F (t)* (N) is normal to F (t) (81iA(t)), all with respect tog (t). Since

ll

p(kr(t)- il (t)* g (t) - hl1l1 k II ::::; _ 1 _

I r(t)-1 Ck(Jl l/kr(t)-1'. h) kr(t)-1

and -k r(t)-1 -^1 - ::::; ~A ( t), one easily concludes that

t-+oo lim llF (t)* g (t) - hi 1-lA(t) II Ce(1lA(t)>h) =^0

for each£ E N.
Moreover, by (33.36) and by Proposition 33.12, it follows that the maps F (t)
depend smoothly on t. In particular, by the uniqueness of harmonic diffeomor-
phisms near the identity, we have the important fact that there is no discontinuity

of F (t) at any t = t(kr). The maps F (t), defined for all t sufficiently large, satisfy

all the conditions of Definition 33.2. Hence Theorem 33.17 is proved assuming the
claim.

STEP 2. Proof of the claim. First observe the following.

LEMMA 33.18 (Almost fiat 2-tori with small area). If '/2 is an embedded 2-torus

in an almost hyperbolic piece in M^3 corresponding to 1i^3 , with bounded intrinsic

and extrinsic curvature and sufficiently small area, then T is contained in an almost

hyperbolic cusp region of the almost hyperbolic piece.