1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

232 33. NONCOMPACT HYPERBOLIC LIMITS

This simple fact follows directly from Corollary 31.47 since the image of such a
2-torus under an almost isometry is a 2-torus with the same properties in H , which
must then be contained in one of the maximal cusp ends of H.

Subclaim. There exists k EN such that for any k' > k 2 k, if pCk'l(t<k'))(1i1jk')

n p(kl(t<k'l)(Hi;k ) =J. 0, then

(33.37) p(k) ( t) (1i1;k) c p(k') ( t) (1i1jk')

for all t E [tCk'l, oo ).

Proof of the subclaim. To prove the subclaim, it suffices to show that the same
statement holds with the conclusion replaced by

(33.38)

i.e., it suffices to show that (33.37) holds for t = tCk'l. To see this, recall t hat

p (kl(t)(81i 1 ;k ) depends continuously on t and is a finite disjoint union of almost
flat CMC almost umbillic tori , with each torus component having area 1 /k. Because
of this and by adjusting k larger if necessary to ensure that (F(k) (t) (Hi; k), g ( t))

is sufficiently close to hyperbolic for any t E [t(k), oo) and k 2 k, it follows from
Lemma 33.18 that we have (33.37) for all t E [tCk'l, oo).
Now we prove (33.38). Let

1i(k) ~ p(kl(tCk'l)(H1;k) and H(k) ~ pCk'l(tCk'l)(H1fk'),

so that by the subclaim hypothesis, H(k) n 1i(k) =J. 0. Now 81i(k) = Ti U · · · U 'Tm is

a disjoint union of embedded 2-tori. Essentially by Lemma 33.18, for each i, either

T;, c int(H(k)) or T;, n H(k) = 0.

In the former case, T;, is contained in an almost hyperbolic cusp region of H(k). We
shall show that t he latter case is not possible.
Suppose that there exists at least one 2-torus; without loss of generality assume
that it is l m, such that l m nH(k) = 0. Let Xo E H(k) nH(k)· Choose any X1 Elm
and let a : [O, L] -+ (M, g( tCk'l)) be a geodesic joining x 0 to x 1 , minimizing length
among all paths in 1i(k). By the geometry of the almost hyperbolic cusp regions ,

a c int(H (k)). Now a intersects H(k)" Indeed , define

uo ~inf{ u E [O, L] : a (u) tt H(k)} < L.

Then a (uo) E 81i(k) and ai(uo,L] does not intersect H(k)· Let 'T' be the 2-torus
boundary component of H(k) which contains a (uo). We then have, essentially by
Lemma 33.18, that 'T' C 1-l(k)· This is a contradiction since then 'T' must be an

almost hyperbolic cusp region of H(k) but it has too small area 1/k' since k' > k.

This completes the proof of the subclaim.

Finally we finish the proof of the claim. By the subclaim, it is sufficient to show
that there exists a subsequence {kr}rEN such that {tCkr)}rEN is increasing and

pCkrl(tCk.))(1i1/kr ) n p(k.l(tCk.))(H1;kJ =J. 0

for all 1 :::; r < s < oo. We leave it as an exercise for the i·eader to verify that this

follows from the immortality of almost hyperbolic pieces (see Proposition 33.16),