- PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 235
is a harmonic diffeomorphism satisfying the CMC boundary conditions and that(33.41) II Fi (Ti)* g(Ti) - hi 1(1i1)A II. :::; _kl'
Ck((1i1)A,h1)
where k E N is as large as we like, by taking i sufficiently large.
By applying Lemma 33.19 to (33.41) and to Mo,Ao(t) (t) being a stable asymp-
totically hyperbolic submanifold, we have for i sufficiently large that
Mo,Ao(Ti)(Ti) n Mi,A(Ti) = 0.
We shall complete Step 1 by establishing the following statement. For any
k EN, by choosing i sufficiently large, one can continue Fi(T 1 ) as harmonic diffeo-
morphisms
F1(t): ((H1)A, h1l(1ii)J-+ (M1,A(t),g(t)), t E [T1,oo),
where Mi,A(t) ~ Fi(t)((Hi)A),(33.42)and(33.43) Mo,Ao(t)(t) n Mi,A(t) = 0.
Hence Mi,A(t) is an immortal almost hyperbolic piece disjoint from the submani-
fold Mo,Ao(t)(t).
The statement is a consequence of the proof of Proposition 33.16 for the fol-
lowing reasons:
(1) By Lemma 33.19 and by taking k sufficiently large, as long as (33.42)
remains true, so does (33.43). To wit, as long as the pieces are close to hyperbolic,
they are disjoint.
(2) As long as (33.43) remains true, we may apply the proof of Proposition
33.16 to derive (33.42).
The reason for (2) is that if we ever have equality in (33.42) at some times
f3i E [ t}, oo) for a sequence of i tending to oo (the dependence on i therein is
implicit since F 1 (T 1 ) = Fi,i), we can then apply the argument in the part of the
proof of Proposition 33.16 between the displays (33.22) and (33.33) to obtain a new
-3 -
hyperbolic limit (Hi, hi) contained in fJIJPi(M,g(t)) (corresponding to (33.23)).
Since (H 1 ,h 1 ) has the minimal number of cusp ends in fJIJPi(M,g(t)), we have
that (Hi, h 1 ) has at least as many cusp ends as (H 1 , h 1 ). We may now argue (using
the Mostow rigidity theorem) as in the paragraph containing (33.33) to obtain a
contradiction.
STEP 2. Existence of a stable asymptotically hyperbolic piece disjoint from
M~,Ao(t) (t). Now, by applying the proof of Theorem 33.17 to the immortal almost
hyperbolic piece Mi,A 1 (t) CM - Mo,Ao(t) (t) corresponding to (H1, h1), we have
that (Hi, hi) is a stable hyperbolic limit. Moreover, we have a corresponding stable
asymptotically hyperbolic submanifold Mi,Ai(t) (t), t E [Ti, oo), with
M1,A 1 (t) C M1,Ai(t) (t) CM - Mo,Ao(t) (t).
This completes the proof of the claim.