230 33. NONCOMPACT HYPERBOLIC LIMITS
conclude that with respect to h and h the map w 00 is a harmonic embedding
satisfying both
(33.32)II - II
1
w* h-h - -
00 1 7-lA Ck(1-lA,h) - kand the CMC boundary conditions. In particular, w 00 is a n almost isometry, but
not an isometry.
To see the claim, recall that
is a harmonic diffeomorphism satisfying the CMC boundary conditions and (33.31).
Since ig (/Ji) converges to h in C^00 as i^1 (Fi (/Ji) (HA)) converges to HA in C^00
(since 8( i^1 (Fi (/Ji) ( 1i A))) consist of CM C tori of area A with respec t to i g (/Ji)),
by the reg ularity theory for elliptic boundary value problems (for the linear theory,
see the proof of Proposition 34 .13), we obtain uniform estimates for the derivatives
of i^1 o Fi (f3i) up to arbitrary order. Hence, by p assing to a subsequence, the
i^1 o Fi (/Ji) converge to W 00 in C^00 (1iA) as claimed.
Since (il, h) has at least as ma ny cusp ends as (1i, h), we may apply Theorem
34.22 (a consequence of the Mostow rigidity theo rem). Hence, for any f E N and
by choosing ko larger if n ecessary, we have that if k ?: ko, then (1i, h) is isometric
to (il, h) and there exists an isometry
(33.33) I: (H, h) -+ (il, h)
with dce(1-lA,h) (w 00 ,I) < t, where this distance is d efined by (K.13). Now, by
the uniqueness part of Proposition 33.12 and by choosing f, large enough , we have
that w 00 = I. However this contradicts (33.32), which completes the proof of
Proposition 33.16. D
Building on Proposition 33. 16 , we next prove that hyp erbolic limits with min-
imal cusp ends are stable.
THEOREM 33.17 (Stability of hyp erbolic limits with minimal cusp ends). Ahyperbolic 3-manifold (1i^3 ,h) contained in .S'Jl)p(M,g(t)) which attains the mini-
mal number of cusp ends is n ecessarily a stable hyperbolic limit (in the sense ofD efinition 33.2) of (M,g(t)) as t-+ oo.
PROOF. By Proposition 33.16, for each k E N such that A ~ k -^1 :::; A, we have
the following. Abbreviate by t(k) ~ tr(k-1,k) and by F(k)(t) ~ Fr(k-1,k) (t). Then
each
(33.34)is a smooth family of harmonic embeddings. Moreover, F(kl(t)(81i 1 ;k) is a disjoint
union of CMC tori, each with area equal to 1 /k, where F (k)(t)* (N) is normal to
F(k)(t)(81i 1 ;k), all with respect tog (t), and where
(33.35) llF(k)(t)*g (t) - hl1-l11k llck(1-l 11 k>h) :::; ~
for all t E [t<k),oo).