220 33. NONCOMPACT HYPERBOLIC LIMITSPROPOSITION 33.13 (Existence of harmonic parametrizations of almost hy-
perbolic pieces). Let { (M~, gi, xi ) };EN be a sequence of complete Riemannian 3-
manifolds converging in the C^00 pointed Cheeger-Gromov sense to a noncompact
finite-volume hyperbolic manifold (H^3 , h, x 00 ). If A E (0, ../3/8] is such that Xoo E
int (HA), then for each i sufficiently large there exists a harmonic embeddingFi : (HA, hl 11 J---+ (Mi, 9i)
satisfying the CMG boundary conditions. Moreover, we havei~~ llFtgi - hl11Jck(HA,h) = 0
for all k E N and we have limi-7 00 d 9 ; (Fi (x 00 ), xi) = 0.
The idea of the proof is simply the following. By the definition of Cheeger-
Gromov convergence, there exist embeddings<I>i : (HA;2, hi.,_, "'A/2 ) ---+ (Mi, gi ) ,which approach isometries in the sense that ll<I>igi - hi.,_, II ---+ 0 for each
"'A/2 Ck(HA12)
k as i ---+ oo. For each topological end E of H and each A E (0, ../3/8] there existsa CMC torus Ti~ c int(HA; 2 ) , with respect to igi. and with area equal to A.
The almost hyperbolic piece (Hi,A, <I>igi), defined to be the submanifold bounded
by the union of the T;~' , limits to (HA, hi.,_, n.A ).
Essentially by Proposition 33.12 we have that for i large enough there exists
a harmonic diffeomorphism Gi : (HA, hl 11 J ---+ (Hi, A, <I>i gi) such that <I>i o Gi :
(HA, hl 11 J ---+ (Mi, gi) satisfies the CMC boundary conditions. These embeddings
<I>i o Gi limit to isometries as i ---+ oo.PROOF. As a consequence of the definition of C^00 pointed Cheeger- Gromov
convergence, for each A E (0, ../3/8] there exists a sequence of embeddings<l>i : (HA;2, hi"' Tl-A/2 ) ---+ (Mi, gi),defined for i large enough, such that i (x 00 ) =Xi and for all k E N,
ll<l>igi - hl1{A/2llCk(1{A/2)-+ 0.By Proposition 33.11, for i sufficiently large, there exists a smooth 1-parameter
family {LJEE£(K) Ts (E, i, A)}sE:I of unions of CMC tori in HA;2, with respect to
i gi. Here, the CMC tori are close to the standard tori slices sweeping out H 2 A; 3 -
H 3 A ; 2 and the subset K C H is a suitable end-complementary compact set. The
areas of the standard CMC slices in each component of H 2 A; 3 - H 3 A; 2 , with respect
to h , take on all values in [2A/3, 3A/2] and the areas of the CMC tori Ts (E, i, A)
in the sweep-out depend continuously on s. Hence, for i large enough and each
EE E (K), there exists a torus Ti~ in the sweep-out which is CMC and which has
area A , both with respect to ig~.
Let Hi,A denote the unique compact 3-dimensional submanifold ofH with CMC
boundary
oHi,A = u 7;~ ·
EE£(K)