236 33. NONCOMPACT HYPERBOLIC LIMITSthe corresponding hyperbolic limits (H~, hb) E ..IJIJPb(M, g (t)), and the correspond-
ing stable asymptotically hyperboli c submanifolds {M~,Ab(t) (t)}tE[Tb,oo) > where
To ::=;Ti ::=; · · · ::=;Ta and Ab (t) decreases to zero.
Case ( A a). If there exists Ta E [Ta, oo) such that
sup max inj g(t) (x) ::=; t:,
tE[Ta,oo) xEM-LJ~=o Mb,Ab(tJ(t)then we let N = a and stop here. For sufficiently la rge times t , the complement of
u~=O Mb,Ab(t) (t) is €-collapsed. So, fort:::::: TN, (M,g (t)) is the union of N stable
asymptotically hyperboli c submanifolds and an t:-coll apsed piece, whose intersec-
tions are comprised of tori.
Case (Ba). Otherwise, we define ..IJIJPa+i (M, g (t)) to be the space of pointed
hyperbolic limits corresponding to sequences {(xi, ti)} with Xi~ u~=O Mb,Ab(t;) (ti)·
There exists (H~+i • ha+i) E ..IJIJPa+i (M, g (t)) with the least number of cusp ends
and with corresponding sequence {(xf+i, tf+i)}iEN· By arguing as in the claim
above, we h ave a corresponding stable asymptotically hyperbolic submanifold
{M~+i,Aa+i(t) (t)}tE[Ta+i,oo)> where Ta+l ::'.::'. Ta. In this case we continue our in-
duction.We now show that there exists N ::=; 2 Volg(t) (M) < oo such that Case (AN)
holds. For otherwise, we have that {M~,Ab(t) (t) }tE[Tb,oo) is defined for 1 ::=; b ::=; a ,where a > 2 Volg(t) (M). Recall from Theorem 31.46 that Vol(Hb, hb) ::'.::'. 1 for each
b = 0, 1, ... , a. Hence, by choosing t large enough, we h ave for each b = 0, 1, ... , a
that Vol 9 (t)(Mb,Ab(t) (t)) ::'.::'. ~-SinceVol 9 (t)(M) ::'.::'. Vol 9 (t) CQ Mb,Ab(t) (t)) ::'.::'.~(a+ 1),
we have a contradiction.
In summary, we have obtained
(1) a finite collection of sequences{(x?, t?)}iEN, {(xL tt)}iEN, ... , {(xf, t f)}iEN,
(2) associated hyperbolic limits(H6,ho), (Hr,hi), ... , (Hfv,hN),(3) corresponding time-dependent pairwise-disjoint 3-dimensional compact sub-
manifolds of M with boundary
{M6,Ao(t) (t)}tE[To,oo)> {Mi,A 1 (t) (t)}tE[Ti,oo)> · · ·, {Mfv,AN(t) (t)}tE(TN,oo)>where 0 ::=;To::=; Ti::=; .. ·::=; TN< oo and Aj: [Tj,oo)---+ [O,A*) for 0 ::=; j ::=; N ,
which are stable asymptotically hyperbolic submanifolds satisfying
Nmax inj g(t) (x) ::=; t:
xEM-LJa=O Ma,Aa(tJ(t)fort::'.::'. TN; that is , for sufficiently large times, the complement of the union of the
immortal almost hyperbolic pieces has small injectivity radius,