- MAIN RESULTS ON HYPERBOLIC PIECES 217
(2) For each t E [T, oo),llF (t)* g (t) - hl1iA llCk(1iA,h) ~ ~·Here, the immortal almost hyperbolic piece MA (t) remains close- by a fixed
distance in Ck- to being hyperbolic for all time, whereas in the definition of sta-
bility the piece MA(t) (t) asymptotically limits to a hyperbolic manifold. Note that
the diffeomorphism types of MA(t) (t) and MA (t) are both independent oft.
In Definition 33.4, we say that the immortal (A, k)-almost hyperbolic piece
corresponds to the hyperbolic 3-manifold (H, h). By the Mostow rigidity theorem
(see Corollary 31.50), the corresponding hyperbolic manifold (H, h) is unique.
The preliminary version of the result that almost hyperbolic pieces cannot jump
around in M is the following.
PROPOSITION 33 .5 (Immortality of almost hyperbolic pieces). Suppose that
( M^3 , g ( t)) is a nonsingular solution satisfying Condition H. If { (Xi, ti)} is a se-quence of points and times with ti ---+ oo such that the sequence (M, gi (t), xi),
where gi (t) = g (t +ti), converges to a finite-volume hyperbolic limit (H^3 , h , x=)as i ---+ oo, then for any A E (0, A] and for any k,^3 there exist T < oo and an im-
mortal (A, k )-almost hyperboli c piece MA ( t) defined for t E [T, oo) corresponding
to (H, h).Using that the parameters (A, k) may be chosen to be arbitrarily good, we shall
show that a ny immortal (Ao, ko)-almost hyperbolic piece MAo (t) corresponding to
(H, h), with (A 0 , k 0 ) sufficiently good, must be eventually contained in a stable
asymptotically hyperbolic submanifold MA(t) (t) corresponding to (H, h). This
improved version of Proposition 33.5 is summarized in the first paragraph of §12 of
[143], where Hamilton wrote:
For large t the metric is as close as we like to hyperbolic; not just
on HB but as far beyond as we like.
PROPOSITION 33.6 (Stability of hyperbolic limits). Let (M^3 , g (t)) be a nonsin-
gular solution satisfying Condition H. If {(xi, ti)} is a sequence of points and times
with ti ---+ oo such that the sequence { (M, gi (t) , xi)}, where gi (t) = g ( t +ti), con-
verges to a finite-volume hyperbolic limit (H^3 , h , x=) as i ---+ oo, then (H, h) is a
stable hyperbolic limit of (M, g (t)).
The following notion, which lies between those of stable hyperbolic limit and
immortal (A, k)-almost hyperbolic piece, shall be useful when we consider the in-
compressibility of tori in §4 of this ch apter.
DEFINITION 33.7. Let (M^3 ,g (t)) be a nonsingular solution satisfying Con-
dition H. Given A E (0, v'3/4], we say that a smooth family of submanifolds
M~ (t) c M, t E [tA, oo), where tA < oo, is an immortal asymptotically
hyperbolic piece corresponding to a hyperbolic limit (H^3 , h) if
(1) 8MA (t) is comprised of CMC tori with area A,
(2) (MA (t), g(t)) converges to (HA, h) inc= as t---+ oo.(^3) Here, A is defined to satisfy both (h5) in Subsection 1.1 of this chapter and the condition
in the paragraph containing (33.18) below.