216 33. NONCOMPACT HYPERBOLIC LIMITS
DEFINITION 33.2 (Stable hyperbolic limit). Let (M^3 , g(t)) be a nonsingular
solution satisfying Condition H. We say that a (connected, complete, and non-
compact) finite-volume hyperbolic limit (H^3 , h) is a stable hyperbolic limit of
(M, g (t)) as t--+ oo provided the following conditions hold. There exist
(1) To < oo and a smooth nonincreasing function A : [To, oo) --+ (0, A] with
limt-; 00 A(t) = 0,
(2) a smooth 1-parameter family of compact submanifolds M~(t) (t) c M
with tori boundary defined for t E [To, oo),
(3) a smooth 1-parameter family of harmonic diffeomorphisms
F (t): (HA(t)l hl1lA(t)--+ (MA(t) (t) ,g (t)))
defined for t E [To, oo) and satisfying the CMC boundary conditions,
with the property that for each m E N(33.3) t~~ llF (t)* g (t) - hl1lA(t) llcm(fi'.A(t)>h) = 0.
We shall call (MA(t) (t), g (t)), t E [To, oo ), a stable asymptotically hyperbolicsubmanifold in (M, g (t)) corresponding to (H, h).
REMARK 33.3 (Regarding the definition of stable hyperbolic limit).
(1) The 1-parameter family of submanifolds MA(t) (t) being smooth is the
same as 8MA(t) (t) depending smoothly on t and MA(t) (t) staying on
the same side of 8MA(t) (t).
(2) Ast--+ oo the 1-parameter family (MA(t) (t) , g (t)) of Riemannian man-
ifolds converges to (H, h) in the C^00 pointed Cheeger- Gromov sense.
(3) For A E (0, v'3/4] and fort sufficiently large with A (t) :::; A, we h ave that
F (t) (8HA) is a disjoint union of embedded concave tori in M and tendsto totally umbillic as t--+ oo, all with respect tog (t).
(4) Even if we fix the function A(t), a corresponding stable asymptotically hy-
perbolic submanifold may not be unique. For example, we could imagine
the existence of a Z 2 -symmetric nonsingular solution to NRF obtained by
smoothly doubling a truncated hyperbolic manifold, which may have two
disjoint isometric stable asymptotically hyperbolic submanifolds. Can one
prove that such an example is possible?
A useful notion to start with, which is a priori weaker than that of stable
hyperbolic limit, is the following.DEFINITION 33.4 (Immortal almost hyperbolic piece). Let (M^3 ,g(t)) be a
nonsingular solution satisfying Condition H. Fix A E (0, v'3/4] and k EN. We say
that a family of compact 3-dimensional submanifolds MA (t), defined fort E [T, oo)
for some T::::: 0 , of (M, g (t)) is an immortal (A, k)-almost hyperbolic piece if
there exists a finite-volume hyperbolic manifold (H^3 , h) and there exist harmonic
diffeomorphisms
(33.4)satisfying the following properties:
(1) The maps F (t) depend smoothly on t E [T, oo) and satisfy the CMC
boundary conditions. Hence each connected component of 8MA (t) de-
pends smoothly on t and has area equ al to A.