226 33. NONCOMPACT HYPERBOLIC LIMITS(33.11). Then, by the fact that F 2 (t) (8HA) = F 1 (t) (8HA) from Statement A
and by the uniqueness part of Proposition 33.12, we have that F2 (t) = F1 (t) for
t E [a , t 1 ]. Proposition 33.14 is now proved. 0
3. Proof of the stability of hyperbolic limits
In this section we discuss the proof of Proposition 33.6 on the stability of
(finite-volume hyperbolic) asymptotic limits of 3-dimensional nonsingular solutions
satisfying Condition H. The difficulty is that we only have sequential convergence
of the corresponding time-translated pointed solutions. The facts which we wish
to exploit to overcome the aforementioned difficulty are (1) in the "negative" Case
III all noncollapsed asymptotic limits are hyperbolic and (2) the Mostow rigidity
theorem (see the discussion surrounding (33.33) below). Throughout this section
(M^3 , g (t)) shall denote a nonsingular solution satisfying Condition H.
We shall prove Proposition 33.6 by induction on the number of cusp ends of
the asymptotic limits. In view of this, we start with the following.
DEFINITION 33.15 (Space of hyperbolic limits). Let fJ1Jp(M^3 , g (t)) denote the
set of all complete hyperbolic 3-manifolds (H^3 , h) which occur as the pointed limit
of the solutions {(M, g (t +ti), xi)} for some sequence {(xi, ti)} with ti-+ oo. We
call (xi, ti) a sequence corresponding to the limit (H, h).
Note that since Vol (g (t)) is constant, if (H, h) E fJl)p(M, g (t)), then by the
Cheeger- Gromov convergence the limit has finite volume:(33.17) Vol(h)::; Vol(g(O)) < oo.
Hence, by Theorem 31.46, the number of cusp ends of each (H, h) is bounded above
by Vol (g (0)).
Now clearly there is an asymptotic limit (H,h) E fJl)p(M,g(t)) such that the
number of cusp ends of (H, h) is minimal among all elements of fJIJp(M, g (t)). In
the next subsection we prove Propositions 33.5 and 33.6 for such an (H, h). Then
in the following subsection we prove the general case by induction on the number
of cusp ends.
3.1. Stability of hyperbolic limits with a minimal number of cusp
ends.
Let (H^3 ,h) E fJl)p(M^3 ,g(t)) be a hyperbolic limit. Recall that we have the
following properties:
(1) There exist a sequence {(xi, ti)}iEN with ti-+ oo and x 00 EH such that the
solutions { (M, g (t +ti), Xi) : t E [O, oo )}iEN converge to the static eternal solution
(H, h, x 00 ) of the NRF in the C^00 pointed Cheeger- Gromov sense.
(2) Let A be as in (h5) in Subsection 1.1 of this chapter and suppose that
A E (0, A]. Then the truncation H A in (33.1) is well defined.
(3) By Proposition 33.12, given A E (0, A], there exists f = f(H , A) E N withthe property that if g is a metric on HA which is close to h in t he sense that
II§ - hl1iA llce(1iA,h) ::; i, then there exists a unique harmonic diffeomorphism
F : (HA, hl1iJ -+ (HA, g)close to the identity map such that F (8HA) = 8HA and F* (N) is normal to 8HA
with respect tog, where N is the unit outward normal vector to 8 HA with respect
to h.