- INCOMPRESSIBILITY OF BOUNDARY TORI 245
so that h (8D) = a. By Sard's theorem, almost every r E (a, b) is a regular
value of h. For su ch r we have that the map f is transversal to the slice L: x { r}
and hence f-^1 (L: x {r}) = f2^1 (r) is a 1-dimensional submanifold of D , whichmust be a disjoint union of loops. Since f is an embedding, for such r, the set
f (D) n (I: x {r}) is a disjoint union of smooth embedded loops.
In particular, we have the following:
LEMMA 33. 29 (Embedded disks intersecting tori slices). Let f be a smooth
embedding of a disk D^2 into a differentiable 3-manifold containing a boundary collar
T^2 x [a, b). Suppose that f (8D) C Tx {a} and write f ~ (! 1 , h) on f-^1 (Tx [a, b)).
Then almost every z E (a , b) is a regular value of h. For such z the map f is
transversal to the torus slice T x {z} and hence f2^1 ({z}) = f-^1 (T x {z}), as wellas f (D) n (T x { z}), is a disjoint union of smooth embedded loops.
We shall apply the above fact to the part of a minimal disk V(t) in an almost
hyperbolic piece of a nonsingular solution. The main idea in what follows is to use
comparison disks (see (33.72) below) to understand the area of the minimal disk
and the length of its boundary circle.
4.4. Bounding length by area for minimal disks.
Let MA (t), where A E (0, A), be an immortal asymptotically hyperbolicpiece corresponding to a hyperbolic limit (ti, h , x 00 ) , as given by Proposition 33.6.
Throughout this subsection we sh all assume, as in the hypothesis of Proposition
33.24, that i: TA(t) '---+ M - MA (t) satisfies ker(i*) "I-{1}; that is, TA(t) is com-
pressible in M - MA (t). As in (33.46) we have a smooth embedded minimal disk
V(t) CM - MA (t) with Areag(t) (V(t)) = A-;g (t). Observe that by Lemma 33. 28
we have
LEMMA 33.30. Let A E (0, A) and c > 0. If t sufficiently large is such that
Lg(t) (8V(t)):::; (1 +c)-^1 A-;gA (t), then
(33.61) _d+A-;g_A (t) < -2w.
dt -
So the proof of Proposition 33.24 is reduced to the complementary case where
(33.62) Lg(t) (8V(t)) > (1 + c)-
1
A-;g (t).The rest of this subsection is devoted to this case. We shall show that by choosing
A E (0, A) small enough and t sufficiently large, we may assume that the length
Lg(t) (8V(t)) is as small as we like (see (33.86) below).
Recall that for any B E (0 , A) and for t large enough, an immortal asymp-
totically hyperbolic piece MB (t) corresponding to the hyperbolic limit (ti, h , x 00 )
satisfies MA (t) c MB (t).
DEFINITION 33.31. Let C~ B (t) C M denote the almost hyperbolic cusp
region bounded by both TA (t)' C 8MA (t) and the corresponding boundary torus
in 8MB (t).
For part of the following discussion, we shall suppress the dependence of t,
which shall be assumed to be sufficiently large, in our notation.
Let NTA denote the normal bundle of TA, and let v be the unit normal field
to TA pointing into CA,B· Given Co > 0, define
L:o ~ {pv(x): x ETA, p E [O, Co]} C NTA·