1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

244 33. NONCOMPACT HYPERBOLIC LIMITS

Since v^0 (to) -7 0 as t 0 -7 oo, there exists t"' (3) < oo such that for to 2 t"' (3),

(33.57) { V^0 (to) ds (to) :Sc L g(to) (8D (to)).

lav(to)

Hence, given A E (0, A) and c > 0, from (33.54), (33.55), (33.56), and (33.57) we

have shown for t 0 2: max { t"' (1) , t"' (2) , t"' (3)} that

d+ A~

T, (to) :S -27r + (1+2c) L g(to) (8D (to)) - (1-c) A~ (to).

To summarize, we have

LEMMA 33.28 (Evolution of the area of a minimizing disk). Assume the hy-

potheses of Proposition 33.24. For any A E (0, A) and c > 0, there exists t"' < oo

such that if t 2 t"', then

(33.58) d+d~ ~ (t) :S -27r + (1 + c)^112 L g(t) (8D (t)) - (1 + c)-^1 /^2 A~ (t),

where the area of D (t) E J (t) equals A~ (t).

Thus we have reduced the proof of Proposition 33.24 to bounding the length
of 8D (t) by the area of D (t); this is accomplished by Lemmas 33.30 and 33.41 in
Subsection 4.4.

4.3. D isks in hyperbolic and almost hyperbolic cusps.
The purpose of this subsection is to present a couple of elementary facts which
help motivate and prepare for the proof of Proposition 33.40 below, which in turn
is used to prove the main Proposition 33.24.

Consider a hyperbolic cusp (V^2 x IR, 9cusp), where 9cusp = dr^2 + e-^2 r 9flat and

9flat is a fl.at metric on a 2-torus V. Let "Y C V be a path. Then "'( x {p} has length
e-P L 9 nat ("Y) and hence

(33.59)

Moreover, the area of the surface "Y x [O, ,8] C V x [O, oo) is given by

Area 9 cusp ("Y x [O, p]) =for; e-P L 9 nat ("Y) dp = (1 - e-P) L 9 nat ("Y).

Thus the quantity

(33.60)

1

1-e-P Area9cusp ("Y X [O,p]) = L9nat ("Y)

is independent of p. Note also that Area 9 cusp ("Y X [O,oo)) = L 9 nat ("'().

Next we shall observe that the intersection of a smooth embedded disk with
tori slices in an almost hyperbolic piece is almost always a finite disjoint union of
smooth embedded loops.
Let N^3 be a compact 3-manifold , let I:^2 be a boundary component, and let
C be an open collar of I:, which we identify via a diffeomorphism with I: x [a, b),

where I: is identified with I: x {a}. Suppose that f : D^2 -7 N is an embedding of

a disk with f (8D) c I:. Let U^2 ~ f-^1 (C) and write

f ~ (11, h): U -7 I: x [a,b),