1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

240 33. NONCOMPACT HYPERBOLIC LIMITS

Fix any time t 0 > 0, which will be chosen sufficiently large. Since the boundary

torus TA(t 0 ) in the hypothesis of Proposition 33.24 is convex and is assumed to be
compressible in M - MA(t 0 ) , we may apply Meeks and Yau's theorem, Theorem
33.25. Hence there exists a smooth embedded minimal disk Jo : D -t M - MA (to)
normal to TA (to) such that JolaD : 8D -t TA (to) represents a nontrivial primitive

element of ker (i*) c ?T 1 (TA (to)), where i : TA (to) -t M - MA (to) denotes the
inclusion map and where t he area of

(33.46) V(to) ~ Jo (D)

is least among all maps in J (to).

For the sake of a comparison argument in estimating ~(t 0 ), we shall now

define smoothly varying embedded disks Disk^0 (t) C M - MA (t) for t -t 0 suffi-
ciently small. First, extend the embedding J 0 to a smooth embedding lo defined in

an 77-neighborhood D^2 (1+77) of D = D^2 (1), where 77 > 0 is small. Then define the

embedded comparison disk at time t to be

(33.47) Disk^0 (t) ~lo (D^2 (1+77)) n M - MA (t).

By definition,

(33.48) -d+ d-A 'J (to):::; -d d I Areag(t) (. Disk^0 (t) ).
t t t=to

Note that Disk^0 (to) = Jo(D) = V(to). For t-t 0 sufficiently small, the disk Disk^0 (t)
bounds an embedded loop

Loop^0 (t) ~ 8Disk^0 (t) c TA (t)

and it is easy to see that Disk^0 (t) E J (t).

immortal asymptotically

hyperbolic piece

at time t 0

MA(to)

FIGURE 33.1. The disk V(to) in M - MA (t 0 ).