1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

252 33. NONCOMPACT HYPERBOLIC LIMITS

(3) Let D~,P• c D^2 be the disk bounded by the loop ft~^1 (SA,p.)· Define the

immersed comparison disk

. - 2 ~~
(33.80) V A,p• ::::;:: ft 0 (D A,p•) U AA,p• U BA,p•.

Here we have taken license with the way we defined V A,p• since this disk is immersed
and may not be embedded. One may easily make this definition technically correct.

0

By definition,

(33.81)

s A ,p,..

BA,p.

FIGURE 33.7. The comparison disk VA,p• ·

Now we proceed to estimate Area(V A,p.) from above.

The following coarse estimate is used to bound the area of AA,p•.

0

LEMMA 33 .39. Let T2 be a fiat torus and let [ex] E ?T 1 (T). If in T an embedded
geodesic loop£ and a smooth embedded loop 51 both represent [ex], then there exists

a geodesic loop £' parallel to £ and an immersed annulus A bounded by 51 U £'

with^14
Area (A) :::; Length(£) · Length(5^1 ).

a

a

FIGURE 33 .8.

(^14) Note that Length (.C') =Length (.C).