1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

250 33. NONCOMPACT HYPERBOLIC LIMITS

Since

ec;p 1P

lim _ L(p)dp=L(O),

p->0 1 - e-P O

we deduce from Lemma 33.34 and (33.67) the following.

COROLLARY 33.35. Let 0 < B < A :::; A. For c > 0 sufficiently small there

exists tc; < oo such that for all t ;::: tc; and all p E (0, PA,B(t)] we have that

ecP 1P e£P

(33.77) Lg(t)(8V(t)) =L(O) :S l _ - L(p)dp:S l _ _ A(p),

-e P 0 -e P

where A (p) is defined by (33.66). In particular,

e£PA,B (t)

L g(t)(8V(t)) :S 1 - e _ PA,B (t) A (PA,B(t)).

An immediate consequence is that area almost bounds length. Namely,

COROLLARY 33.36. Let A E ( 0, Ji). For any 15 > 0 there exist B E (0, A)

and T 0 < oo depending also on B such that at any t ;::: T 0 we have both PA,B(t) ;:::

-ln(l - ~) ~ N and

(33.78) L g(t) (8V(t)) :S (1+15) A (N) :S (1+15) A (PA,B(t)).

PROOF. By Remark 33.33, there exist Tfi < oo and B E (0, A) such that

PA,B(t) ;::: N for t ;::: Tfi. Now choose c > 0 small enough so that ec;N :::; Jf+J.

Then, by Corollary 33.35 and since 1 _;-N = Jf+J, there exists T 0 ;::: Tfi such
that for t ;::: T 0 we have that

ec;N

L g(t)(8V(t)) :S l _ e-N A (N) :S (1+15) A (N). D

Since

A 'JA (t) =Area g(t) (V (t)) ;::: A (PA,B(t)),

it follows from (33.78) that for any 15 > 0 there exists T 0 < oo such that fort;:=: T 0 ,

L 9 ctJ(8V(t)) :S (1+15)A'J(t).

So the length is almost less than the area.
The following says that somewhere far into the cusp, the length of the inter-
section of the minimal disk with a torus slice is not too large.

LEMMA 33 .37. Given A E (0, A) and given c > 0 sufficiently small, let tc; be
given by Lemma 33.28. Let B E (0, A) and T 2 c; < oo be as in Corollary 33.36.
Suppose there exists t<> ;::: max{tc;, T 2 c;} such that (33.62) holds. Then there exists
A E (0, oo) depending only on c with the following property.
For any P# ;::: -ln(l-v'l~ 2 c;) and p* E [P# +A,pA,B(t<>)] there exists P• E

[p#, p*] such that SA,p.,. (to) = V(to) nTA,p.,. (to) is a finite disjoint union of smooth

embedded loops and

(33.79) Lg(to) (p.,) :S (1+2c) e-(p*-p.,.) Lg(to) (0).

REMARK 33.38. Note that for P# +A :::; PA,B(t<>) to hold, we need B to be

sufficiently small.