1547671870-The_Ricci_Flow__Chow

56 2. SPECIAL AND LIMIT SOLUTIONS

Taking C larger if necessary, we may assume by Lemma 5.4 that If; log L :S C
as well. This implies that

1/Jss < 1 - 1/;; C

o/, - o/,2 1 -o/,2

'I-' 'I-' log7

and thus

1 - 1/;2

1/;1/Jss :S C log(l - 1/;;) ~ 2log1/; ·

Hence

1/J1/Jss < C 1

1 - 1/;; - - 2log1/; 1 _ llog(l-7/J;) ·

2 log7/J

Since we have restricted attention to the region where 1/; :Sc: and 1 1/lsl :S c:,

the denominator of the second factor on the right-hand side obeys the bound

l log(l - 1/;;) l log(l - c:^2 )

1-- >1--.

2 log 1/; - 2 log c:

So by choosing c: small enough, we can ensure that

1/J1/Jss < C.

1 - 1/;; - - log 1/;

We now further restrict our attention to the region to the right of the

neck. There 1/Js > 0, which allows us to choose the radius 1/; as a coordinate

and thereby regard all quantities as functions of 1/;. Then we have

d ( ( 2)) 21/Js1/Jss 1/; C

- d log 1/; log^1 - 1/Js = 1 -1/;;. 1/Js :S - log 1/;.

Integrating this differential inequality from the center of the neck (where

1/; = rmin and 1/ls = 0) to an arbitrary point, one gets

• log ( 1 - 1/; 8 2) :S 17/J C ( ) logrmin
  1

d log u = C log

1

1/;.

U=Tmin - 0g U 0g

Using the calculus inequalities x :S - log(l - x) and logx :S x - 1, one then

obtains

(2.63)^01 '!-' ,2 (^8) - <Cl og logrmin log 1/; -< C (logrmin log 1/; _ 1 ) ·
Since we are assuming 1 1/lsl :Sc:, this last inequality will only be useful if the
right-hand side is no more than c:^2. Henceforth we assume that

(2.64) (

1 )c:2/2C

r min :S 1/; :S - .- r min.

rmm

Then using the fact that e-c:^2 /C :S 1 - c:^2 /2C for small c:, one finds that

(2.64) and (2.63) imply that 11/lsl :SE and 1/; < c:, as required.