1547671870-The_Ricci_Flow__Chow

154 5. THE RICCI FLOW ON SURFACES

injectivity radius estimate of Proposition 5.65, we proceed as in Proposi-
tion 5.51 to obtain a uniform upper bound for R. As in Corollary 5.52,
this implies an upper bound for the diameter. Then we use the differential
Harnack estimate of Proposition 5.61 to argue that R eventually becomes
strictly positive within finite time. Once this happens, the proof of Theorem
5.64 goes through exactly as written.
Our first result is an analog of Proposition 5.51.

LEMMA 5.74. If (M^2 , g (t)) is a solution of the normalized Ricci flow on

a closed surface of positive Euler characteristic, then there exists a constant

C > 0 depending only on go such that

sup R :SC.

M^2 x[O,oo)

PROOF. By Proposition 5.18, there exists C1 > 0 depending only on go

such that supM2x[O,l] R :Sr+ Gier. Given T ~ 1, define

K,(T) ~ max R.

M2x[O,T]

We intend to prove that K, is bounded independently of T. We may assume
that /'i,(T) > max{K,1,1/4}, so that T > l. Choose (x 1 ,t 1 ) E M^2 x (1,T]

such that R (x1, t1) = maxM2x[O,T] R = /'\, (T). Following the proof of Propo-

sition 5.51, one shows that

(5.45)

for ally E Bg(ti) (x1, 1/~). Define p ~ inj (M^2 ,go). Then by Proposi-

tion 5.65, one has

inj (M^2 ,g(t)) ~ min{p, ~}.

As long as p :S 7r / .JK,, we have the uniform upper bound K, :S ( 7r / p )^2. So we

may assume p > 7r / .JK,. By Proposition 5.44, there exists C2 > 0 such that

C 2 ~N(g(t 1 ))~ { (R-s)log(R- s)dA

}M2

~ 1 1 (R - s) log (R - s) dA

B 9 (ti) ( x1, V641<)

• ~Area(M^2 ,g(ti)),
  e

where s = s (t1) :S 0 is given by formula (5.28). By the area comparison

theorem (§3.4 of [25]) and estimate (5.45), there exists c > 0 such that

1 1 ( R - s) log ( R - s) dA ~ ~ ( ~ - s) log ( ~ - s)

B 9 (ti) ( x1, ~)

c /'\,

>- 2 -log-. 2