1547671870-The_Ricci_Flow__Chow

13. THE CASE THAT R(·,O) CHANGES SIGN 155

Since Area (M^2 , g (t1)) = Area (M^2 , go), this implies the desired uniform
upper bound. D

We next bound the diameter of the solution as we did in Corollary 5.52.

COROLLARY 5.75. 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

diam (M^2 , g (t)) :S C.

PROOF. Suppose there exist points p 1 , ... ,PN E M^2 such that

dist g(t) (Pi,Pj) ~

2

7r

JRmax (t)

for all 1 :S i =f. j :S N. Define

i5 (t) ~min {inj (M^2 ,go), 7r }.

JRmax (t) /2

By Proposition 5.65, the balls Bg(t) (pi, i5 (t)) are embedded and pairwise
disjoint. Arguing as we did in Corollary 5.52, one estimates that

Area (M^2 ,go)

N:S i5(t).

But by Lemma 5.74, i5 is bounded from below by a constant depending only
on go. The result follows. D

We can now show that R becomes strictly positive within finite time.

LEMMA 5.76. If (M^2 ,g(t)) is a solution of the normalized Ricci flow

on a closed surface of positive Euler characteristic, then there exists T < oo

such that

inf R > 0.

M^2 x[T,oo)

PROOF. We begin by deriving a positive lower bound for R-s, wheres

is given by formula (5.28). Given any (x, t) with t ~ 1, chose x 1 E M^2 such

that

0 < r :SR (x1, t - 1) :S Rmax (t - 1).

Then the differential Harnack estimate (5.43) derived in Proposition 5.61

implies that

(5.46) R (x, t) - s (t) ~ e-c [R (x1, t - 1) - s (t - 1)] e-A/^4 ~ e-cre-A/^4 ,

where C is a constant depending only on the initial metric, and A is defined

by

A (x1, t - 1, x, t) ~inf rt b l^2 dT.

'Y lt-1