1547671870-The_Ricci_Flow__Chow

142 5. THE RICCI FLOW ON SURFACES

Let to ~ t 1 - l"' > 0. Examining the proof of Proposition 5.50, we see that
if JR J:::;: ""on a time interval [to, to+ 1/ (4"")], then
2K,
JV' R (x, t) J :::;: y'f=tQ
t - to

for all x E M^2 and t E (to, to+ 1/ (4"")]. In particular, we have

JV' R (x, tl)J :S 4""^3 /^2

for all x E M^2. Denote by Bg(ti) (x, p) the ball with center x E M^2 and
radius p, measured with respect to the metric g (t1). Let

y E Bg(t1) ( X1, 1/~)'

and choose a unit speed minimal geodesic r ( s) joining y to x1 at time tl.

Then we have

R (x1, tl) - R (y, tl) = 1 :s [R (r (s), tl)] ds

1

4/'\,3/2 K,

:::;: 'Y JV' R (r (s), tl)J ds:::;: 8 "" 112 = 2·

Hence we get

K,

R (y, tl) 2:

2

for ally E Bg(ti) (x1, 1/~).

Now by Klingenberg's Theorem, the injectivity radius of any closed ori-

entable surface whose sectional curvatures k are bounded by 0 < k :::;: k*

satisfies inj (M^2 ,g)?:: n:/Jk*. (See Theorem 5.9 of [ 27 ].) Hence

inj (M^2 ,g(t1))?:: n: > ~-

V Rm2(t1) V K,

By the maximum principle, we have R (·, tl) > 0. So the entropy N (g (t1))

is well defined. Then since R (-, 0) ?:: 0, we can let so /' 0 in Proposition 5.44

to obtain C(go) such that N(g(t1)):::;: C. Since infy>o(ylogy) = -1/e, we

have

C?::N(g(t 1 ))~ { R(logR)dA

}M2

?:: { R(logR)dA-~Area(M^2 ,g(t 1 )).

}Bg(ti)(x1,l/v'64K.) e

But the area comparison theorem (§3.4 of [25]) implies that there is a uni-

versal constant c > 0 such that

{ R(logR) dA?:: ~ (1og~) ·Area(Bg(ti) (x1,l/~))

j B 9 (ti) ( x1,l/v'64K,)

K,

?:: clog

2

.

We conclude that ""(T) has a uniform upper bound. 0