1547671870-The_Ricci_Flow__Chow

12. AN INJECTIVITY RADIUS BOUND 149

for T E [t - 1, t]. Let '"'( be a geodesic joining x and x 1 of constant speed

with respect tog (t), so that

f'"Y' (T)lg(t) = distg(t) (x,xi).

Then we have

A:Sit f'"Y'(T)l;(r)dT

t-1

:S er+C 1~

1

l'"Y' (T)l;(t) dT = er+C [distg(t) (x,x 1 )]

2

.

Hence the desired uniform bound for A is implied by the diameter bound of

Corollary 5.52. D

Applying Corollary 5.35, we conclude that there are constants 0 < c :::;

C < oo such that

Then arguing as we did for the curvature in Sections 5 and 6, it is straight-
forward to obtain estimates for all derivatives of M on a solution of the
modified Ricci flow (5.23).

COROLLARY 5.63. If (M^2 , g (t)) is a solution of the modified Ricci flow

(5.23) on a surface M^2 of positive Euler characteristic, there exist for every

k E N constants Ck and Ck depending only on g 0 such that

l\7kMI :S Cke- ckt.

Following the argument detailed in Section 7, we thus obtain the follow-

ing result.

THEOREM 5.64. Let (M^2 , go) be a closed Riemannian surface with av-

erage scalar curvature r > 0. If R (go) > 0, then the unique solution g ( t)

of the normalized Ricci flow with g (0) = go converges exponentially in any

Ck-norm to a smooth constant-curvature metric g 00 as t----> oo.

12. A lower bound for the injectivity radius
    In the next section, we shall derive uniform upper and lower bounds for

the scalar curvature of a solution (M^2 , g (t)) to the normalized Ricci flow

on a closed Riemannian surface M^2 of positive Euler characteristic whose

curvature is initially of mixed sign. In order to do this, we will require a

lower bound for the injectivity radius of such a solution. This section is

devoted to obtaining a suitable estimate.