9. UNIFORM UPPER BOUNDS FOR R AND l'V RI 139

If to ::::; t ::::; to+ 1/2Rmax (to), then Lemma 5.45 implies that

l

t (r - R(x,T)) dT 2:: -lt R(x,T) dT

to to

l

to+ 2RmaxCtol^1

2:: - 2 Rmax. (to) dT = - 1.

t o

D

It takes only slightly more work to derive an upper bound for the metric.

LEMMA 5.48. Let (M^2 ,g(t)) be any solution of the normalized Ricci

flow on a closed surf ace with r > 0.

• If R (-, 0) 2:: 0, then for any times 0 ::::; to ::::; t < oo,
  g (x, t) :::=; er(t-to)g (x, to).


• If R (-, 0) changes sign, then for any times 0 ::::; to ::::; t < oo,

g (x , t) :::=; [er(t-to) ( 1 - {o) -e-rt l g (x, to).

( 1 - ;o) -e-rto

PROOF. Recall from Section 2 that Rmin (t) is bounded below by the

solution of the ODE

d

dt s = s ( s - r) , S ( O) = { Rmin^0 (0) if if RmRmin in ( (0) 0) 2:: <^0 0,

namely

1 - (1-r/~min (O))ert if R min (0) < 0.

s (t) = {

0 if R min (0) 2:: 0

Hence for any x E M^2 , we have

l

t (r - R (x, T)) dT::::; lt (r - s (T)) dT.

to to

In the case that Rmin (0) 2:: 0, it follows therefore from (5.31) that

g (x, t) :::=; eftto(r- s(r))dr g (x, to):::=; er(t-to)g (x, to).

In the case that R (-, 0) changes sign, we compute

- lt S (T) dT = 1' -rr"' ) dT =log [e-rr - (i -2:_)] r=t

to to e-TT - 1 - ..!._ so So T=to

in order to conclude that

g (x, t) :::=; eftto (r-s(r)) dr g (x, to) :::=; [er(t-to) ( 1 - {o) -e-rt l g (x, to).

( 1 - ;o) -e-rto

D