1547671870-The_Ricci_Flow__Chow

9 UNIFORM UPPER BOUNDS FOR R AND l'V RI 141

at a minimum of R, we have Rmin (t) ~ min {O, Rmin (0)} ~ -K,. And since
r ~ 0, we know Rmax ~ 0, which implies that

_§_R < R^2
at -
at a maximum of R. Combining these observations proves

IRI < _ K,_

• 1 - K,f

for 0 :S t < 1/ K,. This implies that IRI :S 21'\, if 0 :S t :S 1/ (2K,), and hence

that 4t IRI :S 1 if 0 :St :S 1/ (81'\,). Thus equation (5.33) yields the estimate

_§_G at < -b..G + 16/'\,^3

for times 0 :S t :S 1/ (8K,), whence it follows from the maximum principle
that on that interval,
16/'\,^3
G < - K,^2 + 16K,^3 t < - K,^2 + --81'\, = 31'\,^2.

0

9.3. Estimates for solutions with R (-, 0) > O. In the remainder of

this section, we assume that R (-, 0) ~ 0. By the strong maximum principle,

one then has Rmin (t) > 0 for any t > 0 unless R = 0 everywhere, which is

possible only if the initial manifold was a flat torus. Hence (by restarting

the flow after some fixed time c > 0 has elapsed) we may assume that

R (-, 0) > 0. Then it follows from Lemma 5.9 and Proposition 5.18 that

ce-rt :SR (x, t) :S Cert

for some constants c, CE (0, oo) depending only on go. We can now obtain

a uniform upper bound for R by. using the entropy estimate.

PROPOSITION 5.51. Let (M^2 , g (t)) be a solution of the normalized Ricci

fiow on a closed surface of positive Euler characteristic. · If R (-, 0) > 0, then

there exists a constant C E [1, oo) depending only on go such that

sup R :SC.

M^2 x[O,oo)

PROOF. Since we already have time-dependent bounds for R, we may

set

K,1 ~ max R.

M^2 x[O,lj

Given any time TE [1, oo ), define

K, (T) ~ max R ~ K,1.

M2x[O,T]

We want to show that K, is bounded independently of T. Assume that

K, (T) > max { K, 1 , 1/4}, so that T > 1. Let (x1, t1) E M^2 x (1, T] be a point

such that

R (x1, t1) = max R = /'\,.

M2x[O,T ]