1547671870-The_Ricci_Flow__Chow

6. CONVERGENCE WHEN x (M^2 ) = O 123

we obtain

a lk/2J

at (vkR) = 6. \7kR + L (vj R) ®g ( vk-j R) -r (vk R)

j=O

and hence

~ at l\7kRl2 = ~ at (giP ... gJq\7····\7 i p R\7····\7 (^1) q R)
=6.l\7kRl2 -21vk+1R12 - (k+2)rl\7kRl2

• (vkR) ®g [L:]::io^2 J (VJR) ®g (vk-jR)].
  By induction, there exists a time to 2: 0 and a constant C such that for
  t 2: to one has
  gt1\7kRl2 ::;6.l\7kRl2 - (k+2)rl\7kRl2 +cert/2(1+l\7kRI)~
  The induction hypothesis also implies that
  gt lvk-1 Rl2 = 6. l\7k-1 Rl2 - 2 lvk Rl2 -(k + 1) r lvk-1 Rl2


• ( vk-1 R) ®g [ LJ~~l)/2J (vj R) ®g ( vk-1- j R)]
  ::; 6. lvk-1 Rl2 - 2 lvk Rl2 + Cert/2.
  Define 2: I \7k RI^2 by
  ~ lvk Rl2 - (k + 1) r lvk- 1 Rl2.
  Then there exist C' and C" such that
  gt ::; 6-<P + kr lvk Rl2 + C' ert/2 ( 1 + lvk RI)
  < - 6-<P + kr 2 + C" ert.
  As above, the conclusion now follows readily from the maximum principle.
  D
  Our proof of Theorem 5.22 is now complete.

In this section, we will prove the following case of Theorem 5.1.

THEOREM 5.28. Let ( M^2 , go) be a closed Riemannian surface with aver-

Ricci flow with g (0) =go converges uniformly in any Ck-norm to a smooth

constant-curvature metric g 00 as t --+ oo.