1547671870-The_Ricci_Flow__Chow

6. CONVERGENCE WHEN x (M^2 ) = (^0 127)

LEMMA 5.32. Let (M^2 , g (t)) be a solution of the Ricci flow on a closed

surface with r = 0. Then there exists C < oo depending only on g (0) such

that for all t E [O, oo),

sup l\7\7 R (x, t)I^2 :::; c

xEM2 (1 + t)^4.

PROOF. Recall from Lemma 5.25 that

:t l\7\7 R l^2 = Ll l\7\7 Rl^2 - 2 l\7\7\7 Rl^2 + 2R l\7\7 Rl^2

• 2R(ilR)^2 + 2 (V7R, \7 l\7Rl^2 ).

Let f3 > 0 be a constant to be chosen later, and define

1/; ~ t^5 l\7\7 Rl^2 + f3t^4 l\7 R l^2 ,

noticing that 1/; (-, 0) = 0. Using the facts that (ilR)^2 :::; 2 l\7\7 R l^2 and

( \7 R, \7 l\7 Rl^2 ) = 2 (\7\7 R) (\7 R, \7 R),

it is easy to estimate

:t 1/; :::; L'.l'lj; + (6tR + 5 - 2{3) t^4 l\7\7 R l^2

• 4 (t^3 l\7 R l

2

) Vt^4 l\7\7 R l

2

+ 4{3 (tR + 1) (t^3 l\7 R l

2

).

By Proposition 5.30, there is C1 < oo such that tR:::; C1. By Lemma 5.31,

there is C2 < oo such that t^3 l\7 R l^2 :::; C2. So one can choose f3 = f3 ( C1)

large enough and then C = C (C1, C2) such that

[)

at 1/; :::; il'l/; + c.

Hence we have 1/; :::; Ct by the maximum principle. D

We now treat the general case.

PROPOSITION 5.33. Let (M^2 , g (t)) be a solution of the Ricci flow on a

closed surface with r = 0. Then for each positive integer k, there exists a

constant Ck< oo depending only on g (0) such that for all t E [O, oo),

sup \\i'kR(x,t)\2:::; Ckk+2·

xEM 2 (1 + t)

PROOF. The proof is by complete induction on k; we may suppose the

result is known for 0 :::; j :::; k - 1. As in Proposition 5.27, we compute that

:t \i'kR\2 = :t (giP ... gjq\7i .. ·\7pR\7j· .. \7qR)

= Ll \\i'kR\2 - 2 \\7k+1 R \2

• ( \i'k R) ©g [ l::J:!o2J (\7j R) @g ( \i'k- j R)].