1547671870-The_Ricci_Flow__Chow

122 5. THE RICCI FLOW ON SURFACES

Define cp 2 J\7\7 RJ^2 by

Then there exists ti 2 to large enough such that for all t 2 ti, one has
4R - 3r :S ~rand hence

:t 'P:::; t::.cp + 2r l\7\7 Rl^2 + (4R - 3r) (-3r l\7 R1^2 ) + c~ ert/^2 l\7\7 RI

3

:::; t::.cp + 4rcp + c~ ert/2 l\7\7 RI

3

:::; t::.cp + 4rcp + c~ ert/2 v<P

2

:S t::.cp + 3rcp + C~ert'

where c~ :::::: 3 (CD^2 I lrl. By the maximum principle, there exists c~ > 0
such that cp :::; C~ ert/^2 , which implies the result. D

We are now ready for the general case.

PROPOSITION 5.27. Let (M^2 , g (t)) be a solution of the normalized Ricci

flow on a closed surface with r < 0. Then for each positive integer k, there

exists a constant Ck < oo such that for all t E [O, oo),

sup l\7kR (x, t)l2 :S Ckert/2.

xEM^2

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

result is known for 0 :S j :S k - l. There is a commutator of the sort

Lk/2J

\7k t::.R - t::. \7k R = L (\7j R) 0g ( \7k-j R) '

j=O

where l·J is the greatest-integer function, and X 09 Y denotes a finite linear

combination of contractions of the tensors X and Y taken with respect to

the metric g ( t). Similarly, there are formulas

and

Lk/2J

\7k (R2) = L (\7jR) 0g (\7k-jR)

j=O

(:tr) 09 (\7j R) = (\7 R) 09 (\7j R).

Thus by recursive application of the identity

.!! at (\7·\7 i J · · · · \7 q R) = \7· i (.!!\7 at J · · · · \7 q R) - (.!!_rP.) at iJ \7 p · · · \7 q R '