1547671870-The_Ricci_Flow__Chow

126 5. THE RICCI FLOW ON SURFACES

By the maximum principle, R + 2 IV f l^2 :S C /t for all positive times. Be-

cause Proposition 5. 18 and Lemma 5.29 uniformly bound R + IV f l^2 , the
proposition follows easily. D

We now proceed to derive suitable bounds on the derivatives of the
curvature. For pedagogical reasons, our method will again emphasize clarity
of exposition rather than efficiency. The reader interested only in seeing the
general case may proceed directly to Proposition 5.33.

LEMMA 5.31. 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),

2 c

sup IVR(x,t)I :S 3.

xEM2 (1 + t)

PROOF. When r = 0, equation (5.20) takes the form

:t IV Rl

2

=~IV Rl

2

• 2 IVV Rl

2

+ 4R IV Rl

2

:S ~IV Rl

2

+ 4R IV Rl

2

.

Let a be a constant to be chosen later, and consider the quantity

'P ~ t^4 IV Rl^2 + at^3 R^2 ,

which satisfies <p (-, 0) = 0. Observing that

:t (t^4 IV Rl

2

) :S ~ (t^4 IV R1^2 ) + 4t^3 (tR + 1) IV Rl^2

and

we compute

a

at 'P :S ~'P + 4t^3 (tR + 1 - a/2) IV Rl^2 +a (2t^3 R^3 + 3t^2 R^2 ).

By Proposition 5.30, one may choose a < oo such that tR + 1 < a/2.

Therefore

a

at 'P ::; ~'P + c,

where C = a Ga^3 + ~a^2 ). Hence by the maximum principle,

t^4 IV Rl^2 + at^3 R^2 ~ cp :S Ct.

D

This method can clearly be generalized to higher derivatives of the cur-
vature. To motivate the proof for the general case, we will next estimate
IVV RI.