1547671870-The_Ricci_Flow__Chow

6. CONVERGENCE WHEN x (M^2 ) = 0 125

we obtain the inequality

:t (t IV fl

2

+ f

2

) :S Ll (t IV fl

2

+ f^2 ).

Hence there is C1 = C1 (g (0)) such that t IV fl^2 + f^2 :S C 1 ; in particular, we
have IV f l^2 :S Gift. The lemma follows from combining these estimates. D

PROPOSITION 5.30. 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 (IR (x, t)I +IV f (x, t)l^2 ) :S _£_.

xEM2 1 + t

The method of proof is a hybrid of BBS techniques and certain innova-

tions of Hamilton. (See for instance Propositions 5. 16 and 5.50.)

PROOF. Recalling that R^2 = (Llf)^2 :S 2 IVV f l^2 , we compute

:t (R+21Vfl

2

) = Ll (R+21Vfl

2

) + R^2 -4IVVfl

2

:S Ll ( R + 2 Iv f I 2) - R^2.

The motivation to consider this quantity is the favorable -R^2 term on the

right-hand side. Not only do we have R + 2 IV f l^2 :S Co (g (0)) by the maxi-

mum principle, but we can also apply the BBS method to obtain

:t [t ( R + 2 IV fl

2

) ] :S Ll [t ( R + 2 IV f l

2

) ] - tR^2 + R + 2 IV f l

2

:S Ll [ t ( R + 2 Iv f 1

2

) ] - ~ R^2 + R + 2 Iv f 1

2

• ~ ( R + 2 1vf1

2

r + 2t 1vf 1

2

( R + 1vf1

2

).

Since t IV fl^2 :S C1 by Lemma 5.29, there is C' < oo so large that at any

point and time where t ( R + 2 IV f 1^2 ) 2 C', we have R 2 0 and hence

:t [ t ( R + 2 IV f 1

2

) ] :S L1 [ t ( R + 2 Iv f 1

2

) ] - ~ R

2

• ~ ( R + 2 IV f 1

2

r

• (1+2C1) R + 2 (1 + C1) IV f l^2

::; L1 [ t ( R + 2 Iv f 12 ) ] - ;t [ t ( R + 2 IV f I 2) r

• [ ~R - Hi (I+ 2c^1 i]' + ~

2

Thus there is C 2 C' large enough that t ( R + 2 IV fl^2 ) 2 C implies

:t [t (R+21Vf1

2

)] :S Ll [t (R+21Vf!

2

) ].