1547671870-The_Ricci_Flow__Chow

6. THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 195

To prove this estimate, we need to compute several evolution equations,
the first of which is for the square of the norm of the gradient of the scalar
curvature.

LEMMA 6.36. If (Mn, g (t)) is a solution of the Ricci flow, then

(6.42) :t IV' R l

2

=~IV' R l

2

• 2 IV'V' R l

2

+ 4 ( V' R, V' 1 Rcl

2

).

PROOF. By (6.6), one has

a 2 a ..

at IV' RI = at (giJV'iRV'jR)

= 2 Re (V' R, V' R) + 2 ( V' R, V' ( ~R + 2 IRcl^2 )) ,

whence the lemma follows from the Bochner- Weitzenbock formula

~IV' R l^2 = 2 IV'V' R l^2 + 2 (V' R, ~ V' R) + 2 Re (V' R, V' R).

0

Next we calculate the evolution of IV' R \^2 divided by R. This choice of

power is not strongly intuitive, but yields a useful equation.

LEMMA 6.37. Let (Mn,g(t)) be a solution of the Ricci flow such that

R > 0 initially. Then for as long as the solution exists,

(6.43a) :

1 ('":I')= ll ('":I')-

(^2) R [v ("RR)['
(6.43b) - 2 IV'R~\
2
\Rc\^2 + ~ ( V' R, V' \Rc\^2 ).

PROOF. By Corollary 6.8, the inequality R > 0 is preserved. Using (6.6)

and (6.42), one computes that

~ ( \V'Rl2) = _.!._. ~ \V'R\2 - \V'R\2. ~R

at R R at R^2 at

= ~ ( ~ \V'R\^2 - 2 \V'V'R\

2

+ 4 (V' R, V' \Rc\

2

))

- \V'R~\

2

( ~ R + 2 \Rc\^2 ).

Since for any smooth functions u and v,

(

~ -u) = -~u - -u~v - -^2 (V'u, V'v) + 2 -u \V'v\^2 ,

v v v^2 v^2 v^3

we have

~ ( \V'R\2) = ( \V'Rl2)- ( \V'Rl4 - (V' \V'R\2, V'R) \V'V'Rl2)

at R ~ R^2 R^3 R^2 + R

• ~ 2 \V'R\

2

\Rc\

2

+ ~ (V'R, V' \Rc\

2

).