1547671870-The_Ricci_Flow__Chow

6. THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 197

The useful term in the evolution equation for 1Rcl^2 - i R^2 is the quantity

• 2(IV Rcl^2 - i IV Rl^2 ). Our next task is to show that this term dominates

the bad term ~(V R, V 1Rcl^2 ) in the evolution equation (6.43) for IVJ:i2. It

is easy to see that IV Rcl^2 - i IV Rl^2 2: 0, but we need a slightly better
estimate. ·

LEMMA 6.40. In dimension n = 3,

IV Rcl^2 - ~ IV Rl^2 2:

3

1

7

IV Rcl^2.

PROOF. Define a (3, 0)-tensor X by Xijk ~ ViRjk - i ViR9jk and a
(1, 0)-tensor Y by Yk ~ gij Xijk· The contracted second Bianchi identity
shows that

IYl

2

= (Vi Rik - ~VkR) ( V^1 Rj-~VkR) = 3

1

6 IVRl

2

.

But the standard estimate

IZl^2 2: ~(tr 9 Z)^2
n
for any (2, 0)-tensor Z (not necessarily symmetric) implies in dimension
n = 3 that

Hence we get

(6.46) IV Rcl

2

2: ~ ( 1 + 3

1

6 ) IV Rl

2

= 1

3

0

7

8 IV Rl

2

,

from which the result follows easily. D

REMARK 6.41. By decomposing the (3, 0)-tensor V Re into irreducible
components, Hamilton proved a stronger inequality

IV Rcl^2 - ~IV Rl^2 2: 2

1

1 IV Rcl

2

in Lemma 11.6 of [58]; but the estimate above suffices for the proof of
Theorem 6.35. Hamilton observes that V Re may be written as the sum of
two irreducible components

V iRjk = Aijk + Bijki

where

1 3

Aijk = 20 [(VkR) 9ij + (V;R) 9ik] + 10 (ViR) 9Jk·

The constants 1/20 and 3/10 are determined by the condition that all

traces of Bijk are zero. The contracted second Bianchi identity implies that

gik Aijk = V iR and gii Aijk = gij AikJ = ~ V kR· It follows that (A, B) = 0.

Calculating that

IAl

(^2) = (~ 2? · 23) IV Rl (^2) = !._ IV Rl 2
400 + 100 -r 200 20 '