1547671870-The_Ricci_Flow__Chow

6. THE GRADIENT ESTIMATE FOR THE SCALAR CURVATURE 199

The only potentially positive term in the evolution equation for V is

26

W ~ 3R 1Rcl^2 - 8tr 9 (Rc^3 ) - 2R^3.

One expects this quantity to be small when the metric is close to Einstein,
because the formula derived in Corollary 6. 39 shows that W vanishes iden-
tically on an Einstein 3-manifold. We now make this expectation precise.

LEMMA 6.44. On a 3-manifold of positive Ricci curvature, one has

W:::;

5

3 ° R (1Rcl

2

- ~R

2

).

PROOF. If we define

X ~ - 8 \Re -~ Rg, Re^2 ) ,

then X = iR 1Rcl^2 - 8tr 9 (Rc^3 ), and we may write Was

W=X+6R(1Rcl

2

-~R

2

).

Define a (2, 0)-tensor Y by Y ~ Rc^2 -~R^2 g, observing that

Yi]~ Rf Rkj - ~R

2

gij =le ( ~k - ~Rgik) ( Rjc + ~Rgjc).

Then using Cauchy- Schwarz, we can estimate X in dimension n = 3 by

X = -8\Rc-~Rg,Y)

:::; 8 IRc+~Rg\ · \Rc-~Rgl

2

<

32

R · IRc-~Rgl

2

=

32

R (1Rcl

2

- ~R

2

• 3 3 3 3 ).
  We used the positivity of Re to get the last inequality. The lemma follows.
  D

We are ready to prove the main result of this section.

PROOF OF THEOREM 6.35. If (M^3 ,g (t)) is a solution of the Ricci fl.ow

on a closed 3-manifold whose Ricci curvature is positive initially, we may

combine the results of Lemmas 6.43 and 6.44 to obtain

_Q_ V < ~ V - IV' Rcl2 + 7 400v'3 + 925 R (1Rcl2 - ~ R2).

at - 3 3

Applying the result of Theorem 6.30 in the form (6.41) lets us estimate this

further as