1547671870-The_Ricci_Flow__Chow

196 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

The lemma follows by completing the square. 0

The evolution equation for f\7J:i2 has only one bad (potentially positive)

term on the right-hand side, namely it_ (vR, \7 IRc1^2 ). In dimension n = 3,

we can remedy this by adding the quantity 1Rcl^2 - !R^2 to \7Jt, thereby
introducing a good (negative) term that cancels out the bad term. Before
proving this, we compute two more evolution equations valid for a general
n -dimensional solution.

LEMMA 6.38. If (Mn,g(t)) is a solution of the Riccifiow, then

(6.44)

and

(6.45)

PROOF. It follows from (6.6) that the square of the scalar curvature

evolves by

:t R^2 = 2R ( .6. R + 2 IRcl

2

)

= b. (R^2 ) - 2 l\7 Rl^2 +4R1Rcl^2.

On the other hand, ( 6. 7) implies that the square of the norm of the Ricci

tensor evolves by

0

COROLLARY 6.39. If (M^3 ,g (t)) is a solution of the Ricci flow on a

3-manifold, then

:t (1Rcl

2

• lR

2

) = b. (1Rcl

2

• lR

2

) - 2 (1v Rcl

2

• l l\7 Rl

2

)

26

-8tr 9 (Rc^3 ) + 3"R 1Rcl^2 - 2R^3.

PROOF. Formula (6.9) implies that (6.45) can be written in dimension

n = 3 as

:t 1Rcl^2 = b. IRcl^2 - 2 l\7 Rcl^2 - 2R^3 - 8 tr 9 (Re^3 ) + lOR 1Rcl^2.

Alternatively, one may obtain this formula by using (6.10). 0