1547671870-The_Ricci_Flow__Chow

9. PROPERTIES OF THE NORMALIZED RICCI FLOW 215

LEMMA 6.60. There exists a positive constant C such that

PROOF. Let L (f) and V (f) denote the diameter and volume of g ([),

respectively. Since Re = Re > 0, the Bishop-Guenther volume comparison

theorem implies that

(6.58)

On the other hand, Corollary 6.29 shows there is a positive constant (3

depending only on go such that

- 2 2- -

Re= Re~ 2(3 Rmin9 = 2(3 Rmin9·

So by Myers' Theorem,

(6.59)

• 7f
  L < rr;-·


• {Jy Rmin

lim Rmin (f) = lim Rmin (t) = I

[/T Rmax ( f) t/T Rmax ( t)

by Lemma 6.55, there exists a positive constant C such that

(6.60) ---Rm in > -.^1

Rmax - C

Combining estimates (6.58), (6.59), and (6.60), we get the desired result:

and

Rmax ~ CRmin ~ C (;L)

2

~ C (~)

2

(

4

;)

213

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

PROOF OF THEOREM 6.58. Let

p(t) ~ fM^3 Rdμ

fM3dμ

p(f) ~ J M3 Rd-_μ =^1 Rdμ

fM3 dμ M3

D

denote the average scalar curvatures of the unnormalized and normalized

solutions, respectively. Recall that df = 'lj;dt. By Lemma 6.57, we have