1547671870-The_Ricci_Flow__Chow

220

PROOF.

and

6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

Following the proof of Theorem 6.30, we compute that

d

- log ( ,\ - v) = ,\ - μ + v - r

dt •

d. μ2 + 1/2

-d log(μ + v) = ,\ + - r.

t μ+I/

The calculation

d

dt (μ - v) = (μ + v - ,\ - r) (μ - v)

shows that μ - v stays positive. Thus we can compute that

• d log [ ef3t ,\ - I/ ] = a ( ,\ - r) + f3 - (μ - v) - (1 - a) --μ2 + 1/2
  dt (μ+v)l-a μ +v
  Recalling estimates (6.62) and (6.63), and applying Lemma 6.65, we estimate
  this as

!!: log [ef3t ,\ - ~ ] :::; aB (μ + v) + f3 -^1 - a (μ + v).

dt (μ + v) °' 2

If a > 0 is chosen smaller than 1/ [2 (1 + 2B)], then

• d l og [ e f3t ,\ - v ] < f3 - 1 - a (1 + 2B) ( μ + v ) < f3 - --μ + v.

dt (μ+v)l-a - 2 - 4

By Lemmas 6.65 and (6.67), there exists E > 0 such that

4E:::; ,\ + μ + v:::; (1 + B) (μ + v).

Hence

dl [{3t A-1/] E

dt og e (μ+v)l-a '.S/3- l+B·

For sufficiently small f3 > 0 depending only on B and E, the right-hand side

is negative. So there exists a constant C = C (go) large enough that Ml (t)

starts inside and hence remains in the convex set

K = {IP' : ef3t [ ,\ (IP') - v (IP')] - C [μ (IP') + v (IP')] 1-a :::; 0}.

By Lemma 6.65 and estimate (6.62), the proposition implies that

,\ - v < CR^1 - ae-f3t < Ace-f3t

- - '

D

whence formula (6.34) and Lemma 6.67 imply that g (t) is exponentially

becoming a metric of constant positive sectional curvature.

COROLLARY 6.69. If (M^3 , g( t)) is a solution of the normalized Ricci

flow on a closed 3 -manifold of initially positive Ricci curvature, then there

exist positive constants f3 and B such that

IRc-~R · gl = !Rel :::; Be-f3i