1547671870-The_Ricci_Flow__Chow

8. FINITE-TIME BLOWUP

By Theorem 6. 35 , there are positive constants A , B , and a such that

IV R l

2

::::: ~A

2

R~2°' + B

2

.

211

Since 1Rcl^2 :S R^2 and R is positive, it follows from (6.53) that there exists
TE [O, T) such that

I \7 R I :S AR~~-a

at all times t E (T, T). Consider any time t E (T, T). Since M^3 is compact,

there exist x (t) E M^3 such that Rmax (t) = R (x, t). Given E: > 0, consider

the geodesic ball B (x, L), where

L (t) ~ l

c./Rmax (t)

If/ is any minimizing geodesic from x to ~ E B ( x, L), we can estimate that

Rmax - R (~) :S j 1 \7 R I ds ::; ALR~~-a ::; AR~-;::_.

~ c

Hence on B (x, L), we have the lower bound

(6.54) R ~ Rmax ( 1 - ~ R;:;;~x).

It follows that there exists [ E ( T , T) depending only on A , a , and E: such

that

(6.55) R~(l-c)Rmax

on B ( x, L) for all t E [t , T). Now by Corollary 6. 29, there exists f3 > 0

depending only on 9o such that the inequality

Re~ 2/3^2 R9

holds at all points of M^3. By Myers' Theorem, this implies that the minimiz-

ing geodesic I emanating from x must encounter a conjugate point within

the distance

/3.jinfB(x,L) R

Now when c > 0 is sufficiently small, estimate (6.55) implies that

n < n < 1 = L

/3JinfB(x,L) R - /3)(1 -c) Rmax - c)Rmax '

hence that B (x, L) is all of M^3.

To prove claim (2), recall that Theorem 6.30 implies there exist positive

constants C and o < 1 depending only on 9o such that

v ~A - C (μ + v)^1 -^8 ~A - C (.A+μ+ v)^1 -^8

at all points on the manifold. Thus we have the pointwise inequality

(6.56) !:_ A > - 1 - 3CR-^8 > - 1 - 3CR-? mm.