1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH

one calculates

L SS = -2 'l/J'l/Jsss 1f;2 + 6 'l/J; 1f;2 (K + L) + 2 ('lf;; 1f;2 -L) K - 2K^2.

Combining these equations yields

Lt= Lss - 2 (n + 2) ~~ (K + L) + 2 [K^2 + (n - l)L^2 J,

whence the result follows.

45

D

COROLLARY 2.26. Lmin (t) is nondecreasing; moreover, if Lmin (0) # 0

then

L · (t) > l

mm - L~~n(0)-2(n-l)t"

We can now show that the radius 'lf; is strictly decreasing in time. We

will make use of the observation that 'l/Jt, a, and the sectional curvatures

satisfy the relations

'l/Jt = -'l/J [Ko+ (n - 1) K1] = 'lf; (Ko - ~R) = ~ - n'l/JK1.

LEMMA 2.27. Let g (t) be a solution of the Ricci flow having the form

(2.43) and satisfying l'l/Js I :::; 1 initially. If the scalar curvature is positive

initially, then it remains so, and one has

'l/Jt < 0

for as long as the solution exists.

PROOF. That R ;:::: Rmin (0) is a general fact we shall prove in Lemma

6.8. To show that 'l/Jt < 0, first note that the bound l'l/Jsl :::; 1 forces K1 =

(1 - 'l/J;) /'l/;^2 to be nonnegative everywhere. There are now two cases. If

'l/Jss < 0, then one has

'l/Jt = 'l/Jss - (n - 1) 'l/;K1 < 0.

On the other hand, if 'l/Jss 2: 0, then Ko = -'l/Jss/'l/J :::; 0 and hence

• 'l/Jt = 'lf; [Ko+ (n - 1) K1] = 'lf; (~R-Ko) > 0.

D

We conclude this part of the analysis by showing that 'l/;^2 is a uniformly

Lipschitz continuous function of time.

LEMMA 2.28. Let g (t) be a solution of the Ricci flow having the form

{2.43) and satisfying l'l/Js I ::::; 1 initially. Then

j('l/J2)t l::::; 2(a+n).