1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

126 20. COMPACTNESS OF THE SPACE OF tv-SOLUTIONS

The assumption that 9(t) is K-noncollapsed at all scales^3 implies that injec-

tivity radii of 9i (0) at Xi have a uniform lower bound, i.e., inj 9 i(o) (xi) ~ <5

for some constant <5 > 0. Applying Hamilton's Cheeger-Gromov-type com-

pactness theorem, we conclude

(Bgi(O) (xi,VR(xi,to)r;) ,9i(t),xi)--+ (M~,9oo(t),xoo),

where t :::; 0 and R 900 (x 00 , 0) = 1. The limit 900 (t) is a complete noncom-

pact K-solution with R 900 (x, t) :S 2. The limit (M 00 , 900 (t), x 00 ) must also

contain a line by Theorem 18.17 and it splits as a product:

(M~,9oo (t)) = (~xwn-^1 ,du^2 + 9W (t)),

where (wn-1, 9w (t)) is a K-solution.

Next we shall derive a contradiction by showing that (W, 9W ( t)) has

positive AVR.^4 Since AVR is independent of the choice of basepoint, by

the Bishop-Gromov volume comparison theorem we have for any r E (0, oo)

and i EN,

Vol Bi(o)Bgi(o) (xi, r)

rn

_ Vol g(to)Bg(to) (Xi, R(xi, to)-^112 r)

(R(xi, to)-^1!^2 rt

~ wnAVR(to).

Taking the limit as i--+ oo ' we obtain Volgoo(D) Bgoo(o)(xoo,r) rn > _ w n AVR(to) for

all r E (0, oo ); hence

AVR(9 00 (0)) ~ AVR(to) > 0.

Denote Xoo = (O,xw) E ~xW. For the product metric 900 (0) = du^2 +

9W (0), we have for any r E (0, oo)

(-r,r) x Bgw(O) (xw,r) ~ Bdu2+gw(O) (x 00 ,r),

Voldu2+gw(O) ((-r,r) x B 9 w(O) (xw,r)) = 2rVol 9 w(O) Bgw(O) (xw,r).

Hence

Vol~~---,-1~- 9 w(O) B 9 w(O) (xw, r) > Voldu2+.9w(O) Bdu2+gw(O) (x 00 , r).

rn- - 2rn

This implies that

AVR(9w (0)) ~ ~ AVR(9 00 (0)) > 0.

2Wn-1

On the other hand, by the induction assumption, any ( n - 1 )-dimensional

K-sol ution (YV, 9W ( t)) must have zero AVR. This is a contradiction, so Case

A is ruled out.

(^3) Actually in Case A, the property that g(to) is 11;-noncollapsed at all scales is automatic
since 4 we assume AVR(to) > 0. This is all we need to obtain the injectivity radius estimate.
Aside: AVR > 0 implies the manifold is noncompact.