1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

130 20. COMPACTNESS OF THE SPACE OF ,,;-SOLUTIONS

2.1. Volume collapsing on large space-time scales.

The following is Corollary 11.5 in Perelman's [152]. We may think of

this result as a quantitative or approximate version of the fact that nonfiat

ancient solutions (not necessarily 11;-noncollapsed) with bounded Rm 2: 0

have AVR = 0 (Corollary 20.2). As is usual for such a result, this is proved

by contradiction.

PROPOSITION 20.4 (Almost ancient solutions with bounded Rm 2: 0

are collapsed at large scales). For every s > 0 there exists A < oo with the

following property: Suppose we have a sequence of (not necessarily complete)

solutions (Mk, gk ( t)), t E [tk, OJ, n 2: 2, with nonnegative curvature operator

and with Xk E Mk and rk E (0, oo) such that

(1) (relatively compact balls) for each k the ball Bgk(o)(xk,rk) is com-

pactly contained in )Vlk,

(2) (curvature bounds on parabolic cylinders)

1

(20.5) 2,Rgk(x,t):::; Rgk(xk,O) ~ Qk

for all (x, t) E Bgk(O) (xk, rk) x (tk, O], and

(3) (parabolic cylinders are large) ask-+ oo,

(20.6) tkQk -+ -oo, r~Qk -+ oo.

Then the volume ratios satisfy

Volgk(o) Bgk(o) (xk, AQk

112

)

( )

n < c

AQk1/2 -

(20.7)

provided k is sufficiently large. That is, we have small volume ratios at large

scales.

REMARK 20.5. Properties (20.5) and (20.6) say that the dilated solutions

9k(t) ~ Qk · gk(Q"k^1 t), t E [tkQk, 0],

are 'almost ancient' (since tkQk -+ -oo) and have uniformly bounded cur-

vatures on larger and larger balls (since rkQk/^2 -+ oo)

R 9 k (x, t) :::; 2

for all (x, t) E B 9 k(o)(xk, rkQk^12 ) x (tkQk, OJ.

PROOF. We prove the proposition by contradiction. If the proposition is
false, then by diagonalization there exist n 2: 2 and sequences (Mk,gk(t)),
t E [tk, OJ, with Rmgk(t) 2: 0, Xk E Mk and rk E (0, oo) satisfying hypotheses

(1)-(3) and there exist so > 0 and a sequence Ak -+ oo such that

Vol gk(o)Bgk(o) (xk, AkQk1/2)

( I )

n 2: so for all k.

AkQk1 2