1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. ASCR AND AVR OF ,,;-SOLUTIONS 125

PROOF. We prove the corollary by contradiction. Suppose that there

exists a complete noncompact nonflat ancient solution (Mn,g(t)) to the

Ricci flow with bounded nonnegative curvature operator such that there

exists to ::; 0 with

Wn AVR(to) ~ 11,0 > 0.

Then by the Bishop-Gromov volume comparison theorem we have that g(to)

is 11,0-noncollapsed at all scales (in fact, we get a lower bound for volume ra-

tios without assuming curvature bounds on balls). By the proof of Theorem

20.1 we have

AVR(to) = O;

this is a contradiction.^1 D

REMARK 20.3. The corresponding result in the Kahler case, conjectured
by H.-D. Cao, was proved by one of the authors (see Theorem 2 in [140]).
The statement is as follows. Let (Mm,g(t)), t E (-oo,O], be a nonflat

ancient solution to the Kahler-Ricci flow gtga:S = -Ra;s with nonnegative

bisectional curvature and SUP(x,t)EMx(-oo,o] R(x, t) < oo. Then the asymp-

totic volume ratio satisfies AVR(t) = 0 for all t.

We shall prove Theorem 20.1 by contradiction and by induction on the

dimension n.^2 First observe that Corollary 19.43 implies that when n = 2

there are no noncompact 11,-solutions with Harnack, and hence we are done
in this case. Then suppose that the theorem holds in dimension n - 1,
where n 2": 3. We shall prove by contradiction that in dimension n we have
ASCR ( t) = oo and AVR ( t) = 0 for all t. In particular, if this last statement
is false, then there exists to such that one of the following three mutually
exclusive cases holds.

Case A. ASCR(to) = oo and AVR(to) > 0.

Case B. ASCR(to) E (O,oo).

Case C. ASCR(to) = 0.

Case A. ASCR(to) = oo and AVR(to) > 0 for some to :S 0. In this case

we can perform dimension reduction and use mathematical induction.

Since ASCR(to) = oo, by Theorem 18.10, there exist a sequence of points

{xi}: 1 with dg(to) (xi,P)-+ oo and radii ri > 0 such that R (xi, to) r[-+ oo,
d 9 (to) (xi,P) /ri-+ oo, and

sup R (x, to) ::; 2R (xi, to).

B 9 (to) (xi ,ri)

Let

(20.1) gi(t) ~ R(xi, to)g (to+ R(x~, to)) , t E (-oo, OJ.

(^1) We leave it to the reader to check that in the proof below of Theorem 20.1 one only
needs that g(to) is K;o-noncollapsed on all scales to conclude AVR(to) = 0.
(^2) For another proof that AVR(t) = 0, by Hamilton, see Theorems 9.30 and 9.32 in
[45]. See also Theorem 1 in [140] by one of the authors.