- THE K;-NONCOLLAPSED CONDITION 87
by (19.9) there exists a constant C < oo such that R (x) d (x, 0)^2 :::; C for
all x EM. That is, ASCR(g) < oo.
2.1.3. A relation between AVR and ASCR.
Let (Nn, h) be a complete Riemannian manifold with Rm 2'.: 0 and which
is f);-noncollapsed at all scales. Suppose that there exists C E [1, oo) such
that
JRmJ (x) :::; Cd (x, 0)-^2 for all x EN - B (0, 1).
Then for all x EN - B (0, 2) we haveJRmJ (y):::; 4Cd (x, 0)-^2 for ally EB ( x,
2~d (x, 0)).
Since (Nn, h) is f);-noncollapsed at all scales, for such x we haveVolB (x, ~d (x, o)) > /'\; 12 d (x, O)n.
2vC - 2ncn
Therefore, for r E [2, oo ), by taking any x with d (x, 0) = r, we have
VolB ( o, ( 1 +
2~) r) 2'.: VolB (x,
2rc) 2'.: 2 n~n 12 rn.
Thus AVR ( h) 2:: 2 n 0 : 12 wn > 0. In particular, we have shown that if
(Nn, h) is f);-noncollapsed at all scales and AVR ( h) = O, then we conclude
ASCR (h) = oo; compare with the related Theorem 20.1.
REMARK 19.20. Although being f);-noncollapsed at all scales and having
AVR = 0 are certainly not contradictory, there is some tension between
these two properties.
2.1.4. AVR and ASCR under Ricci flow.
Next we consider some basic properties of AVR and ASCR for solutions
of Ricci fl.ow.
In Theorem 19.1 of [92] Hamilton proved the following (see also Theorem
8.32 in [45]).
THEOREM 19.21 (ASCR is independent oft). If (Mn, g (t)) is a com-
plete ancient solution on a noncompact manifold with bounded nonnegative
curvature operator, then ASCR(g (t)) is independent oft.
In Theorems 18.2 and 18.3 of [92] Hamilton proved the following (see
also Proposition 8.37 in [45]).
PROPOSITION 19.22 (AVRis independent oft). If(Mn,g(t)) is a com-
plete solution with bounded nonnegative Ricci curvature and
(19.10) lim JRm (x, t)J = 0
dt(x,0)-toofor all t, then AVR (g (t)) is independent oft. Moreover, if (19.10) holds at
some time, then it holds for all later times.