112 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS
be a point such that£ (qn 7) :::; n/2. Then for any E > 0 and A > 1, there
exists o > 0 such that for any 7 > 0,
(19.56) £(q,T):::; 6-^1 and fR(q,f):::; 6-^1
for all (q,T) E B 9 ( 7 ) (qnVC^1 7) x [A-^1 7,A7].
5.2. Review of the reduced volume.
In this subsection we recall some results which were proved in Chapter
8 of Part I.
The following definition is partially motivated by the integral of the heat
kernel and partially motivated as a space-time volume (see Definition 8.14
in Part I).
DEFINITION 19.48 (Reduced volume for Ricci fl.ow). Let (Mn, g (7)),
7 E [O, T], be a complete solution to the backward Ricci fl.ow with bounded
curvature. The reduced volume functional is defined by
(19.57) V(7) ~JM (4n7)-n/^2 exp(-£(q,7))dμ 9 ( 7 ) (q)for 7 E (0, T).
The reduced volume is monotone nondecreasing under the Ricci fl.ow
(see Corollary 8.17 in Part I).THEOREM 19.49 (Reduced volume monotonicity). Suppose (Mn,g(7)),
7 E [O, T], is a complete solution to the backward Ricci flow with the curva-
ture bound IRm(x,7)1:::; Co< oo for (x,7) EM x [O,T]. Then
(i) lim 7 -tO+ V ( 7) = 1.
(ii) The reduced volume is nonincreasing:(19.58) V ( 71) 2: V ( 72)
for any 0 < 71 < 72 < T, and V (7):::; 1 for any 7 E (0, T).(iii) Equality in (19.58) holds if and only if (M,g(7)) is isometric to
Euclidean space (IR.n, gm;).The reduced volume monotonicity formula may be used to prove a weak-
ened version of no local collapsing. Recall Definition 8.23 in Part I.
DEFINITION 19.50 (Strongly 11:-collapsed). Let 11: > 0 be a constant. We
say that a solution (Mn,g(t)), t E [O,T), to the Ricci fl.ow is strongly
11:-collapsed. at (qo, to) EM x (0, T) at scaler> 0 if
(1) (curvature bound in a parabolic cylinder) IRm_g (x, t)I :::; r^12 for all
x E B_q(to)(qo, r) and t E [max {to - r^2 , O}, to] and
(2) (volume of ball is 11:-collapsed)Vol9(to) --'-'--'----'-''--'---B_q(to) ( qo, r) < A:.
rn