402 8. APPLICATIONS OF THE REDUCED DISTANCE
(2) there exist r 1 > 0 and v1 > 0 such that Vol_g(o) B_g(o) (x, r1) ;::: v1 for
all x EM.
Then there exists /'i, > 0 depending only on r1, v1, n, T, and SUPMx[O,T/2] Rc_g(t)
such that g (t) is weakly /'i,-noncollapsed at any point (p, t) EM x (T /2, T)
at any scale r < V'f72.
Note that if Mis a closed manifold, then assumptions (1) and (2) of the
theorem are automatically true.
PROOF OF THEOREM 8.26. Let c1 (n) be as in Theorem 8.24. Suppose
g (t) is strongly /'i,-collapsed at a point (p, t) E M x (T /2, T) at a scale
r < V'f72 with /'i,l/n :S: c 1 ( n). Let E ~ /'i,l/n. Consider the backward solution
of the Ricci flow
g ( T) ~ g ( t - T)
with basepoint p. Since cr^2 :S: r^2 :S: t, by the monotonicity of the reduced
volume v*, we have
1% (t) :s: v (cr^2 ).
By Lemma 8.22(ii), choosing (po, To) = (p, t), we have V (t) ;::: C2 > 0.
Applying Theorem 8.24 yields
0 < C2 (ri, v1, n, T, sup Re g(t)).
Mx[O,T/2]< exp (-6^1 n (n - 1)) En /2 + w _ (n - 2)2 n-2 e--2-n-2 exp ( ---1 ).
- (47rt/2 n 1 2ylc
This implies a positive lower bound for E = /'i,l/n and the theorem is proved.
D
REMARK 8.27. Under the assumptions of Theorem 8.26, (Mn,9 (t)),t E [O, T /2] , has bounded geometry; so it is easy to see that there exists
/'i,1 > 0 depending only r1, v1, n, T, and SUPMx(O,T/ 2 ] Re g(t) such that g (t) is
weakly /'i, 1 -noncollapsed at (p, t) EM x (0, T/2] at any scaler< V'f72.
3.3. Bounding reduced volume from above when the solution
is strongly /'i,-collapsed. This subsection will be devoted to the proof of
Theorem 8.24, which follows directly from Propositions 8.28 and 8.30 below.
Note that the assumptions on g (t) translate to the following assumptions
on g* (r):
(8.36)(8.37)