70 18. GEOMETRIC TOOLS AND POINT PICKING METHODS5.2. Proof of Theorem 18.36.
The proof is similar to the proof of the 'weakened no local collapsing
theorem' in §7.3 of [152] (see also Theorem 8.26 in Part I). We may assume
that r 0 = 1 by replacing g (t) by the solution r 02 g (r5t).
Let (Mn,g(t)), t E [O, 1], be a complete solution to the Ricci fl.ow with
bounded curvature and suppose x 0 E M is such that
JRmJ ~ 1 in B 9 (o) (xo, 1) x [O, 1]and Vol 9 (o) B 9 (o)(xo, 1) 2:: A-^1. Let
r ( t) = 1 - t and g ( r) = g ( t).
Then g (r), TE [O, 1], is a solution to the backward Ricci fl.ow. Suppose that
(18.60) x E Bg(l)(xo,A)
and r E (0, 1] are such that JRmJ ~ r-^2 in Bg(l)(x, r) x [1-r^2 , 1]. We define"' = "'(x, r) ~ Vol 9 (1) Bg(l) (x, r).
rn
Note that, by Definition 19.50 below, g (t) is strongly ("' + c)-collapsed at(x, 1) at scale 1 for any c > 0. The theorem shall follow from bounding "'
from below by a positive constant depending only on n and A.
In the following, the reduced distance f, and reduced volume V aredefined with respect to the solution g (r) and the basepoint (x, 0). Let
c1 (n) E (0, !J be the constant given in Theorem 8.24 in Part I (see also
Theorem 19.51 below).
(1) If "'l/n 2:: c1 (n), then there is nothing to prove since we have a lower
bound for "' depending only on n.(2) On the other hand, if K}/n < ci (n), then by Theorem 8.24 in Part I
the reduced volume has the upper bound
(18.61) V ( "'l/nr^2 ) ~ c2 ( n, "') ,
where
c2 (n, "') ~ exp(ln(n-1))^6 i n-2 n-2 (^1 )
12K,2 + Wn-1 (n - 2)_2_ e--2-exp --- 1.
( 47f r 2K,2n
Observe that
(18.62) lim c2 (n, "') = 0.
fl;-+O+We shall show that there exists a constant c3 (n, A) > 0 independent of
"'such that
(18.63) V (1) 2:: c3 (n, A).
Then, by the monotonicity of the reduced volume (see Corollary 8.17 in Part
I) and (18.61), we have
(18.64)