1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY 401

3.2. The weakened no local collapsing theorem. In Lemma 8.9 we

saw how the reduced volume of a static metric bounds the volume ratios of

balls from below. Similarly, the reduced volume monotonicity for solutions

of the Ricci flow enables one to prove a weakened form of the no local

collapsing theorem, which we first encountered in Chapter 6 using entropy

monotonicity.

DEFINITION 8.23 (Strongly K-collapsed). Let K > 0 be a constant. We

say that a solution (Mn,g(t)), t E [O,T), to the Ricci flow is strongly

K-collapsed at (qo, to) EM x (0, T) at scaler> 0 if

( 1) (curvature bound in a parabolic cylinder) I Rm_g ( x, t) I :::; / 2 for all

x E Hw 0 ) ( qo, r) and t E [max {to - r^2 , 0} , t 0 ] and

(2) (volume of ball is K-collapsed)

Vol_g(to) B_g(to) ( qo, r)

-------<K. rn

Given an r > 0, if for any to E [r^2 , T) and any qo EM the solution g (t)

is not strongly K-collapsed at (qo, t 0 ) at scaler, then we say that (M, g (t))

is weakly K-noncollapsed at scale r.

Recall that the reduced volume V ( T) has the upper bound 1. When the

solution is strongly K-collapsed, we shall obtain a better upper bound for V

which tends to 0 as K tends to 0.

THEOREM 8.24 (Main estimate for weakened no local collapsing). Let

(Mn, g ( t)) , t E [O, T), be a complete solution to the Ricci flow with T < oo

and suppose SUPMx[O,ti] IRml < oo for any ti < T. Then there exists c1

= ci (n) E (0, ~] depending only on n such that if for some Kl/n :::; ci (n),

the solution g (t) is strongly K-collapsed at (p*, t*) at scale r, where t* > t

and r < .;r;, then the reduced volume v of g (r) ~ g (t* - r) with basepoint

p* has the upper bound

where

E:::::: Kl/n

and

</> (c:, n) ~ exp ( ln^6 ( n - 1))^12 · n-2 n-2 ( 1 )

12

En + Wn-1 (n - 2)_2_ e--2-exp -

2

r;::. •

(47rt ye

REMARK 8.25. Note that limc-->O </> (c:, n) = 0 ..

We will prove this theorem in the next subsection. This theorem gives
a proof of the following weakened no local collapsing theorem.

THEOREM 8.26 (Weakened no local collapsing). Let (Mn, g (t)), t E

[O, T), be a complete solution to the Ricci flow with T < oo. Suppose

(1) SUPMx[O,ti] IRml < oo for any ti < T and