1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. LOCALIZED NO LOCAL COLLAPSING THEOREM 69

5.1. Statement of the main theorem.

Recall that, as compared to Definition 19.1 below, there is another ver-

sion of r;;-noncollapsing for solutions of the Ricci fl.ow which uses a stronger

assumption (see also Definition 8.23 in Part I).

DEFINITION 18.34 (Weakly r;;-noncollapsed). Let (Mn, g (t)) be a com-

plete solution to the Ricci fl.ow with bounded curvature for each time t E

[O,T). Given r;;,p E (O,oo), we say that g(t) is weakly r;;-noncollapsed

at (xo, to) E M x [O, T) below the scale p if for any r E (0, p] satisfying

IRml :::; r-^2 in Bg(to)(xo, r) x [to - r^2 , to], we have

Volg(to) Bg(to)(xo,r) 2: r;;rn.

REMARK 18.35. Since the function

'11 (r) ~ r^2 _ max 1Rm 9 (t)(x)I

Bg(to) (xo,r) x [to-r^2 ,to]

is nondecreasing and since limr-+O '11 ( r) = 0, there exists a maximum radius

r E (0, oo] such that '11 (r) :::; 1 for r E (0, r). We .have that g(t) is weakly

r;;-noncollapsed at (xo, to) below the scale p if and only if

Volg(to) Bg(to)(xo,r) 2: r;;rn for all r E (O,min{p,r}].

The main result of this section is the following, which is Theorem 8.2 in

Perelman [152].

THEOREM 18.36 (Localized no local collapsing). For any n 2: 2 and

A > 0 there exists r;;(n, A) > 0 satisfying the following property. For any

complete solution g(t), t E [O, rg], of the Ricci flow on a manifold Mn with

bounded curvature and any xo E M, if

JRml:::; r 02 in B 9 (o)(xo,ro) x [o,r5J

and ifVolg(o)Bg(o)(xo, ro) 2: A-^1 rg, then g(t) is weakly r;; (n, A)-noncollapsed

below the scale ro at any point in Bg(r5)(xo,Aro) x {r5}.

REMARK 18.37. Given a nonfiat complete solution (Mn, g (t)), t E [O, T],

and xo E M, let r 0 :::; VT be the maximum radius such that

r5 _ max 1Rm 9 (t)(x)I:::; 1.

B 9 (o) (xo,ro) x [o,r6]

Define

rn

A (ro) ~

0
Vol ( )"
9 (o) B 9 (o) xo, ro
In this respect the theorem says that for any ro E (O; fo] the solution g(t) is
weakly r;; (n, A (ro))-noncollapsed below the scale ro at any point in the ball
Bg(r5)(xo,A(ro)ro) x {r5}.