1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. NO FINITE TIME LOCAL COLLAPSING 251

Now

Ln£ (log IF (x) I) IF (x) 12 dx

= 1n£ t (f (xk)

2

log lfk (xk) I IT! (xi)

2

) dx1 · · · dx.e,

~ k=l i#k

= e f (log If (x)I) J (x)^2 dx.

}~n

Hence

log (Anef f l\7f(x)l^2 dx) ~ _± f f(x)^2 loglf(x)ldx

J~n n J~n

for all f EN. Recall that Ane rv 1f1ne' which by taking the limit as f--+ oo,

implies

log (_2__ r l\7f(x)l^2 dx) ~ i r f (x)^2 loglf(x)ldx.

7ren }~n n }~n

This completes the proof of the proposition.

5. No finite time local collapsing: A proof of Hamilton's little

loop conjecture

0

In this section we first define the notion of A;-noncollapsed at scale r
and show its equivalence to the injectivity radius estimate. We then prove
Perelman's celebrated no local collapsing theorem and indicate its equiva-
lence to Hamilton's little loop conjecture. We end this section by showing
the existence of singularity models for solutions of the Ricci flow on closed
manifolds developing finite time singularities corresponding to sequences of
points and times with curvatures comparable to their spatial maximums.
Perelman's no local collapsing theorem solves a major stumbling block
in Hamilton's program for the Ricci flow on 3-manifolds. In particular, it
provides a local injectivity radius estimate which enables one to obtain singu-
larity models when dilating about finite time singular solutions of the Ricci
flow on closed manifolds of any dimension. The no local collapsing theorem
also rules out the formation of the cigar soliton as a singularity model.^14
The above two consequences of the no local collapsing theorem, together
with Hamilton's singularity theory in dimension 3, imply that necks exist in
all finite time singular solutions on closed 3-manifolds. This, together with
Hamilton's analysis of nonsingular solutions, leads one to hope/expect that

(^14) More precisely, in dimension 3 it rules out singularity models which are quotients
of the product of the cigar soliton and the real line.