256 6. ENTROPY AND NO LOCAL COLLAPSING
5.2. The no local collapsing theorem and its proof.
5.2.l. No local collapsing theorem and little loop conjecture. One of the
major breakthroughs in Ricci fl.ow is the following.
THEOREM 6.58 (No local collapsing-A). Let g(t), t E [O, T), be a smoothsolution to the Ricci flow on a closed manifold Mn. If T < oo, then for
any p E (O,oo) there exists K, = K,(n,g(O),T,p) > 0 such that g(t) is K,-
noncollapsed below the scale p for all t E [O, T).
We shall prove this theorem in the next subsubsection. Actually Theo-
rem 4.1 of [297] states the result a bit differently.
THEOREM 6.59 (No local collapsing-B). If M is closed and g (t) is any
solution on [O, T) with T < oo, then g (t) is not locally collapsing at T.
REMARK 6.60. We leave it as an exercise to show that Theorems 6.58
and 6.59 are equivalent.
Hamilton's little loop conjecture says the following (see §15 of
[186]). Let (Mn, g (t)), t E [O, T), be a smooth solution to the Ricci fl.ow on
a closed manifold. There exists o = o ( n, g ( 0)) > 0 such that for any point
(x, t) E M x [O, T) where
IRm (g (t))I :S ~ 2 in Bg(to) (x, W)for some W > 0, we have
injg(t) (x) 2: oW
Note that the role of the positive number K, in the definition of K,-
noncollapsed is similar to the role of o in the injectivity radius lower bound
which is used in the statement of Hamilton's little loop conjecture.
Rephrasing the little loop conjecture (LLC) a little differently, we have
the following equivalence between no local collapsing (NLC) at T and the
little loop conjecture.
LEMMA 6.61 (NLC and LLC are equivalent). Let (Mn, g (t)), t E [O, T),
be a smooth complete solution to the Ricci flow where TE (0, oo]. The fol-
lowing two statements are equivalent.
(i) (Little loop conjecture) For any C > 0 there exists o > 0 such that
if (x, t) E M x [O, T) and W E (0, VCt] satisfy
1
IRm (t)I :S w2 in Bg(t) (x, W)'then(6.81) injg(t) (x) 2: oW.
(ii) (No local collapsing) The solution g(t) is not locally collapsing at
T.