1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. NO FINITE TIME LOCAL COLLAPSING 257

PROOF. (i) ===?(ii). We prove (ii) by contradiction. Suppose g (t) is

locally collapsing at T. Then there exists a sequence of times tk /' T and a

sequence of metric balls Bg(tk)(xk, rk) such that

r2

(1) t: ::=:; C for some C < oo,

(2) IRm(g(tk))I ::=:; r'k^2 in Bg(tk)(xk,rk),

(3) Volg(tk) B9~k) (xk, rk) ""'0 as k ---too.

rk

Hence by (i) we have injg(tk) (xk) 2:: Ork for all k, where o > 0 is indepen-

dent of k. By Lemma 6.54(ii), the volume collapsing statement (3) above

cannot be true, a contradiction.

(ii)===? (i). We also prove (i) by contradiction. If (i) is not true, then

there exists C > 0 and a sequence of points and times (xk, tk) EM x [O, T)

and Wk E (0, JCtk] satisfying

and

1.

IRm(tk)I::::; w2 m Bg(tk) (xk, Wk)

k

injg(tk) (xk)

wk ""'

0

·

Lemma 6.54(i) implies that Volg(tk) B~~)(xk,Wk) ""'0 as k ---too. Thus g (t) is

k

locally collapsing at T and we have a contradiction. The lemma is proved.

D

It follows from Theorem 6.59 and Lemma 6.61 that Hamilton's little loop

conjecture holds for solutions of the Ricci flow on closed manifolds forming

finite time singularities.

COROLLARY 6.62. Let g(t), t E [O, T), be a smooth solution to the Ricci

flow on a closed manifold Mn. If T < oo, then the little loop conjecture

holds. That is, 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::::; w2 in Bg(t) (x, W)'

then we have injg(t) (x) 2:: oW

The little loop conjecture illustrates the essence of no locally collapsing

from the injectivity radius perspective. For convenience we give the following

DEFINITION 6.63 (Local injectivity radius estimate). We say that a com-
plete solution (Mn, g (t)), t E [O, T), to the Ricci flow satisfies a local in-
jectivity radius estimate if for every p E (0, oo) and C < oo, there exists
c = c (p, C, g (t)) > 0 such that for any (p, t) E M x [O, T) and r E (0, p]
which satisfy

IRm (·, t)I ::::; cr-^2 in Bg(t) (p, r)'