1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY 403

Shi's local derivative estimate implies that there is a constant c 2 = c 2 (n)

depending only on n such that

(8.38) l\7 9 ,R(x,r)i Sc~ for x E B 9 .(o) (p*, ;) and r E [o, !r^2 ].

r

From (8.19) we can write the reduced volume of g* (r) as

v* (r) =Vi (r) + V-2 (r),

where

V1 (r) ~ r (47rT)-n/^2 e-£('/'v(T),T) .c Jv (r) dx (V)'

lwig,(O)~e-1/4.

\/2 (r) ~ r (47rr)-n/^2 e-£('/'v(T),T) £ Jv (r) dx (V).

lw1g, (O) >e-1/4

Here both the reduced distance .e ("Iv ( T) , T) and the .C-J acobian .CJ v ( r)

are defined with respect to g* ( T).

PROPOSITION 8.28. Under the assumptions of Theorem 8.24, there exists

c1 = c1 (n) E (0, !J, depending only on n, such that if E = Kl/n S ci (n),

then

• ( 2) expan(n-1)) n/2
  Vi Er s /2 f;
  (27rt
  The idea of the proof is to show that for some choice of c 1 , 'YV ( T) is con-
  tained in Bg(O) (p*, r /2) and .e ("iv ( Er^2 ) , Er^2 ) has a lower bound independent

of E when IVlg.(O) S c^114. The proposition then follows easily.

LEMMA 8.29. Suppose (p*, t*), where t* :> f, and r < yff:;. are such that

(8.36) holds, i.e., such that

1 2

1Rm 9 • (x,r)I S 2 for all x E B 9 ,(o)(p*,r) and TE [O,r ].

r.

Then there exists C1 = ci (n) E (0, !J depending only on n such that if

E = Kl/n S c1 (n), then

'Yv (r) E B 9 .(o) (p*,r/2) for any VE B;.(p.,o) (o,E-r/^4 ) and r E [0,Er^2 ].

PROOF. We prove the lemma by contradiction. Suppose

VE B 9 .(p.,o) (o, E-l/^4 )

and r' E (0, Er^2 ] is the first time such that 'YV ( r') E 8B 9 .(o) (p*, r /2). Let

V (r) ~ JT'Yv(r) = JTX (r), so that lim 7 --+0 V (r) = V Since 'YV ([O, r']) E

Bg.(O) (p*, r /2) and r' E [o, !r^2 ] , (8.36) and (8.38) imply

n-l

Re g. >---- r2 and