1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

178 21. PERELMAN'S PSEUDOLOOALITY THEOREM

Fix a time in the interval (t', 0). Since, by (21.64), f 00 is a proper strictly con-

cave function on M 00 ,^15 we have M 00 is diffeomorphic to ffi.n, and hence by

(21.65), (M 00 , fj 00 (t)), t E (-oo, OJ, is isometric to the stationary Euclidean

space IEn,^16 contradicting the fact that inj_g=(o) (x 00 ) = 1. This proves Claim

4, which in turn implies that Claim 31 is true for Case 2. In conclusion, both

Claim 3' and the equivalent Claim 3 are proven. D

4. Contradicting the almost Euclidean logarithmic Sobolev
   inequality

Now we complete the proof of Theorem 21.9 by obtaining a contradiction

with the almost Euclidean logarithmic Sobolev inequality.

Recall what we have so far accomplished: Let a E ( 0, 13 (n~l)y'ri:) and

(Mi,gi (t) ,xoi), t E [O,cfl,

for i EN, be as in Counterstatement A and satisfying (21.18). In particular,

by (21.13), the balls B 9 i(o) (xoi, 1) are bi-almost Euclidean isoperimetrically.

Let (xi, ti) be the well-chosen a-large curvature space-time points given by

(21.21) and satisfying (21.23)-(21.25), where A ~ 100 ;,"oi. By Claim 3,

passing to a subsequence, there exists /31 > 0 such that for all i E N

(21.66) r (- _ _ l/ 2 ) Vi (ti) dμgi(ti) :'S -(31

}Bgi(ti) Xi,(ti-ti)

for some ti E [ti - ~Qi1,ti) c [O,cfl, where Qi is defined by (21.22).

STEP 4. Contradicting the almost Euclidean logarithmic Sobolev inequal-

ity at time zero.

By Lemma 22.13 below we shall relate the local entropy bound (21.66),

which is at a time ti, to the corresponding bound at time 0. This will then

contradict the isoperimetric assumption at time 0 in Theorem 21.9.

Using (21.18) and (21.23)-(21.25) and taking i large enough so that Ai ~

67n, for each i we may apply Lemma 22.13(2) below to (Mi,gi (t) ,xoi),

t E [o, cfl, (xi, ti), and ti, to obtain the existence of a nonnegative function

it on Mi satisfying

(21.67)

/Mi (-til\7hl

2

-h+n)Hidμ 9 i(o) ~/31 (1- ~;)-~;-er~ ~

1

(^15) If 'Y : JR -+ M= is a unit speed geodesic, then f.^22 J"" ('y ( s)) = -~, so that for all
s E JR
A 82
foo ('y(s)) = - 4 t +01s+02
for some 16 01, 02 ER This shows that J:O is strictly concave and proper.
Alternatively, since the reduced volume of fl= ( t) is independent of t, by Corollary
8.17 on p. 392 in Part I, ( M~, fl= ( t)) is the stationary Euclidean space.