1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. NOTES AND COMMENTARY 179

with JM Hi dμgi(o) E (0, 1] and with supp Hi C Bgi(o) (xoi, 20AiEoi) c

Bgi(o) (xoi, 1). Here the quantities in the integral on the LHS of (21.67)

are at time t = 0.

Let

and let

'l/J'f (x) ~ (2fit/^2 Hi(x, 0) = (27r)-n/^2 e-Mx,o).

We have supp 'I/Ji c Bi'Ji(o) ( xoi, 20 (2fi)-^1 l^2 Acoi), f Mi 'I/Jr dμi'Ji(o) E (0, 1],

and h = -~log (27r) - log'l/J'f. Hence, using Sn~ ~ log(27r) + n, inequality

(21.67) implies

!Mi (-~IV' log ( 1fJ'f) IEi(O) + ~log (27r) +log 'I/Jr + n) 'I/Jr dμi'Ji(O) ~ ~

1

'

so that

(21.68)

JMi ( 21\71/JilEi(o) - 'I/Jr log ( 1fJ'f) - sn'l/J'f) dμi'Ji(o) :::: -~

1

:::: -

13

; JMi 'I/Jr dμMo).

On the other hand, since

{ 'l/J'f dμi'Ji(O) ::; 1

}Mi

and since supp 'I/Ji c Bi'Ji(o) ( Xoi, (2fi)-^1!^2 ) from the choice of A, by using the

isoperimetric inequality (21.13) in the logarithmic Sobolev Theorem 22.19

(applied to 9i (0) on Bi'Ji(o) (xoi, (2fi)-^1 l^2 )) we have

{ (21\71/JilEi(0)-1/J'flog'l/J'f) dμi'Ji(O) ~ (sn+log(l-8i)) { 'l/J'fdμi'Ji(O)·

}Mi }Mi

This contradicts (21.68) since /31 > 0 is fixed, whereas bi ---+ 0.

REMARK 21.18. The assumption that R (x, 0) ~ -1 in Bg(O) (xo, 1) in

Theorem 21.9 is used in the proof of (22.82).

5. Notes and commentary

The original reference for pseudolocality and its variants is §10 of Perel-
man [152]; with regret, in this chapter we only discuss §10.1 and we do not
discuss §§10.2-10.5. Regarding further work on Perelman's pseudolocality,
one may also see the papers by Chau, Tam, Yu [26], Kleiner and Lott [110],
and Topping [179]. See also Chen and Yin [35], Hsu [101], Y. Wang [184],
and one of the authors [124].
§1. For large time, the bound IRml (x, t) ::; % can be a strong assump-
tion. By the proof of Theorem 16.2 in Hamilton [92] (see also Lemma 8.9
in [45]), we have the following.