- UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ 201
3.3. Proof of part (1) of Lemma 22.13.
We localize the entropy monotonicity formula. Recall that v defined in
(22.56) satisfies(22.65) D*v = -2 (f-t) JR-·+ \7·\7 ·J-
1
g··l2iJ i J 2 ( t - t) iJ H < -^0
by Lemma 6.8 in Part I.
By (22.65) and the fact that h 2: 0 has compact support, we compute
using Green's second identity (compare with Lemma 26.1)
! JM (-v) hdμ 9 (t) =JM (-vDh + hD*v) dμg(t)
:S JM ( -v) Oh dμg(t)
(22.66) :S^10 b 2 JM f (-v) hdμg(t)'
where b = 10.::oA and in the last inequality we have used (22.47) and Perel-
man's Harnack estimate, i.e., v ::::; 0.
Integrating (22.66) in time, we have
f ( -v) h dμg(t) I 2: exp (-~~ t) f (-v) h dμ 9 (t) I ,
JM t=O JM t=t(22.67) 2: exp (-
1
~~2) JM (-v) hdμ 9 (t) lt=i
for any i E [o, .::^2 ].
Now let (x, f) and t be as in the hypothesis of Lemma 22.13. Note that
~::::; J~CJ-1 < ~Vt"::::; .fi.· We shall later show that b = 10.::oA
satisfies
(22.68)Since h (-, t) = 1 on B 9 (i) (xo, b-200nJt) and since v::::; 0, using assump-
tion (22.62), we then have
-/31 2: f (- F-:;.) v dμg(i)
J B g(t) x,y t-t
2: f v dμg(i)
J B 9 (t) ( xo,b-200nvt)
2: JM v h dμ 9 (i)(10.::2