(^172) 21. PERELMAN'S PSEUDOLOCALITY THEOREM
3.1.2. A uniform negative upper bound for the local entropies.
We shall prove the following, which says that the local entropies of the
adjoint heat kernels, based at the points (xi, ti) given by (21.21), have a
certain amount of concentration (negativity). Let a E ( 0, 13 (n2l)vfn) be as
in Counterstatement A.
Claim 3. For the sequence of Harnack quantities Vi defined in (21.31),
by passing to a subsequence (still indexed by i), we have that for all i E N
there exists times
ti E [ti - ~Qi1, ti)
at which the 'xi-centered local entropies' of the adjoint heat kernels based at
(xi, ti) have a certain amount of negativity:
(21.33) r (- __ 112 ) Vi (ti) dμgi(ti) ~ -/31,
}Bgi(ti) Xi,(ti-ti)
where /31 > 0 is a constant independent of i, where Qi is defined by (21.22),
and where (xi, ti) is given by (21.21).
First we note that the above statement is invariant under the following
space-time rescalings:(21.34a) gi (x, t) f--t ?Ji (x, t) ~ Qigi ( x, ti+ Qi^1 t) ,
(21.34b) Hi (x, t) f--t Hi A (x, t) ::::;=Qi • A -n/2 Hi ( x, ti+ - Qi A -1 t ) ,
(21.34c) fi (x, t) f--t Ji (x, t) ~ fi ( x, ti+ Qi^1 t) ,
(21.34d) Vi (x, t) f--t Vi (x, t) ~ Q-;n/^2 vi (x, ti+ Q;^1 t),
where the 'dilation factors' Qi > 0 are any constants.^9
For the two cases in the proof of Claim 3' below, we shall make different
choices for Qi.
REMARK 21.14. Note that ?Ji (t) is defined fort E [-Qiti, Qi (c:z -ti) J
and ti E (aQi^1 , c:ZJ, so that in particular §i(t) is defined fort E [-aQiQi1, 0 J.
On the other hand, Hi (t), Ji (t), and Vi (t) are defined fort E [-Qiti, 0).
By the above rescalings, Claim 3 is equivalent to the following statement.Claim 31 • There exist /31 > 0 and a subsequence such that for all i E N
we have
(21.35)
1
v· (t.) dμ (v) < -/31
BYi(ti) (Xi,(-ti) - v 1/2) i i {ji ti -