- UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ 197
The following problem is not necessary for our discussion but is of inde-
pendent interest.MINI-PROBLEM 22.11 (H 00 is the adjoint heat kernel). Prove that H 00
in Lemma 22. 9 satisfiesi.e., Hoo is the minimal positive fundamental solution to the adjoint heat
equation.Note that when the curvatures of (Mf, gi (t)) are uniformly bounded
(independent of i), this was proved by S. Zhang [196].3. Upper bound for the local entropy JB v dμ
In this section we show that if there is a negative upper bound for the
local entropy (i.e., the integration of Perelman's Harnack quantity v in a
ball) at some time i, then there is a bound for a corresponding local entropy
at time 0. The idea of the proof is to localize the entropy monotonicity
formula.
3.1. A nice time-dependent cutoff function.
We shall calculate a local form of the entropy monotonicity formula
(17.12) via multiplying by a suitably nice time-dependent cutoff function h,
which we now proceed to define.
Let </> : IR---+ [O, 1] be a smooth function which is strictly decreasing on
the interval [1, 2] and which satisfies
(22.42)
and
(22.43a)
(22.43b)for s ER
</>(s)={ 1 ifsE(-oo,1],
0 if s E [2, oo),(</>'(s))^2 ~ 10</>(s),
<f>"(s) :2: -10</>(s)Let (Mn,g (t) ,x 0 ), t E [O,c:^2 J, where c: E (O,c: 0 ], be a complete pointed
solution to the Ricci fl.ow. Given positive constants a and b, define the cutoff
function h : M x [ 0, c:^2 J ---+ [O, 1] by
(22.44) h( x, t) ~ <P ( dg(t) ( x, x:) + av'i).
LEMMA 22.12. Suppose
a 2(22.45) I Rm I (x, t) ~ - + 2 whenever t ~ (0, c:^2 ] and d 9 (t) (x, xo) ~ c:o,
t Eo