- HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 193
be Perelman's differential Harnack quantity. Recall that by (21.32), since
9i ( T) has bounded curvature, we have
Vi::; 0 on Mix (0,w].
Lemma 21.16 is a consequence of the following property of heat kernels
under pointed Cheeger-Gromov convergence of the underlying manifolds.
LEMMA 22.9 (Convergence of heat kernels under Cheeger-Gromov lim-
its). Under the above setup, for a subsequence we have the following C^00
convergences to C^00 functions on M 00 :
Hi ~Hi o <f?i ---+Hoo,
A~ fi o <Pi---+ Joo,
f\ ~ Vi 0 <f?i ---+ Voo.
Moreover, H 00 = (47rT)-nl^2 e-f^00 is a positive solution to the. adjoint heat
equation(22.30)and
(22.31)satisfies the equation(22.32) D:Ov= ~ -2r IR.cg_.+V'-V'-f= - 2 ~9=C H=.
We now give a proof of this lemma.
(1) To prove the C^00 convergence of Hi to a nonnegative C^00 function
H 00 for a subsequence, we shall show that for any 6 E (0, w /2] we have for
all i
(22.33)where C < oo is independent of i. The C^00 convergence of Hi to H 00 on
M 00 x (0, w] then follows from the Bernstein local derivative estimates for
heat-type equations (see Ladyzenskaja, Solonnikov, and Uralceva [113]).
(2) To show that H 00 > O, we shall show that for any 6 E (0, w] we have
for all i
(22.34)where c > 0 is independent of i. Then the positivity of H 00 follows from the
strong maximum principle.
Proof of (1). First note that, by Corollary 26.15 below, we have(22.35) r Hi (x, T) dμgi(r) (x) = 1