192 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY(2) The limit (M~, 900 (t), Xoo), the exhaustion {UihEN' the solutions{(Mf,9i(t),xi)}iEN' the embeddings <l'>i: (Ui,Xoo) -t (Mi,xi),
and their images Vi =<Pi (Ui) are as in Lemma 21.15.The way that we obtain setup (1) from setup (2) is to let Ui c M 00 be
the same and to let
(i) 900 (t) = 900 (t), Xoo = Xoo,
(ii) 9i (t) = <l'>i [9i (t)lvJ on ui,
(iii) Mi= (Mi 1I Ui) I,....,,, where ,..._,,identifies x E ui with <l'>i (x) E Mi
(this way we have both a natural diffeomorphism Ii : Mi -t Mi
and a natural inclusion of Ui into Mi), and
(iv) ffi (t) =Ii (9i (t)).2.2. Estimates for the adjoint heat kernel.
In this subsection, based on estimates for the adjoint heat kernel, we
give a proof of Lemma 21.16. For convenience, we formulate a general result
concerning Cheeger-Gromov limits and heat kernels.
Let{(Mf, 9i(r), (xi, O))}, r E [O, w],
be a pointed sequence of complete solutions of the backward Ricci flow with
bounded curvature; here the bounds for the curvatures of the solutions 9i ( r)on Mi may depend on i. Suppose that the above sequence converges, in the
C^00 pointed Cheeger-Gromov sense, to a smooth complete limit solution(M~n 900 (r), (x 00 , 0)), TE [O, w].
By Definition 3.6 in Part I, this means that there exist an exhaustion {UihEN
of Moo by open sets with x 00 E Ui and a sequence of diffeomorphisms<l'>i : Ui -t Vi ~ <l'>i (Ui) c Mi
with <l'>i (x 00 ) =Xi such that (Ui,gi (r)), where
9i (r) ~<Pi [9i (r)IVi],converges pointwise in C^00 to (M 00 , 9 00 ( r)) uniformly on compact subsets
of M 00 X [O,w].
Let Hi : Mi x (0, w] -t (0, oo) be the adjoint heat kernel centered at
(xi, 0), i.e., Hi is the minimal positive function such that(22.29) Di Hi ~ ( : 7 - !J. 9 i + R 9 i) Hi = 0,
T\,0 lim Hi ( · , r) = bx i ..
The existence of Hi will be proved in Chapter 23. As in (21.30) and (21.31),
let Hi~ (47rr)-n/^2 e-fi and let
Vi~ (r ( Rgi + 2!J.gJi - l\79ifi1^2 ) +°Ji - n) Hi