270 24. HEAT KERNEL FOR EVOLVING METRICS
have that equation (24.12) is equivalent to the following equation:
(24.18)
(
0 = -rT8logaT - --rT°;;) ~"'' L......J <pk ( T - V )k-l + ~ L......J rT--8'¢k ( T - V )k-l
2 8rT (^2) k=O k=O 8rT
N N
- L Lx,T ('lfJk-l) (r - v)k-l + L k'l/Jk(r - v)k-l
k=l k=O
(
= rT-+ 8'¢0 rT8logaT"'' 'PO---'Po rT°;;"'·)( r-v )-1
8rT 2 8rT 2
~
+ N......_ ( T - V )k-l ( rT--8'¢k + (rT -^8 log aT - --rT °;; + k) "'' lf/k )
k=l -' orT^2 8rT^2
N
+ L (r - v)k-l Lx,T ('l/Jk-l).
k=l
If the {'¢k}k':,o solve (24.15)-(24.16), then the coefficients of (r - v)-^1 and
(r - v)k-l for 1 s ks Nin (24.18) are zero. Hence part (2) follows. D
REMARK 24.6. By (24.16) we have
(24.19) 'l/Jk (y,y,r,v) = -~Lx,T ('lfJk-1) (x,y,r,v)lx=y ·
Note that, in regards to the quantity 8;; on the RHS of (24.17), we have
the elementary estimate:
LEMMA 24. 7 (Elementary bound for changing distances).
(24.20) I~; (x)I s CrT (x).
PROOF. If -y is a minimal geodesic joining y to x at time r, then under
(24.1) we have
: 7 L g(T) ("() = 1 RT ("f/y) ds,
where ds is the arc length element of -y ( s) with respect to g ( r) and where
RT is the symmetric 2-tensor in (24.1) at time r.^1 In particular,
(24.21)
(^1) In the above notation, R simultaneously denotes a symmetric 2-tensor Rij and its
trace gijRij· Here, as elsewhere, it will be clear from the context which quantity R
denotes.